305 04 Krishnan Probability and Random Processes 2006

B PROBABILITY AND RANDOM PROCESSES

Venkatarama Krishnan Professor Emeritus of Electrical Engineering University of Massachusetts Lowell

A John Wiley & Sons, Inc., Publication

B

PROBABILITY AND RANDOM PROCESSES

WILEY SURVIVAL GUIDES IN ENGINEERING AND SCIENCE Emmanuel Desurvire, Editor Wiley Survival Guide in Global Telecommunications: Signaling Principles, Network Protocols, and Wireless Systems Emmanuel Desurvire Wiley Survival Guide in Global Telecommunications: Broadband Access, Optical Components and Networks, and Cryptography Emmanuel Desurvire

Fiber to the Home: The New Empowerment Paul E. Green, Jr. Probability and Random Processes Venkatarama Krishnan

B PROBABILITY AND RANDOM PROCESSES

Venkatarama Krishnan Professor Emeritus of Electrical Engineering University of Massachusetts Lowell

A John Wiley & Sons, Inc., Publication

Copyright # 2006 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of proﬁt or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Krishnan, Venkatarama, Probability and random processes / by Venkatarama Krishnan. p. cm. — (Wiley survival guides in engineering and science) Includes bibliographical references and index. ISBN-13: 978-0-471-70354-9 (acid-free paper) ISBN-10: 0-471-70354-0 (acid-free paper) 1. Probabilities. 2. Stochastic processes. 3. Engineering—Statistical methods. 4. Science—Statistical methods. I. Title. II. Series QA273.K74 2006 519.2–dc22

2005057726

Printed in the United States of America 10 9 8 7

6 5 4 3

2 1

Contents

Preface, xi

4.6 Hypergeometric Distribution, 46 4.7 Poisson Distribution, 48

CHAPTER 1 Sets, Fields, and Events, 1 1.1

Set Deﬁnitions, 1

1.2

Set Operations, 3

1.3

Set Algebras, Fields, and Events, 8

4.8 Logarithmic Distribution, 55 4.9 Summary of Discrete Distributions, 62

CHAPTER 5 Random Variables, 64 5.1 Deﬁnition of Random Variables, 64

CHAPTER 2 Probability Space and Axioms, 10 2.1

Probability Space, 10

2.2

Conditional Probability, 14

2.3

Independence, 18

2.4

Total Probability and Bayes’ Theorem, 20

5.2 Determination of Distribution and

Density Functions, 66 5.3 Properties of Distribution and Density

Functions, 73 5.4 Distribution Functions from Density

Functions, 75 CHAPTER 6

CHAPTER 3 Basic Combinatorics, 25

Continuous Random Variables and Basic Distributions, 79

3.1

Basic Counting Principles, 25

6.1 Introduction, 79

3.2

Permutations, 26

6.2 Uniform Distribution, 79

3.3

Combinations, 28

6.3 Exponential Distribution, 80

CHAPTER 4 Discrete Distributions, 37

6.4 Normal or Gaussian Distribution, 84

CHAPTER 7

4.1

Bernoulli Trials, 37

Other Continuous Distributions, 95

4.2

Binomial Distribution, 38

7.1 Introduction, 95

4.3

Multinomial Distribution, 41

7.2 Triangular Distribution, 95

4.4

Geometric Distribution, 42

7.3 Laplace Distribution, 96

4.5

Negative Binomial Distribution, 44

7.4 Erlang Distribution, 97

vii

viii

Contents

7.5 Gamma Distribution, 99

10.4

Higher-Order Moments, 153

7.6 Weibull Distribution, 101

10.5

Bivariate Gaussian, 154

7.7 Chi-Square Distribution, 102 7.8 Chi and Other Allied Distributions,

104 7.9 Student-t Density, 110 7.10 Snedecor F Distribution, 111

CHAPTER 11 Characteristic Functions and Generating Functions, 155 11.1

Characteristic Functions, 155

11.2

Examples of Characteristic Functions, 157

11.3

Generating Functions, 161

11.4

Examples of Generating Functions, 162

11.5

Moment Generating Functions, 164

11.6

Cumulant Generating Functions, 167

11.7

Table of Means and Variances, 170

7.11 Lognormal Distribution, 112 7.12 Beta Distribution, 114 7.13 Cauchy Distribution, 115 7.14 Pareto Distribution, 117 7.15 Gibbs Distribution, 118 7.16 Mixed Distributions, 118 7.17 Summary of Distributions of

Continuous Random Variables, 119

CHAPTER 12 Functions of a Single Random Variable, 173

CHAPTER 8

12.1

Random Variable g(X ), 173

12.2

Distribution of Y ¼ g(X), 174

12.3

Direct Determination of Density fY (y) from fX (x), 194

12.4

Inverse Problem: Finding g(x) Given fX(x) and fY (y), 200

12.5

Moments of a Function of a Random Variable, 202

Conditional Densities and Distributions, 122 8.1 Conditional Distribution and Density

for P(A) = 0, 122 8.2 Conditional Distribution and Density

for P(A) ¼ 0, 126 8.3 Total Probability and Bayes’ Theorem

for Densities, 131 CHAPTER 9 Joint Densities and Distributions, 135

CHAPTER 13

9.1 Joint Discrete Distribution

Functions of Multiple Random Variables, 206

Functions, 135

13.1

Function of Two Random Variables, Z ¼ g(X,Y), 206

13.2

Two Functions of Two Random Variables, Z ¼ g(X,Y ), W ¼ h(X,Y ), 222

13.3

Direct Determination of Joint Density fZW(z,w ) from fXY (x,y), 227

13.4

Solving Z ¼ g(X,Y) Using an Auxiliary Random Variable, 233

13.5

Multiple Functions of Random Variables, 238

9.2 Joint Continuous Distribution

Functions, 136 9.3 Bivariate Gaussian Distributions, 144

CHAPTER 10 Moments and Conditional Moments, 146 10.1 Expectations, 146 10.2 Variance, 149 10.3 Means and Variances of Some

Distributions, 150

Contents

CHAPTER 14 Inequalities, Convergences, and Limit Theorems, 241 14.1

Degenerate Random Variables, 241

14.2

Chebyshev and Allied Inequalities, 242

14.3

Markov Inequality, 246

14.4

Chernoff Bound, 248

14.5

Cauchy–Schwartz Inequality, 251

14.6

Jensen’s Inequality, 254

14.7

Convergence Concepts, 256

14.8

Limit Theorems, 259

17.4 Diagonalization of

Covariance Matrices, 330 17.5 Simultaneous Diagonalization of

Covariance Matrices, 334 17.6 Linear Estimation of Vector

Variables, 337 CHAPTER 18 Estimation Theory, 340 18.1 Criteria of Estimators, 340 18.2 Estimation of Random

Variables, 342 18.3 Estimation of Parameters (Point

Estimation), 350 CHAPTER 15 Computer Methods for Generating Random Variates, 264

18.4 Interval Estimation (Conﬁdence

Intervals), 364 18.5 Hypothesis Testing (Binary), 373

15.1

Uniform-Distribution Random Variates, 264

15.2

Histograms, 266

15.3

Inverse Transformation Techniques, 269

Random Processes, 406

15.4

Convolution Techniques, 279

19.1 Basic Deﬁnitions, 406

15.5

Acceptance –Rejection Techniques, 280

19.2 Stationary Random Processes, 420

CHAPTER 16 Elements of Matrix Algebra, 284 16.1

Basic Theory of Matrices, 284

16.2

Eigenvalues and Eigenvectors of Matrices, 293

16.3 16.4

Vectors and Matrix Differentiations, 301 Block Matrices, 308

18.6 Bayesian Estimation, 384

CHAPTER 19

19.3 Ergodic Processes, 439 19.4 Estimation of Parameters of

Random Processes, 445 19.5 Power Spectral Density, 472

CHAPTER 20 Classiﬁcation of Random Processes, 490 20.1 Speciﬁcations of Random

Processes, 490 20.2 Poisson Process, 492

CHAPTER 17 Random Vectors and Mean-Square Estimation, 311

20.3 Binomial Process, 505 20.4 Independent Increment Process, 507

17.1

Distributions and Densities, 311

20.5 Random-Walk Process, 512

17.2

Moments of Random Vectors, 319

20.6 Gaussian Process, 521

17.3

Vector Gaussian Random Variables, 323

20.7 Wiener Process (Brownian

Motion), 523

ix

x

Contents

20.8 Markov Process, 527

CHAPTER 23

20.9 Markov Chain, 536

Probabilistic Methods in Transmission Tomography, 666

20.10

Martingale Process, 551

23.1

Introduction, 666

20.11

Periodic Random Process, 557

23.2

Stochastic Model, 667

20.12

Aperiodic Random Process (Karhunen – Loeve Expansion), 563

23.3

Stochastic Estimation Algorithm, 671

23.4

Prior Distribution PfMg, 674

23.5

Computer Simulation, 676

23.6

Results and Conclusions, 678

23.7

Discussion of Results, 681

23.8

References for Chapter 23, 681

CHAPTER 21 Random Processes and Linear Systems, 574 21.1 Review of Linear Systems, 574 21.2 Random Processes through

Linear Systems, 578 21.3 Linear Filters, 592 21.4 Bandpass Stationary Random

Processes, 606

APPENDIXES A

A Fourier Transform Tables, 683

B Cumulative Gaussian Tables, 687 C Inverse Cumulative Gaussian Tables, 692 D

Inverse Chi-Square Tables, 694

Weiner and Kalman Filters, 625

E

Inverse Student-t Tables, 701

22.1 Review of Orthogonality

F Cumulative Poisson Distribution, 704

CHAPTER 22

Principle, 625

G

Cumulative Binomial Distribution, 708

22.2 Wiener Filtering, 627 22.3 Discrete Kalman Filter, 647 22.4 Continuous Kalman Filter, 660

References, 714 Index, 716

Preface

Many good textbooks exist on probability and random processes written at the undergraduate level to the research level. However, there is no one handy and ready book that explains most of the essential topics, such as random variables and most of their frequently used discrete and continuous probability distribution functions; moments, transformation, and convergences of random variables; characteristic and generating functions; estimation theory and the associated orthogonality principle; vector random variables; random processes and their autocovariance and cross-covariance functions; stationarity concepts; and random processes through linear systems and the associated Wiener and Kalman ﬁlters. Engineering practitioners and students alike have to delve through several books to get the required formulas or tables either to complete a project or to ﬁnish a homework assignment. This book may alleviate this difﬁculty to some extent and provide access to a compendium of most distribution functions used by communication engineers, queuing theory specialists, signal processing engineers, biomedical engineers, and physicists. Probability tables with accuracy up to nine decimal places are given in the appendixes to enhance the utility of this book. A particular feature is the presentation of commonly occurring Fourier transforms where both the time and frequency functions are drawn to scale. Most of the theory has been explained with ﬁgures drawn to scale. To understand the theory better, more than 300 examples are given with every step explained clearly. Following the adage that a ﬁgure is worth more than a thousand words, most of the examples are also illustrated with ﬁgures drawn to scale, resulting in more than 400 diagrams. This book will be of particular value to graduate and undergraduate students in electrical, computer, and civil engineering as well as students in physics and applied mathematics for solving homework assignments and projects. It will certainly be useful to communication and signal processing engineers, and computer scientists in an industrial setting. It will also serve as a good reference for research workers in biostatistics and ﬁnancial market analysis. The salient features of this book are .

Functional and statistical independence of random variables are explained.

.

Ready reference to commonly occurring density and distribution functions and their means and variances is provided.

.

A section on Benford’s logarithmic law, which is used in detecting tax fraud, is included.

.

More than 300 examples, many of them solved in different ways to illustrate the theory and various applications of probability, are presented. Most examples have been substantiated with graphs drawn to scale.

.

xi

xii

Preface .

More than 400 ﬁgures have been drawn to scale to improve the clarity of understanding the theory and examples.

.

Bounds on the tails of Gaussian probability have been carefully developed. A chapter has been devoted to computer generation of random variates.

. .

.

Another chapter has been devoted to matrix algebra so that some estimation problems such as the Kalman ﬁlter can be cast in matrix framework. Estimation problems have been given considerable exposure.

.

Random processes deﬁned and classiﬁed. A section on martingale processes with examples has been included.

.

Markov chains with interesting examples have been discussed.

.

Wiener and Kalman ﬁltering have been discussed with a number of examples. Important properties are summarized in tables.

.

. . .

.

The ﬁnal chapter is on applications of probability to tomographic imaging. The appendixes consist of probability tables with nine-place decimal accuracy for the most common probability distributions. An extremely useful feature are the Fourier transform tables with both time and frequency functions graphed carefully to scale.

After the introduction of functional independence in Section 1.2, the differences between functional and statistical independences are discussed in Section 2.3 and clariﬁed in Example 2.3.2. The foundation for permutations and combinations are laid out in Chapter 3, followed by discrete distributions in Chapter 4 with an end-of-chapter summary of discrete distributions. Random variables are deﬁned in Chapter 5, and most of the frequently occurring continuous distributions are explained in Chapters 6 and 7. Section 6.4 discusses the new bounds on Gaussian tails. A comprehensive table is given at the end of Chapter 7 for all the continuous distributions and densities. After discussion of conditional densities, joint densities, and moments in Chapters 8 –10, characteristic and other allied functions are explained in Chapter 11 with a presentation of an end-of-chapter table of means and variances of all discrete and continuous distributions. Functions of random variables are discussed in Chapters 12 and 13 with numerous examples. Chapter 14 discusses the various bounds on probabilities along with some commonly occurring inequalities. Computer generation of random variates is presented in Chapter 15. Elements of matrix algebra along with vector and matrix differentiations are considered in Chapter 16. Vector random variables and diagonalization of covariance matrices and simultaneous diagonalization of two covariance matrices are taken up in Chapter 17. Estimation and allied hypothesis testing are discussed in Chapter 18. After random processes are explained in Chapter 19, they are carefully classiﬁed in Chapter 20, with Section 20.10 presenting martingale processes that ﬁnd wide use in ﬁnancial engineering. Chapter 21 discusses the effects of passing random processes through linear systems. A number of examples illustrate the basic ideas of Wiener and Kalman ﬁlters in Chapter 22. The ﬁnal chapter, Chapter 23, presents a practical application of probability to tomographic imaging. The appendixes include probability tables up to nine decimal places for Gaussian, chi-square, Student-t, Poisson, and binomial distributions, and Fourier transform tables with graphs for time and frequency functions carefully drawn to scale. This book is suitable for students and practicing engineers who need a quick reference to any probability distribution or table. It is also useful for self-study where a number of

Preface

xiii

carefully solved examples illustrate the applications of probability. Almost every topic has solved examples. It can be used as an adjunct to any textbook on probability and may also be prescribed as a textbook with homework problems drawn from other sources. During the more than four decades that I have been continuously teaching probability and random processes, many authors who have written excellent textbooks have inﬂuenced me to a considerable extent. Many of the ideas and concepts that are expanded in this book are the direct result of this inﬂuence. I owe a scientiﬁc debt of gratitude to all the authors who have contributed to this ubiquitous and exciting ﬁeld of probability. Some of these authors are listed in the reference, and the list is by no means exhaustive. Most of the graphs and all the probability tables in this book were created with Mathcad software. All algebraic calculations were veriﬁed with Mathcad software. Mathcad and Mathsoft are registered trademarks of Mathsoft Engineering and Education, Inc., http://www.mathcad. com. While facing extreme personal difﬁculties in writing this book, the unequivocal support of my entire family have been a source of inspiration in ﬁnishing it. I acknowledge with great pleasure the support given to me by the Wiley staff. George Telecki welcomed the idea of this book; Rachel Witmer did more than her share of keeping me in good humor with her cheerful disposition in spite of the ever-expanding deadlines; and Kellsee Chu who did an excellent job in managing the production of this book. Finally, the enthusiastic support given to me by the Series Editor Emmanuel Desurvire was very refreshing. This is my second book with Wiley, and the skills of their copyeditors and staff who transform highly mathematical manuscript into a ﬁnished book continue to amaze me. I have revised this book several times and corrected errors. Nevertheless, I cannot claim that it is error-free since correcting errors in any book is a convergent process. I sincerely hope that readers will bring to my attention any errors of commission or omission that they may come across. VENKATARAMA KRISHNAN Chelmsford, Massachusetts March 2006

CHAPTER

1

Sets, Fields, and Events

B 1.1

SET DEFINITIONS

The concept of sets play an important role in probability. We will deﬁne a set in the following paragraph. Deﬁnition of Set

A set is a collection of objects called elements. The elements of a set can also be sets. Sets are usually represented by uppercase letters A, and elements are usually represented by lowercase letters a. Thus A ¼ {a1 ;a2 ; . . . , an }

(1:1:1)

will mean that the set A contains the elements a1,a2, . . . , an. Conversely, we can write that ak is an element of A as ak [ A

(1:1:2)

ak A

(1:1:3)

and ak is not an element of A as

A ﬁnite set contains a ﬁnite number of elements, for example, S ¼ f2,4,6g. Inﬁnite sets will have either countably inﬁnite elements such as A ¼ fx : x is all positive integersg or uncountably inﬁnite elements such as B ¼ fx : x is real number 20g. Example 1.1.1

The set A of all positive integers less than 7 is written as A ¼ {x : x is a positive integer ,7}: finite set:

Example 1.1.2

The set N of all positive integers is written as N ¼ {x : x is all positive integers}: countably infinite set:

Example 1.1.3

The set R of all real numbers is written as R ¼ {x : x is real}: uncountably infinite set:

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

1

2

Sets, Fields, and Events

Example 1.1.4

The set R 2 of real numbers x, y is written as R2 ¼ {(x,y) : x is real, y is real}

Example 1.1.5

The set C of all real numbers x,y such that x þ y 10 is written as C ¼ {(x,y) : x þ y 10}: uncountably infinite set:

Venn Diagram

Sets can be represented graphically by means of a Venn diagram. In this case we assume tacitly that S is a universal set under consideration. In Example 1.1.5, the universal set S ¼ fx : x is all positive integersg. We shall represent the set A in Example 1.1.1 by means of a Venn diagram of Fig. 1.1.1. Empty Set

An empty is a set that contains no element. It plays an important role in set theory and is denoted by 1. The set A ¼ f0g is not an empty set since it contains the element 0. Cardinality

The number of elements in the set A is called the cardinality of set A, and is denoted by jAj. If it is an inﬁnite set, then the cardinality is 1. Example 1.1.6 The cardinality of the set A ¼ f2,4,6g is 3, or jAj ¼ 3. The cardinality of set R ¼ fx : x is realg is 1. Example 1.1.7

The cardinality of the set A ¼ fx : x is positive integer ,7g is jAj ¼ 6.

Example 1.1.8 The cardinality of the set B ¼ fx : x is a real number ,10g is inﬁnity since there are inﬁnite real numbers ,10. Subset

A set B is a subset of A if every element in B is an element of A and is written as B # A. B is a proper subset of A if every element of A is not in B and is written as B , A.

FIGURE 1.1.1

1.2 Set Operations

3

FIGURE 1.1.2 Equality of Sets

Two sets A and B are equal if B # A and A # B, that is, if every element of A is contained in B and every element of B is contained in A. In other words, sets A and B contain exactly the same elements. Note that this is different from having the same cardinality, that is, containing the same number of elements. Example 1.1.9 The set B ¼ f1,3,5g is a proper subset of A ¼ f1,2,3,4,5,6g, whereas the set C ¼ fx : x is a positive even integer 6g and the set D ¼ f2,4,6g are the same since they contain the same elements. The cardinalities of B, C, and D are 3 and C ¼ D. We shall now represent the sets A and B and the sets C and D in Example 1.1.9 by means of the Venn diagram of Fig. 1.1.2 on a suitably deﬁned universal set S. Power Set

The power set of any set A is the set of all possible subsets of A and is denoted by PS(A). Every power set of any set A must contain the set A itself and the empty set 1. If n is the cardinality of the set A, then the cardinality of the power set jPS(A)j ¼ 2n. Example 1.1.10 If the set A ¼ f1,2,3g then PS(A) ¼ f1, (1,2,3), (1,2), (2,3), (3,1), (1), (2), (3)g. The cardinality jPS(A)j ¼ 8 ¼ 23.

B 1.2

SET OPERATIONS

Union

Let A and B be sets belonging to the universal set S. The union of sets A and B is another set C whose elements are those that are in either A or B, and is denoted by A < B. Where there is no confusion, it will also be represented as A þ B: A < B ¼ A þ B ¼ {x : x [ A or x [ B}

(1:2:1)

4

Sets, Fields, and Events

FIGURE 1.2.1

Example 1.2.1 The union of sets A ¼ f1,2,3g and B ¼ f2,3,4,5g is the set C ¼ A B. Where there is no confusion, it will also be represented by AB. A > B ¼ AB ¼ {x : x [ A and x [ B}

(1:2:2)

Example 1.2.2 The intersection of the sets A and B in Example 1.2.1 is the set C ¼ f2,3g Examples 1.2.1 and 1.2.2 are shown in the Venn diagram of Fig. 1.2.1. Mutually Exclusive Sets

Two sets A and B are called mutually exclusive if their intersection is empty. Mutually exclusive sets are also called disjoint. A>B¼1

(1:2:3)

One way to determine whether two sets A and B are mutually exclusive is to check whether set B can occur when set A has already occurred and vice versa. If it cannot, then A and B are mutually exclusive. For example, if a single coin is tossed, the two sets, fheadsg and ftailsg, are mutually exclusive since ftailsg cannot occur when fheadsg has already occurred and vice versa. Independence

We will consider two types of independence. The ﬁrst is known as functional independence [42]. Two sets A and B can be called functionally independent if the occurrence of B does not in any way inﬂuence the occurrence of A and vice versa. The second one is statistical independence, which is a different concept that will be deﬁned later. As an example, the tossing of a coin is functionally independent of the tossing of a die because they do not depend on each other. However, the tossing of a coin and a die are not mutually exclusive since any one can be tossed irrespective of the other. By the same token, pressure and temperature are not functionally independent because the physics of the problem, namely, Boyle’s law, connects these quantities. They are certainly not mutually exclusive.

Numbers in brackets refer to bibliographic entries in the References section at the end of the book.

1.2 Set Operations

5

Cardinality of Unions and Intersections

We can now ascertain the cardinality of the union of two sets A and B that are not mutually exclusive. The cardinality of the union C ¼ A Bj twice. Hence we have to subtract once the cardinality jA > Bj as shown in Fig. 1.2.2. Or, in other words jA < Bj ¼ jAj þ jBj jA > Bj

(1.2.4a)

In Fig. 1.2.2 the cardinality jAj ¼ 9 and the cardinality jBj ¼ 11 and the cardinality jA < Bj is 11 þ 9 2 4 ¼ 16. As a corollary, if sets A and B are mutually exclusive, then the cardinality of the union is the sum of the cardinalities; or jA < Bj ¼ jAj þ jBj

(1.2.4b)

The generalization of this result to an arbitrary union of n sets is called the inclusion– exclusion principle, given by n n n n X [ X X Ai > A j þ A i > A j > Ak jA i j Ai ¼ i¼1 i¼1 i, j¼1 i, j, k¼1 i=j

(+1)n

i=j=k

n X A i > A j > A k > An

(1:2:5a)

i, j, k¼1 i=j=k...=n

If the sets fAig are mutually exclusive, that is, Ai > Aj ¼ 1 for i = j, then we have n n X [ jA j (1:2:5b) A i ¼ i¼1 i¼1 i This equation is illustrated in the Venn diagram for n ¼ 3 in Fig. 1.2.3, where if jA Bj, jB > Cj and jC > Aj. However, if we subtract once jA > Bj, jB > Cj, and jC > Aj and write jA Bj jB > Cj jC > Aj then we have subtracted jA > B > Cj thrice instead of twice. Hence, adding jA > B > Cj we get the ﬁnal result: jA Bj jB > Cj jC > Aj jA > B > Cj (1:2:6)

FIGURE 1.2.2

6

Sets, Fields, and Events

FIGURE 1.2.3

Example 1.2.3 In a junior class, the number of students in electrical engineering (EE) is 100, in math (MA) 50, and in computer science (CS) 150. Among these 20 are taking both EE and MA, 25 are taking EE and CS, and 10 are taking MA and CS. Five of them are taking EE, CS and MA. Find the total number of students in the junior class. From the problem we can identify the sets A ¼ EE students, B ¼ MA students, and C ¼ CS students, and the corresponding intersections are A > B, B > C, C > A, and A > B > C. Here, jAj ¼ 100, jBj ¼ 50, and jCj ¼ 150. We are also given jA > Bj ¼ 20, jB > Cj ¼ 10, and jC > Aj ¼ 25 and ﬁnally jA > B > Cj ¼ 5. Using Eq. (1.2.6) the total number of students are 100 þ 50 þ 150 2 20 2 10 2 25 þ 5 ¼ 250. Complement

If A is a subset of the universal set S, then the complement of A denoted by A is the elements in S not contained in A; or A ¼ S A ¼ {x : x A , S and x [ S}

(1:2:7)

Example 1.2.4 If the universal set S ¼ f1,2,3,4,5,6g and A ¼ f2,4,5g, then the complement of A is given by A ¼ f1,3,6g. Difference

The complement of a subset A , S as given by Eq. (1.2.7) is the difference between the universal set S and the subset A. The difference between any two sets A and B denoted by A 2 B is the set C containing those elements that are in A but not in B and, B 2 A is the set D containing those elements in B but not in A: C ¼ A B ¼ {x : x [ A and x B} D ¼ B A ¼ {x : x [ B and x A}

(1:2:8)

A 2 B is also called the relative complement of B with respect to A and B 2 A is called the relative complement of A with respect to B. It can be veriﬁed that A 2 B = B 2 A. Example 1.2.5 If the set A is given as A ¼ f2,4,5,6g and B is given as B ¼ f5,6,7,8g, then the difference A 2 B ¼ f2,4g and the difference B 2 A ¼ f7,8g. Clearly, A 2 B = B 2 A.

1.2 Set Operations

7

FIGURE 1.2.4 Symmetric Difference

The symmetric difference between sets A and B is written as ADB and deﬁned by ADB ¼ (A B) < (B A) ¼ {x : x [ A and x B) or {x : x [ B and x A) (1:2:9) Example 1.2.6 The symmetric difference between the set A given by A ¼ f2,4,5,6g and B given by B ¼ f5,6,7,8g in Example 1.2.5 is ADB ¼ f2,4g < f7,8g ¼ f2,4,7,8g. The difference and symmetric difference for Examples 1.2.5 and 1.2.6 are shown in Fig. 1.2.4. Cartesian Product

This is a useful concept in set theory. If A and B are two sets, the Cartesian product A B is the set of ordered pairs (x,y) such that x [ A and y [ B and is deﬁned by A B ¼ {(x,y) : x [ A, y [ B}

(1:2:10)

When ordering is considered the cartesian product A B = B A. The cardinality of a Cartesian product is the product of the individual cardinalities, or jA Bj ¼ jAj . jBj. Example 1.2.7 If A ¼ f2,3,4g and B ¼ f5,6g, then C ¼ A B ¼ f(2,5), (2,6), (3,5), (3,6), (4,5), (4,6)g with cardinality 3 2 ¼ 6. Similarly, D ¼ B A ¼ f(5,2), (5,3), (5,4), (6,2), (6,3), (6,4)g with the same cardinality of 6. However, C and D do not contain the same ordered pairs. Partition

S A partition is a collection T of disjoint sets fAi, i ¼ 1, . . . , ng of a set S such that Ai over all i equals S and Ai Aj , with i = j empty. Partitions play a useful role in conditional probability and Bayes’ theorem. Figure 1.2.5 shows the concept of a partition.

FIGURE 1.2.5

8

Sets, Fields, and Events

FIGURE 1.2.6

TABLE 1.2.1

Table of set properties

Property

Equation

Identity

AS¼A

Domination

A>1¼1 AA¼A A (B > C ) ¼ (A > B) > C

Distributive

A > (B < C ) ¼ (A > B) < (A > C ) A < (B > C ) ¼ (A < B) > (A < C )

Noncommutative

A2B = B2A

De Morgan’s law

AB A>B¼A A2 [ F. Also A1 2 A2 [ F and (A1 2 A2) < (A2 2 A1) [ F. Thus the conditions listed above are sufﬁcient for any ﬁeld F to be closed under all set operations. Sigma Field

The preceding deﬁnition of a ﬁeld covers only ﬁnite set operations. Condition 3 for ﬁnite additivity may not hold for inﬁnite additivity. For example, the sum of 1 þ 2 þ 22 þ 23 þ 24 ¼ (25 2 1)/(2 2 1) ¼ 31, but the inﬁnite sum 1 þ 2 þ 22 þ 23 þ 24 þ . . . diverges. Hence there is a need to extend ﬁnite additivity concept to inﬁnite set operations. Thus, we have a s ﬁeld if the following extra condition is imposed for inﬁnite additivity: 4. If A1, A2, . . . An, . . . [ F then B ¼ 1, then P{A < B} ¼ P(A) þ P(B): Boolen additivity:

(2:1:3)

Axiom 3b: If fA1, A2, . . . g is a sequence of sets belonging to the ﬁeld 1 such that Ai > Aj ¼ 1 for all i = j, then PfA1 < A2 < < Ai < g ¼

1 X

PfA j g: sigma additivity:

(2:1:4)

i¼1

From these axioms we can write the following corollaries: Corollary 1: P{1}

(2:1:5)

Note that the probability of the empty event is zero, but if the probability of an event A is zero, we cannot conclude that A ¼ 1. All sets with zero probability are deﬁned as null sets. An empty set is also a null set, but null sets need not be empty. Corollary 2. Since A < A ¼ S and A > A ¼ 1, we can write P(A) ¼ 1 P(A)

(2:1:6)

12

Probability Space and Axioms

Corollary 3. For any A and B that are subsets of S, not mutually exclusive, we can write Pf A < Bg ¼ P{A < A > B} ¼ P{A} þ P{A > B} P{B} ¼ P{(A > B) < (A > B)} ¼ P{A > B} þ P{A > B} Eliminating P{A < B} between the two equations, we obtain P{A < B} ¼ P{A} þ P{B} P{A > B}

(2:1:7)

Clearly, if P{A > B} ¼ 0 implying A and B are mutually exclusive, then Eq. (2.1.7) reduces to Kolmogorov’s axiom 3a. If the probabilities P{A} and P{B} are equal, then sets A and B are equal in probability. This does not mean that A is equal to B.

The probability of the union of three sets can be calculating from the Venn diagram for the three sets shown in Fig. 1.2.3 and redrawn in Fig. 2.1.2. The results of the analysis for the cardinalities given in Eq. (1.2.6), can be extended to ﬁnding the P(A B) þ P(B > C) þ P(C > A)} from {P(A) þ P(B) þ P(C)} and add P(A > B > C), resulting in

P{A B} P{B > C} P{C > A} þ P{A > B > C}

FIGURE 2.1.2

(2:1:8)

2.1 Probability Space

13

Using the inclusion– exclusion principle of Eq. (1.2.5a), we can also generalize Eq. (2.1.8) to the probabilities of the union of n sets A1, A2, . . . , An as follows: Pfni¼1 Ai g ¼

n X

P{Ai }

n X

P{Ai > A j } þ

i, j¼1 i=j

i¼1

(+1)n

n X

n X

P{Ai > A j > Ak }

i, j, k¼1 i=j=k

P{Ai > A j > Ak > An }

(2:1:9)

i, j, k¼1 i=j=k...=n

Example 2.1.6 A fair coin is tossed 3 times. Since each toss is a functionally independent trial and forms a Cartesian product, the cardinality of the three tosses is 2 2 2 ¼ 8 and the sample space consists of the following 8 events: S ¼ {½HHH, ½HHT, ½HTT, ½HTH, ½THH, ½TTH, ½THT, ½TTT} In this sample space P{2 heads} ¼ P{½HHH, ½HHT, ½HTH, ½THH} ¼

4 8

P{both heads and tails} ¼ {½HHT, ½HTT, ½HTH, ½THH, ½TTH, ½THT} ¼

6 8

Example 2.1.7 A telephone call occurs at random between 9:00 a.m. and 11:00 a.m. Hence the probability that the telephone call occurs in the interval 0 , t t0 is given by t0 =120. Thus the probability that the call occurs between 9:30 a.m. and 10:45 a.m. is given by: P{30 , t 105} ¼

105 30 75 5 ¼ ¼ 120 120 8

Example 2.1.8 Mortality tables give the probability density that a person who is born today assuming that the lifespan is 100 years as f (t) ¼ 3 109 t2 (100 t2 )u(t) and the distribution function that the person will die at the age t0 years as ð t0 F(t0 ) ¼ P{t t0 } ¼

3 109 t2 (100 t2 ) dt

0

These probabilities are shown in Fig. 2.1.3. The probability that this person will die between the ages 70 to 80 is given by P{70 , t 80} ¼ 0:1052, obtained from the following integral: ð 80 3 109 t2 (100 t2 ) dt ¼ 0:1052 P{70 , t 80} ¼ F(80) F(70) ¼ 70

The density and distribution functions are shown in Fig. 2.1.3.

14

Probability Space and Axioms

FIGURE 2.1.3

B 2.2

CONDITIONAL PROBABILITY

The conditional probability of an event B conditioned on the occurrence of another event B is deﬁned by

P{B j A} ¼

P{B > A} P{A}

P{B > A} ¼ P{B j A}P{A}

9 > = > ;

if P{A} = 0

(2:2:1)

If P{A} ¼ 0, then the conditional probability P{B j A} is not deﬁned. Since P{B j A} is a probability measure, it also satisﬁes Kolmogorov axioms. Conditional probability P{B j A} can also be interpreted as the probability of the event B in the reduced sample space S1 ¼ A. Sometimes it is simpler to use the reduced sample space to solve conditional probability problems. We shall use both these interpretations in obtaining conditional probabilities as shown in the following examples.

2.2

Conditional Probability

15

FIGURE 2.2.1

Example 2.2.1 In the tossing of a fair die, we shall calculate the probability that 3 has occurred conditioned on the toss being odd. We shall deﬁne the following events: A ¼ {odd number}: B ¼ 3 1 2

1 6

P{A} ¼ and P{B} ¼ and the event A > B ¼ {3} and hence 1 p{3} 1 P{3 j odd} ¼ ¼6¼ 1 P{odd} 3 2 Note that P{2 j odd} ¼ 0 since 2 is an impossible event given that an odd number has occurred. Also note that P{3 j 3} ¼ 1. The conditional probability of 3 occurring, given that 3 has occurred is obviously equal to 1. We can also ﬁnd the solution using the concept of reduced sample space. Since we are given that an odd number has occurred, we can reduce the sample space from S ¼ {1,2,3,4,5,6} to S1 ¼ {1,3,5} as shown in Fig. 2.2.1. The probability of 3 in this reduced sample space is 13, agreeing with the previous result. Example 2.2.2 Consider a family of exactly two children. We will ﬁnd the probabilities (1) that both are girls, (2) both are girls given that one of them is a girl, (3) both are girls given that the elder child is a girl, and (4) both are girls given that both are girls. The sample space S has 4 points {gg, gb, bg, bb}, where g is girl and b is boy. The event B ¼ fboth children are girlsg. The conditioning event A will depend on the following four cases: 1. The event A ¼ S. Hence P{B} ¼ 14. 2. The event A ¼ fone child is a girlg. The reduced sample space is S1 ¼ fgg, gb, bgg and the conditional probability P{B j A} ¼ 13. 3. The event A ¼ felder child is a girlg. In this case, the reduced sample space is S2 ¼ {gg, gb} and P{B j A} ¼ 12. 4. The event ¼ fboth are girlsg. The reduced sample space has only one point, namely, S3 ¼ {gg} and P{B j A} ¼ 1. As our knowledge of the experiment increases by the conditioning event, the uncertainty decreases and the probability increases from 0.25, through 0333, 0.5, and ﬁnally to 1, meaning that there is no uncertainty.

16

Probability Space and Axioms

Example 2.2.3 The game of craps as played in Las Vegas has the following rules. A player rolls two dice. He wins on the ﬁrst roll if he throws a 7 or a 11. He loses if the ﬁrst throw is a 2, 3, or 12. If the ﬁrst throw is a 4, 5, 6, 8, 9, or 10, it is called a point and the game continues. He goes on rolling until he throws the point for a win or a 7 for a loss. We have to ﬁnd the probability of the player winning. We will solve this problem both from the deﬁnition of conditional probability and the reduced sample space. Figure 2.2.2 shows the number of ways the sums of the pips on the two dice can equal 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and their probabilities. Solution Using Deﬁnition of Conditional Probability. The probability of winning in the ﬁrst throw is P{7} þ P{11} ¼

6 2 8 þ ¼ ¼ 0:22222 36 36 36

The probability of losing in the ﬁrst throw is P{2} þ P{3} þ P{12} ¼

1 2 1 4 þ þ ¼ ¼ 0:11111 36 36 36 36

To calculate the probability of winning in the second throw, we shall assume that i is the point with probability p. The probability of not winning in any given throw after the ﬁrst is given by r ¼ P{not i and not 7} ¼ 1 p 16. We compute the conditional probability P{win j i in first throw} ¼ p þ rp þ r2 p þ ¼

p p ¼ 1 r p þ 16

as an inﬁnite geometric series: P{win in second or subsequent throws} ¼

p2 p þ 16

The probability p of making the point i ¼ 4,5,6,8,9,10 is obtained from Fig. 2.2.2.

FIGURE 2.2.2

2.2

Conditional Probability

17

Thus, for i ¼ 4,5,6, we obtain

2

3 1 1 þ ¼ 36 6 36

2

4 1 2 þ ¼ 36 6 45

2

5 1 25 þ ¼ 36 6 396

3 P{win after point i ¼ 4} ¼ 36 4 36

P{win after point i ¼ 5} ¼

P{win after point i ¼ 6} ¼

5 36

with similar probabilities for 8,9,10. Thus the probability of winning in craps is P{win} ¼ P{win in roll 1} þ P{win in roll 2,3, . . .} 8 1 2 25 244 ¼ þ2 ¼ þ þ ¼ 0:4929 36 36 45 396 495 Solution Using Reduced Sample Space. We can now use the reduced sample space to arrive at the same result in a simpler manner. After the point i has been rolled in the ﬁrst throw, the reduced sample space consists of (i 2 1 þ 6) points for i ¼ 4,5,6 and (13 2 i þ 6) points for i ¼ 8,9,10 in which the game will terminate. We can see from the ﬁgure that there are 6 ways of rolling 7 for a loss, (i 2 1, i ¼ 4,5,6) ways of rolling the point for a win, and (13 2 i, i ¼ 8,9,10) ways of rolling the point for a win. Thus P{win in roll 2,3, . . . j i in roll 1} ¼ P{W j i} 8 i1 > , < ¼ i1þ6 > : 13 i , 13 i þ 6

i ¼ 4,5,6 i ¼ 8,9,10

4 5 5 4 Thus, P{W j 4} ¼ 39 , P{W j 5} ¼ 10 , P{W j 6} ¼ 11 , P{W j 8} ¼ 11 , P{W j 9} ¼ 10 , 3 P{W j 10} ¼ 9. Hence probability of a win after the ﬁrst roll is P{W j i} P{i}: Hence

3 P{win after first roll} ¼ 2 9 3 P{win} ¼ 2 9

3 4 4 5 5 134 þ þ ¼ 36 10 36 11 36 495 3 4 4 5 5 8 134 244 þ þ þ ¼ 0:4929 ¼ þ 36 10 36 11 36 36 495 495

a result obtained with comparative ease. Example 2.2.4 We shall continue Example 2.1.8 and ﬁnd the probability that a person will die between the ages 70 and 80 given that this individual has lived upto the age 70. We will deﬁne the events A and B as follows: A ¼ {person has lived upto 70} B ¼ {person dies between ages 70 and 80}

18

Probability Space and Axioms

The required conditional probability equals the number of persons who die between the ages 70 and 80 divided by the number of persons who have lived upto the age 70:

P{B j A} ¼

P{70 , t 80 and t 70} P{70 , t 80} ¼ P{t 70} P{t 70} Ð 80

70 ¼ Ð 100 70

3 109 t2 (100 t2 )dt 3

109 t2 (100

t2 )dt

¼

0:105 ¼ 0:644 0:163

This type of analysis is useful in setting up actuarial tables for life insurance purposes.

B 2.3

INDEPENDENCE

We have already seen the concept of functional independence of sets in Chapter 1. We will now deﬁne statistical independence of events. If A and B are two events in a ﬁeld of events F, then A and B are statistically independent if and only if P{A > B} ¼ P{A}P{B}

(2:3:1)

Statistical independence neither implies nor is implied by functional independence. Unless otherwise indicated, we will call statistically independent events “independent” without any qualiﬁers. Three events A1 ,A2 ,A3 are independent if they satisfy, in addition to P{A1 > A2 > A3 } þ P{A1 }P{A2 }P{A3 }

(2:3:2)

the pairwise independence P{Ai > A j } ¼ P{Ai }P{A j },

i, j ¼ 1,2,3, i = j

(2:3:3)

Note that if A and B are mutually exclusive, then, from Kolmogorov’s axiom 3a, we obtain P{A < B} ¼ P{A} þ P{B}

(2:3:4)

The concepts of independence and mutual exclusivity are fundamental to probability theory. For any two sets A and B, we recall from Eq. (2.1.7) that P{A < B} ¼ P{A} þ P{B} P{A > B}

(2:1:7)

If A and B are mutually exclusive, then P{A > B} ¼ 0, but if they are independent, then P{A > B} ¼ P{A} P{B}. If this has to be zero for mutual exclusivity, then one of the probabilities must be zero. Therefore, we conclude that mutual exclusivity and independence are incompatible unless we are dealing with null events. We shall now give examples to clarify the concepts of functional independence and statistical independence. Example 2.3.1 The probability of an ace on the throw of a die is 16. The probability of a head on the throw of a coin is 12. Since the throw of a die is not dependent on the throw of a coin, we have functional independence and the combined sample space is the Cartesian product with 12 elements as shown in Table 2.3.1.

2.3 Independence

19

TABLE 2.3.1 1, H

2, H

3, H

4, H

5, H

6, H

1, T

2, T

3, T

4, T

5, T

6, T

We assume as usual that all the 12 elements are equiprobable events and the probability of 1 since there is only one occurrence of f1, Hg among the 12 points as seen ace and head is 12 in the ﬁrst row and ﬁrst column in Table 2.3.1. Using Eq. (2.3.1), we have 1 , illustrating that in this example, functional indeP{1, H} ¼ P{1} P{H} ¼ 16 12 ¼ 12 pendence coincides with statistical independence.

Example 2.3.2 Two dice, one red and the other blue, are tossed. These tosses are functionally independent, and we have the Cartesian product of 6 6 ¼ 36 elementary events in the combined sample space, where each event is equiprobable. We seek the probability that an event B deﬁned by the sum of the numbers showing on the dice equals 9. There are 4 ¼ 19. We now condition the four points {(6,3), (5,4), (4,5), (3,6)}, and hence P{B} ¼ 36 event B with an event A deﬁned as the red die shows odd numbers. The probability of 1 the event A is P{A} ¼ 18 36 ¼ 2. We want to determine whether the events A and B are statistically independent. These events are shown in Fig. 2.3.1, where the ﬁrst number is for the red die and the second number is for the blue die. 2 1 ¼ 18 and ¼ P{A} P{B} ¼ 12 19 ¼ From Fig. 2.3.1 P{A > B} ¼ P{(3,6), (5,4)} ¼ 36 1 18 showing statistical independence. We compute P{B j A} from the point of view of reduced sample space. The conditioning event reduces the sample space from 36 points to 18 equiprobable points and the event {B j A} ¼ {(3,6), (5,4)}. Hence 2 ¼ 19 ¼ P{B}, or, the conditioning event has no inﬂuence on B. Here, even P{B j A} ¼ 18 though the events A and B are functionally dependent, they are statistically independent. However, if another set C is deﬁned by the sum being equal to 8 as shown in 5 . Fig. 2.3.2, then P{C} ¼ 36 Here the events C and A are not statistically independent because P{C} P{A} ¼ 5 1 5 2 4 36 2 ¼ 72 = P{C > A} ¼ P{(3,5), (5,3)} ¼ 18 ¼ 72. In this example, we have the case where the events A and C are neither statistically independent nor functionally independent.

FIGURE 2.3.1

20

Probability Space and Axioms

FIGURE 2.3.2

B 2.4

TOTAL PROBABILITY AND BAYES’ THEOREM

Let {Ai , i ¼ 1, . . . , n} be a partition of the sample space and let B an arbitrary event in the sample space S as shown in Fig. 2.4.1. We will determine the total probability PfBg, given the conditional probabilities P{B j A1 }, P{B j A2 }, . . . , P{B j An }. The event B can be given in terms of the partition as B ¼ {B > S} ¼ B > {A1 < A1 < < An } ¼ {B > A1 } < {B > A2 } < < {B > An }

(2:4:1)

The events {B > A1 }, {B > A2 }, . . . , {B > An } in Eq. (2.4.1) are mutually exclusive since A1, A2, . . . , An are mutually exclusive and we have from Kolmogorov’s axiom 3a: P{B} ¼ P{B > A1 } þ P{B > A2 } þ þ P{B > An }

(2:4:2)

Using the deﬁnition of conditional probability from Eq. (2.2.1), we can rewrite Eq. (2.4.2) as P{B} ¼ P{B j A1 }P{A1 } þ P{B j A2 }P{A2 } þ þ P{B j An }P{An } ¼

n X

P{B j Ai }P{Ai }

(2:4:3)

i¼1

Equation (2.4.3) is the expression for the total probability P{B} in terms of the conditional probabilities P{B j Ai , i ¼ 1, . . . , n}. We shall give an example to illustrate this concept.

FIGURE 2.4.1

2.4 Total Probability and Bayes’ Theorem

21

Example 2.4.1 Four computer companies A, B, C, D supply microprocessors to a company. From previous experience it is known that the probability of the processor being bad if it comes from A is 4%, B is 2%, C is 5%, and D is 1%. The probabilities of picking supplier A is 20%, B is 30%, C is 10%, and D are 40%. We want to ﬁnd the probability that a processor chosen at random is bad. Let us deﬁne X as the event that the processor is bad. We need to ﬁnd P{X} from the conditional probabilities P{X j A} ¼ 0:04, P{X j B} ¼ 0:02, P{X j C} ¼ 0:05, and P{X j D} ¼ 0:01. We are also given that P{A} ¼ 0:2, P{B} ¼ 0:3, P{C} ¼ 0:1, and P{D} ¼ 0:4. Substituting these quantities in Eq. (2.4.3), we obtain P{X} ¼ P{X j A}P{A} þ P{X j B}P{B} þ P{X j C}P{C} þ P{X j D}P{D} ¼ 0:04 0:2 þ 0:02 0:3 þ 0:05 0:1 þ 0:01 0:4 ¼ 0:023 Thus the probability that the processor is bad is 2.3%.

Bayes’ Theorem

The next logical question to ask is, “Given that the event B has occurred, what are the probabilities that A1, A2, . . . , An are involved in it?” In other words, given the conditional probabilities P{B j A1 }, P{B j A2 }, . . . , P{B j An }, we have to ﬁnd the reverse conditional probabilities P{A1 j B}, P{A2 j B}, . . . , P{An j B}. Using the total probability P{B} from Eq. (2.4.3), we can write P{Ai j B} ¼

P{Ai B} P{B j Ai }P{Ai } P{B j Ai }P{Ai } ¼ ¼ Pn P{B} P{B} j¼1 P{B j A j }P{A j }

(2:4:4)

This is the celebrated Bayes’ theorem. This theorem connects the a priori probability P{Ai } to the a posteriori probability P{Ai j B}. This useful theorem has wide applicability in communications and medical imaging, as we will see in Chapter 23. Example 2.4.2 We will now continue Example 2.4.1 and ﬁnd the probability that the processor came from companies A, B, C, or D given that the processor is bad. Using the result from Example 2.4.1 that P{X} ¼ 0:023, we can write from Eq. (2.4.4) P{X j A} ¼

P{X j A}P{A} 0:04 0:2 ¼ ¼ 0:348 P{X} 0:023

P{X j B} ¼

P{X j B}P{B} 0:02 0:3 ¼ ¼ 0:261 P{X} 0:023

P{X j C} ¼

P{X j C}P{C} 0:05 0:1 ¼ ¼ 0:217 P{X} 0:023

P{X j D} ¼

P{X j D}P{D} 0:01 0:4 ¼ ¼ 0:174 P{X} 0:023

Note that the sum of the resulting probabilities, namely, 0.348 þ 0.261 þ 0.217 þ 0.174 ¼ 1, as it should be because the bad processor has to come only from any one of the four companies A, B, C, or D resulting in the probability being 1. Example 2.4.3 A communication receiver sends numbers 1,2,3. As a result of inherent noise in the channel, the conditional probabilities of receiving 1,2,3 under the condition that 1,2,3 were sent are shown in Fig. 2.4.2.

22

Probability Space and Axioms

FIGURE 2.4.2

Transition Probabilities: P{1R j 1S } ¼ 0:60

P{2R j 1S } ¼ 0:25

P{3R j 1S } ¼ 0:15

P{1R j 2S } ¼ 0:35

P{2R j 2S } ¼ 0:55

P{3R j 2S } ¼ 0:10

P{1R j 3S } ¼ 0:05

P{2R j 3S } ¼ 0:30

P{3R j 3S } ¼ 0:65

The total probability that 1 was received is obtained from Eq. (2.4.3): P{1R } ¼ P{1R j 1S }P{1S } þ P{1R j 2S }P{2S } þ P{1R j 3S }P{3S } ¼ 0:6 0:25 þ 0:35 0:3 þ 0:05 0:45 ¼ 0:2775 From Bayes’ theorem, Eq. (2.4.4), the a posteriori probabilities of 1, 2, and 3 being sent conditioned on 1 received are found as follows: P{1S j 1R } ¼

P{1R j 1S }P{1S } 0:6 0:25 ¼ ¼ 0:5405: P{1R } 0:2775

P{1S } ¼ 0:25

P{2S j 1R } ¼

P{1R j 2S }P{2S } 0:35 0:3 ¼ ¼ 0:3784: P{1R } 0:2775

P{2S } ¼ 0:3

P{3S j 1R } ¼

P{1R j 3S }P{3S } 0:05 0:45 ¼ ¼ 0:0811: P{1R } 0:2775

P{3S } ¼ 0:45

The sum of these probabilities is equal to 1, as it should be. Note that the a priori probability that 3 was sent is 0.45 and after 1 has been received, the a posteriori probability is 0.0811, a very small probability. Similar posterior probabilities of 1, 2, and 3 being sent conditioned on 2 and 3 received can be found. Example 2.4.4 (Monty Hall Problem) This classic problem is called the “Monty Hall problem” because of the game show host Monty Hall who designed this game. There are three doors A, B, and C, and behind one of them is a car and behind the other two are goats. The contestant is asked to select any door, and the host Monty Hall opens one of the other two doors and reveals a goat. He then offers the choice to the contestant of switching to the other unopened door or keep the original door. The question now is whether the

2.4 Total Probability and Bayes’ Theorem

23

probability of winning the car is improved if she switches or it is immaterial whether she switches or not. The answer is counterintuitive in the sense that one may be misled into thinking that it is immaterial whether one switches or not. We will analyze this problem in two different ways. Mathematical Analysis. Let us analyze this problem from the point of view of Bayes’ theorem. We shall ﬁrst assume that the host has prior knowledge of the door behind which the car is located. Let us also assume that the host opens door B given that the contestant’s choice is door A. The a priori probabilities of a car behind the doors A, B, and C are 1 1 1 P{A} ¼ ; P{B} ¼ ; P{C} ¼ 3 3 3 We can make the following observations. If the contestant’s choice is door A and the car is behind door A, then the host will open the door B with probability of 12 (since she has a choice between B and C ). On the other hand, if the car is behind door B, then there is zero probability that she will open door B because of her prior knowledge. If the car is behind door C, then she will open door B with probability 1. Thus we can write the following conditional probabilities for the host opening door B: 1 P{B j A} ¼ ; P{B j B} ¼ 0; P{B j C} ¼ 1 2 We can now calculate the total probability P(B) of the host opening door B. P{B} ¼ P{B j A}P{A} þ P{B j B}P{B} þ P{B j C}P{C} 1 1 1 1 1 ¼ þ0 þ1 ¼ 2 3 3 3 2 Using Bayes’ theorem [Eq. (2.4.4)] we can now ﬁnd the a posteriori probabilities of the car behind the doors A or C (B has already been opened by the host) conditioned on B: 1 1 P{BA} P{B j A}P{A} 2 3 1 P{A j B} ¼ ¼ ¼ ¼ 1 P{B} P{B} 3 2 1 P{BC} P{B j C}P{C} 1 3 2 P{C j B} ¼ ¼ ¼ ¼ 1 P{B} P{B} 3 2 Similar analysis holds for other cases of opening the doors A or C. We can see from the result above that the contestant must switch if she wants to double her probability of winning the car. TABLE 2.4.1 First Choice

Host Opens

Second Choice

Probability

A

B

A

P{B} ¼ 12 13 ¼ 16

A

C

A

P{C} ¼ 12 13 ¼ 16

B

C

B

P{C} ¼ 1 13 ¼ 13

C

B

B

P{B} ¼ 1 13 ¼ 13

24

Probability Space and Axioms

TABLE 2.4.2 First Choice A A B C

TABLE 2.4.3 First Choice A A B C

For not switching Host Opens

Second Choice

B C C B

A A B B

Win or Loss

Probability P{B} ¼ 12 13 ¼ 16 P{C} ¼ 12 13 ¼ 16 P{C} ¼ 1 13 ¼ 13 P{B} ¼ 1 13 ¼ 13

Win Win Loss Loss

For switching Host Opens

Second Choice

B C C B

C B A A

Win or Loss

Probability P{B} ¼ 12 13 ¼ 16 P{C} ¼ 12 13 ¼ 16 P{C} ¼ 1 13 ¼ 13 P{B} ¼ 1 13 ¼ 13

Loss Loss Win Win

Heuristic Analysis. In this case we shall enumerate all the possibilities and determine which possibility gives a better chance. Assume that the car is behind door A. The probabilities of A,B,C are once again 1 P{A} ¼ ; 3

1 P{B} ¼ ; 3

P{C} ¼

1 3

Conditioned on the contestant choosing door A, the host has two equiprobable choices of opening either door B or door C. However, conditioned on the contestant choosing B or C, the host has only one choice of either opening C or B. Thus 1 P{B j A} ¼ ; 2

1 P{C j A} ¼ ; 2

P{C j B} ¼ 1;

P{B j C} ¼ 1

From these conditional probabilities we can evaluate the unconditional probabilities as shown in Table 2.4.1. We can now enumerate the various win or loss possibilities for not switching and for switching in Tables 2.4.2 and 2.4.3. From the Table 2.4.2 we can see that the probability of winning the car by not switching is 16 þ 16 ¼ 13. From Table 2.4.3 we can see that the probability of winning the car by switching is 13 þ 13 ¼ 23. Thus the probability of winning the car doubles with switching.

CHAPTER

3

Basic Combinatorics

B 3.1

BASIC COUNTING PRINCIPLES

Combinatorics is the study of arrangement and manipulation of mathematical elements in sets and essentially involves counting. We will discuss two basic counting principles. The Addition Rule

A task A consists of m subtasks A1, A2, . . . , Am that can be done in n(A1), n(A2), . . . , n(Am) ways, respectively. If no two of them can be done at the same time, then the total number of ways n(A) of doing the task A is n(A) ¼ n(A1 ) þ n(A2 ) þ þ n(Am )

(3:1:1)

This rule is the same as in set theory [Eq. (1.2.5a)] where the cardinality of the union of mutually exclusive sets is the sum of the cardinalities of the individual sets. Thus, the addition rule is a restatement of one of the properties of mutually exclusive sets. Example 3.1.1 An engineering student undertaking a capstone project T has a choice of 4 professors A,B,C,D who can give him a list of projects from which he can choose one. A has 5 projects, B has 7 projects, C has 6 projects, and D has 2 projects. Since the student can not take any other project once he has chosen a project, the number of choices he has is n(T) ¼ n(A) þ n(B) þ n(C) þ n(D) ¼ 5 þ 7 þ 6 þ 2 ¼ 20 The Multiplication Rule

A task A consists of m sequential subtasks A1, A2, . . . , Am that can be performed only after the previous one is completed. The numbers of ways n(A1), n(A2), . . . , n(Am) of performing the subtasks are functionally independent; that is, n(A2) ways are needed for the second step regardless of n(A1) of the ﬁrst step. The number of ways n(A) of performing the task A is n(A) ¼ n(A1 ) n(A2 ) n(Am )

(3:1:2)

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

25

26

Basic Combinatorics

This is very similar to the Cartesian product in set theory, where the cardinality of the product is the product of the cardinalities of the individual sets. These sets are also functionally independent. They may or may not be statistically independent. Example 3.1.2 A professor assigns a capstone project T to senior engineering students consisting of four parts A,B,C,D that must be done in sequence independent of the previous ones. Part A can be done in 2 ways, part B in 3 ways, part C in 5 ways, and part D in 4 ways. Since A,B,C,D are functionally independent, the project can be done in n(T) ¼ n(A) n(B) n(C) n(D) ¼ 2 3 5 4 ¼ 120 ways

B 3.2

PERMUTATIONS

Many problems in combinatorics can be modeled as placing r balls into n cells. The two ways of considering permutations are (1) sampling with replacement and with ordering and (2) sampling without replacement and with ordering. Sampling with Replacement and with Ordering

We consider placing r balls into n cells with replacement and with ordering. The ﬁrst ball can be placed in n cells in n ways. The second ball can be placed also in n ways. Since the placing of the second ball is functionally independent of that of the ﬁrst ball, there are now n 2 ordered 2-permutations. Thus, for placing r balls into n cells, there will be nr ways of placing r balls into n cells with replacement and ordering: Number of distinct ordered r-tuples ¼ nr

(3:2:1)

The probability of each one of these r-tuples assuming that they are equally likely is n – r. Clearly we can place 100 balls into one cell or 1 ball in 100 cells. Hence in Eq. (3.2.1) it does not matter whether r . n, r ¼ n, or r , n. Example 3.2.1 The number of distinct ordered ways we can draw 2 balls from an urn containing 4 balls numbered from 1 to 4 is obtained as follows. The cells correspond to n ¼ 4, the total number of balls. The balls correspond to r ¼ 2. The number of ways 2 balls can be placed into 4 cells is, from Eq. (3.2.1), 42 ¼ 16 ways. These 16 ordered ways are shown in Table 3.2.1 for clarity.

Sampling without Replacement and with Ordering

We now consider placing r balls into n cells without replacement and with ordering. This situation is different from the previous case. The ﬁrst ball can be placed in n cells in

TABLE 3.2.1 1,1 3,1

1,2 3,2

1,3 3,3

1,4 3,4

2,1 4,1

2,2 4,2

2,3 4,3

2,4 4,4

3.2 Permutations

27

n different ways. Since one of the cells is occupied, the second ball can be placed only in (n – 1) ways. The rth ball can be placed in (n – r þ 1) ways. Since placements of the balls are again functionally independent, we have Number of distinct r-tuples ¼ n(n 1) (n r þ 1)

(3:2:2)

These will be called r-permutations and will be deﬁned by (n)r or nPr. Using the factorial notation, we can write (n)r ¼

n! (n r)!

(3:2:3)

We cannot place 5 balls into 2 cells without replacement because at least some cells will have more than 1 ball, violating the nonreplacement condition. Hence we deﬁne (n)r ¼ 0 whenever r . n. If r ¼ n, then (n)r ¼ (n)n ¼ n!. Example 3.2.2 We continue Example 3.2.1 and now draw 2 balls from 4 balls without replacement and with ordering. The number of ways this can be done is obtained by applying Eq. (3.2.3) with n ¼ 4 and r ¼ 2 which is equal to 4.3 ¼ 12 ways instead of the 16 ways that we obtained in Table 3.2.1. The 12 ways are shown in Table 3.2.2. In Table 3.2.2, the arrangement with ordering, (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) represents an ordering different from (2,1), (3,1), (4,1), (3,2), (4,2), (4,3). We can now summarize the preceding results as follows. The number of different ways in which r balls can be placed into n cells with ordering and replacement is n r, and with ordering and without replacement is (n)r. Example 3.2.3 In the Olympic 100-m dash, there are 9 runners. In how many ways can the gold, silver, and bronze can be given to these runners? Here the 9 runners are the cells n and the 3 medals are the balls r. Hence, from Eq. (3.2.3), the number of ways that the medals can be distributed is (9)3 ¼ 9 8 7 ¼ 504 ways. Example 3.2.4 (Birthdays Problem) We want to determine the probability that in a class size of r two or more birth dates match. We shall assume that the year consists of 365 days and the year of birth is inconsequential. Here the 365 days in a year correspond to n cells and the class size, to r. If the class size is over 365, then two or more birth dates will match with probability 1. The number of ways r students can be placed into 365 birth dates with replacement is 365r. The number of ways they can be placed without replacement is (365)r, meaning that all the birth dates are different. Thus, the probability of no two birth dates match is P{no birthdays match} ¼

(365)r 365r

TABLE 3.2.2 1,2 3,1

1,3 3,2

1,4 3,4

2,1 4,1

2,3 4,2

2,4 4,3

28

Basic Combinatorics

FIGURE 3.2.1

TABLE 3.2.3 #

Probability

#

Probability

#

Probability

20 21 22 23 24 25 26

0.411438 0.443668 0.475695 0.507297 0.538344 0.5687 0.598241

30 31 32 33 34 35 36

0.706316 0.730455 0.753348 0.774972 0.795317 0.814383 0.832182

44 45 46 46 48 49 50

0.932885 0.940976 0.948253 0.954774 0.960598 0.96578 0.970374

The probability P(r) that two or more birth dates match is the complement of the preceding equation and is given by P{r} ¼ 1

(365)r 365r

P(r) is graphed for r ¼ 0– 50 in Fig. 3.2.1 and tabulated for r ¼ 20– 26, 30– 36, and 44 – 50 in Table 3.2.3. Note that for two or more birthdays to match, there is about 50% chance for 23 students, 71% chance for 30 students, and 95% chance for 46 students. In other words, we need only 46 students for 2 or more birthdays to match with 95% probability, but to get to 100% probability, we need to go to 366 students!

B 3.3

COMBINATIONS

Similar to the case of permutations we have two ways of considering combinations: 1. Sampling without replacement and without ordering 2. Sampling with replacement and without ordering

3.3 Combinations

29

Sampling without Replacement and without Ordering

We now discuss the case where we want to obtain a sample of r balls from n balls without replacement and without ordering. We have already seen that the number of ways to get r balls from n balls with ordering is (n)r. If we disregard ordering, then the number of ways is called the r-combination from the n balls. We can deﬁne this quantity as n {r-combination from n balls} ¼ (3:3:1) r n By ordering the elements in each r-combination, we can obtain from n r the rpermutations (n)r. This can be done in r! ways. Hence we can determine r from either of the following formulas: n r! ¼ (n)r r (3:3:2) n (n)r n! ¼ ¼ r! r!(n r)! r The quantity nr is also called the binomial coefﬁcient since it occurs as the coefﬁcient of the binomial expansion, given by n n n n1 n nr r n n n (a þ b) ¼ a þ a b þ þ a b þ þ b 0 1 r n Properties of Binomial Coefﬁcients

n n ¼ r nr

1.

(3:3:3)

Expanding the righthand side of Eq. (3.3.3), we obtain n! n! n n ¼ ¼ ¼ r nr (n r)!½n (n r)! (n r)!r! 2. Pascal’s Identity:

nþ1 r

¼

n r1

n þ r

(3:3:4)

Expanding the righthand side of Eq. (3.3.4), we obtain

n r1

þ

n r

n! n! þ (r 1)!(n r þ 1)! r!(n r)! n! r þ1 ¼ r!(n r)! n r þ 1 nþ1 n! nþ1 (n þ 1)! ¼ ¼ ¼ r!(n r)! n r þ 1 r!(n r þ 1)! r ¼

Pascal’s identity shows that if two adjacent binomial coefﬁcients are added, the binomial coefﬁcient in the next row in between these two coefﬁcients results, as shown in Fig. 3.3.1. It is called the Pascal triangle.

30

Basic Combinatorics

FIGURE 3.3.1

3. Sum of Binomial Coefﬁcients: n X n r¼0

r

¼ 2n

(3:3:5)

Substituting in the binomial expansion (1 þ x)n ¼ 1 þ

n 2 n n n xþ x x þ þ 2 1 n

x ¼ 1, we obtain Eq. (3.3.5). 4. Vandermonde’s Identity:

nþm r

r X n m ¼ k rk k¼0

(3:3:6)

Let us assume that there are two urns with n balls in the ﬁrst and m balls in the second. nþm The total number of ways of picking r balls from both of these urns is r . We can also choose these r balls by ﬁrst picking k balls from the ﬁrst urn and (r 2 k) balls from the second urn. Since thechoice m is functionally independent, . The realizations for every the number of ways of accomplishing this is nk rk 0 k r are mutually exclusive, and hence these numbers add, resulting in Eq. (3.3.6). This is shown in Fig. 3.3.2.

FIGURE 3.3.2

3.3 Combinations

31

Example 3.3.1 We continue Example 3.2.2 with sampling 2 balls from 4 balls without replacement and without ordering. Thus means that there is no distinction between (12) and (21), (23) and (32), (34), and (43). Thus we have only 6 ways and this is obtained by applying Eq. (3.3.1) with n ¼ 4 and r ¼ 2, yielding (4 3)/(2 1) ¼ 6. These are shown in Table 3.3.1. TABLE 3.3.1 1,2 2,3

1,3 2,4

1,4 3,4

Example 3.3.2 1. A committee of 3 women and 4 men must be formed from a group of 6 women and 8 men. The number of ways 3 women can be chosen from 6 women is 63 from 8 Eq. (3.2.2) and the number of ways 4 men can be chosen from 8 men is 4 . Since these choices are functionally independent, the total number of ways the committee can be formed is 63 84 ¼ 1400. 2. We modify the problem above by having 2 particular men be always on the committee. In this case we have a choice of only 2 men from the 6 men after taking out the 2 required men. Thus, there are 63 62 ¼ 300 ways. 3. We again modify the problem by requiring that two particular women should not be on the committee. In this case, we calculate the number of ways that the two women will be together and subtract it from the total number of ways. There are 51 84 ¼ 350 ways that the two women will be together, and hence there are 63 84 51 84 ¼ 1400 350 ¼ 1050 ways in which two given women will not be together. Example 3.3.3 (Catalan Numbers) There are n objects to be placed in computer stacks in the following way. These objects are ﬁrst stored and deleted later with the result that when all the n objects are deleted, the stack is empty. Hence, the operation store (S ) has to precede the operation delete (D), and at any point along the stack the number of stores should always be more than or equal to the number of deletes. Finally, the stack should begin with store and end with delete. We need to ﬁnd the number C(n) of operations required for n objects. For example, if there are n ¼ 3 objects, the number of ways of ordering the store and delete operations are 2 3 ¼ 6 given by SSSDDD, SDSDSD, SSDDSD, SDSSDD, SSDSDD, SDSDSSD. Thus, the operation starts with an S and ends in a D, and at no point along the chain can the number of D operations exceed the S operations. We will show that the number C(n) is given by C(n) ¼

1 (2n)! n þ 1 n!n!

Since there are a total of 2n operations, the total number of ways that we can obtain an n-combination is 2n n . Out of these combinations there will be many realizations in which the number of Ds will exceed the number of Ss. We will now ﬁnd the number of ways in which the Ds will exceed the Ss. If we are at the point (n þ 1)

32

Basic Combinatorics

TABLE 3.3.2 #

Stacks

#

Stacks

#

Stacks

1 2 3 4 5 6 7

1 2 5 14 42 132 429

8 9 10 11 12 13 14

1430 4862 16796 58786 208012 742900 2674440

15 16 17 18 19 20

9694845 35357670 129644790 477638700 1767263190 6564120420

2n 2n along the stack, the number of ways that D will exceed S will be nþ1 ¼ n1 . Hence 2n 2n 1 2n the required number of stacks is n n1 ¼ nþ1 n , which is the desired result. The Catalan numbers are tabulated in Table 3.3.2 for n ¼ 1 to 20.

Sampling with Replacement and without Ordering

We shall now consider forming r-combinations with replacement and without ordering. In this case we pick r balls with replacement from balls in n distinct categories such as red, blue, green, and brown. We do not distinguish among balls of the same color, and the distinction is among different colors. For example, if we are picking 6 balls from the 4 color categories, then it can be arranged as shown in Figure 3.3.3. There are now 2 red balls, 1 blue ball, no green balls, and 3 brown balls. The bar symbol | separates the four categories. The arrangement cannot start with a bar or end with a bar; hence we have 4 2 1 ¼ 3 bars. We now have to choose 6 balls from 6 balls and 3 bars without replacement and without ordering. There are then 987 9 9 ¼ 84 ¼ ¼ 6 3 321 ways of picking 6 balls from 4 different categories. Generalizing this result, if there are n distinct categories of balls, and if we have to pick r balls with replacement and without regard to order, then Number of r-combinations ¼

n1þr r

(3:3:7)

This is the celebrated Bose –Einstein statistics, where the elementary particles in the same energy state are not distinguishable.

FIGURE 3.3.3

3.3 Combinations

33

Example 3.3.4 An urn contains red, blue, green, yellow, white, and black balls. In how many ways can we choose 1. 10 balls? This is a direct application of Eq. (3.3.7) with n ¼ 6, r ¼ 10 and the number of ways is 6 1 þ 10 15 15 ¼ ¼ ¼ 3003 10 10 5 2. 20 balls with least 2 colors each? If we draw 2 balls of each color, then we have drawn 12 balls, leaving us to draw 20 – 12 ¼ 8 balls to be drawn without restriction. Hence, applying Eq. (3.2.7) with n ¼ 6 and r ¼ 8, we obtain the number of ways as 61þ8 13 13 ¼ ¼ ¼ 1287 8 8 5 3. 20 balls with no more than 3 blue balls? We ﬁrst calculate the number of unrestricted ways of drawing 20 balls as 6 1 þ 20 25 ¼ ¼ 53,130 20 5 We next draw at least 4 blue balls out of 20 balls. Having drawn the 4 blue balls, the number of ways to draw the remaining 16 balls is 6 1 þ 16 21 21 ¼ ¼ ¼ 20,349 26 16 5 Thus the number of ways of drawing 20 balls with no more than 3 blue balls is 53,130 2 20,349 ¼ 32,781. 4. 20 balls with at least 4 red balls, at least 2 green balls, at least 6 white balls, and no more than 3 brown balls? We ﬁrst draw the lower bounded balls, namely, 4 þ 2 þ 6 ¼ 12, before we go to the upper-bounded brown balls. We then draw the remaining 8 balls in an unrestricted manner, and the number of ways is 61þ8 13 ¼ ¼ 1287 8 5 We have to subtract from this number the number of ways 4 brown balls can be picked from the 6 colors. This number is 61þ4 9 ¼ ¼ 126 4 4 Hence the number of ways in which all the restrictions will be satisﬁed is 1287 2 126 ¼ 1161. Example 3.3.5 the equation

We tackle a different kind of a problem. How many solutions are there to x1 þ x2 þ x3 þ x4 þ x5 þ x6 ¼ 24

where fxk, k ¼ 1, . . . , 6g are positive integers such that

34

Basic Combinatorics

1. xk . 2 for all k? We need xk 3 for all k. This takes up 6 3 ¼ 18 of the required total of 24, leaving us to solve the equation x1 þ x2 þ x3 þ x4 þ x5 þ x6 ¼ 6 in an unrestricted manner. Here we identify, in Eq. (3.2.7), n ¼ 6 corresponding to the number of xk values and r ¼ 6 corresponding to the righthand side of the equation. Hence the number of solutions is 61þ6 11 11 ¼ ¼ ¼ 462 6 6 5 2. x1 . 2, x2 1, x3 . 3, x4 4, x5 . 2, and x6 2? Using x1 3, x3 4, x5 3, the number of restrictions add upto 3 þ 1 þ 4 þ 4 þ 3 þ 2 ¼ 17. This leaves 24 2 17 ¼ 7 unrestricted choices. Hence the number of solutions is 61þ7 12 12 ¼ ¼ ¼ 792 7 7 5 3. x2 , 5, x3 . 10? The number of balls that can be drawn with the restriction x3 11 corresponding to n ¼ 6 and r ¼ (24 2 11) ¼ 13 is given by 6 1 þ 13 18 18 ¼ ¼ ¼ 8568 13 13 5 The number of draws that violate the additional restriction x2 , 5 is obtained by ﬁrst ﬁnding the number of draws with the additional restriction x2 6. The number of ways corresponding to n ¼ 6 and r ¼ (24 2 11 2 6) ¼ 7 is given by 61þ7 12 12 ¼ ¼ ¼ 792 7 7 5 Hence the required result is 8568 2 792 ¼ 7776 ways. Example 3.3.6 Here is an interesting example to illustrate the versatility of the formula in Eq. (3.2.7). Four dice, each with only 4 faces marked 1,2,3,4, are tossed. We have to calculate the number of ways in which the tosses will result in exactly 1 number, 2 numbers, 3 numbers, and 4 numbers showing on the faces of the dice without regard to order and with replacement. The total number of ways is tabulated in Table 3.3.3.

TABLE 3.3.3 1,1,1,1 1,1,1,2 1,1,2,2 1,2,2,2 1,1,1,3 1,1,3,3

2,2,2,2 2,2,2,3 2,2,3,3 2,3,3,3 2,2,2,4 2,2,4,4

3,3,3,3 3,3,3,4 3,3,4,4 3,4,4,4 3,3,3,1

4,4,4,4 4,4,4,1 4,4,1,1 4,1,1,1 4,4,4,2

1,1,2,3 1,1,3,4 1,1,4,2

2,2,3,4 2,2,4,1 2,2,1,3

3,3,4,1 3,3,1,2 3,3,2,4

4,4,1,2 4,4,2,3 4,4,3,1

1,2,3,4

)

4 single digits

18 double digits

12 triple digits 1 quadruple digit

3.3 Combinations

35

We can see from Table 3.3.3 that there are 4 single digits, 18 double digits, 12 triple digits, and 1 quadruple digit totaling 35 arrangements without regard to order and with replacement. The problem can be stated as follows. Four dice each with 4 faces are tossed without regard to order and with replacement. Find the number of ways in which 1. This can be achieved. With n ¼ 4 and r ¼ 4 in Eq. (3.2.7), the number of ways is

41þ4 4

7 7 ¼ ¼ ¼ 35 4 3

which agrees with the total number in the table. 2. Exactly one digit is present. With n ¼ 4 and r ¼ 1, the number of ways is

41þ1 1

¼

4 ¼4 1

which agrees with the table. 3. Exactly 2 digits are present. The number of ways 2 digits can be drawn from 4 digits is 42 . The number of ways any particular 2 digits, for example f1,2g, will occupy a 4-digit number is a restricted case as in Example 3.2.4 with both 1 and 2 occupying two slots and the rest of the two remaining unrestricted. With n ¼ 2 and r ¼ 4 2 2 ¼ 2 in Eq. 21þ2 (3.2.7), the number of ways for this particular 3 ¼ sequence f1,2g is 2 2 ¼ 3. The total number using all the combinations are 42 32 ¼ 6 3 ¼ 18 ways. This agrees with the table. 4. Exactly 3 digits are present. The number of ways 3 digits can be drawn from 4 digits is 43 ¼ 4 ways. The number of ways 3 digits can be arranged in 4 slots such that all the three digits 3 are present is again given by the restriction 4formula 3 ¼ ¼ 3. Hence the total number of ways is Eq. (3.2.7) 31þ(43) 2 1 3 1 ¼ 12 ways, agreeing with the table. 5. Exactly 4 digits are present. The number of ways 4digits will be present is ¼1 obtained from the argument in item 3 above, giving 44 41þ(4þ4) 0 Generalization

The result in Example 3.3.6 can be generalized to the tossing of n n-sided dice. The total number of sample points without regard to order and with replacement is, from Eq. (3.2.7)

n1þn n

¼

2n n

(3:3:8)

Out of these, the number of ways exactly r digits will occur can be derived as follows: The number of ways r digits can be chosen from n is nr , and if any particular sequence of r digits has to ﬁll n slots, then the number of ways this can be done is obtained by substituting r ¼ (n 2 r) in Eq. (3.2.7), yielding

r 1 þ (n r) nr

¼

n1 nr

(3:3:9)

36

Basic Combinatorics

FIGURE 3.3.4

Since all the nr sequences are functionally independent, the number of ways N(n,r) is as follows, from Eq. (3.1.2): n n1 N(n,r) ¼ ¼ (3:3:10) r nr Using Eq. (3.3.10), N(n,r) tabulated for n ¼ 1, . . . , 8 and r ¼ 1, . . . , 8 in Fig. 3.3.4. In addition, we have the following interesting identity. Combining Eqs. (3.3.8) and (3.3.9), we have X n n n1 2n 1 (3:3:11) ¼ r nr n r¼0 Equation (3.3.11) can also be expressed by substituting m ¼ n – 1, r ¼ n, and k ¼ r in the Vandermonde identity of Eq. (3.3.6) X X r n m n1 n n nþn1 nþm ¼ : (3:3:12) ¼ ¼ rk nk k k n r k¼0 k¼0

CHAPTER

4

Discrete Distributions

B 4.1

BERNOULLI TRIALS

We will elaborate on the Cartesian product already discussed in Chapter 1. Consider the tossing of an unfair coin with success given by fhead ¼ sg with probability p and failure given by ftail ¼ fg with probability q, and p þ q ¼ 1. The sample space is S1 ¼ fs, fg with ﬁeld F ¼ ½S1 , 1, {s}, { f }. If the coin is tossed twice, then the sample space S2 ¼ fss, sf, fs, ffg is an ordered set from the Cartesian product S1 S1. The cardinality of S2 is 2 2 ¼ 4. The probability of two successes is p 2 and two failures is q 2. If we toss the coin n times, then the resulting sample space is Sn ¼ S1 S1 S1 S1 and the cardinality of Sn is 2n. The probability of n successes is p n.

Bernoulli Trials

Bernoulli trials are repeated functionally independent trials with only two events s and f for each trial. The trials are also statistically independent with the two events s and f in each trial having probabilities p and (1 2 p), respectively. These are called independent identically distributed (i.i.d.) trials. Example 4.1.1 Tossing an unfair coin 4 times constitutes 4 Bernoulli trials with heads ¼ s and tails ¼ f with Pfsg ¼ p and Pf fg ¼ 1 2 p. The sample space S is the Cartesian product with ordered 4-tuples. The cardinality of S is 24 ¼ 16 points. Example 4.1.2 Tossing a die 5 times can be considered Bernoulli trials if we deﬁne getting an ace as success, s with Pfsg ¼ 16 and not getting an ace as failure, f with Pf fg ¼ 56. The resulting sample space S is the Cartesian product with ordered quintuples with number of points equaling the cardinality 25 ¼ 32.

Bernoulli Distribution

Bernoulli distribution is the simplest of a discrete distribution. A single event is Bernoulli distributed if the probability of success is p and probability of failure is q with p þ q ¼ 1.

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

37

38

Discrete Distributions

Example 4.1.3 A single die is thrown. The probability of getting an ace is p ¼ 16 and the probability of not getting an ace is q ¼ 56. Hence this experiment is Bernoulli distributed.

B 4.2

BINOMIAL DISTRIBUTION

We are given an experiment S with an event A , S with PfAg ¼ Pfsuccessg ¼ p and PfAg ¼ Pffailureg ¼ q with p þ q ¼ 1. We repeat this experiment n times, and the resulting sample space Sn is the Cartesian product Sn ¼ {S S S} n times

containing 2n elementary events. We will now determine the probability b(k; n, p) that the event A occurs exactly k times in n Bernoulli trials. We shall address this problem in two parts. In the ﬁrst part we will ﬁnd the probability of k successes in a particular sequence in n trials. For example, one such sequence can be {s, s, f , s, f , f , . . . , s, f } exactly k successes

For the k independent successes the probability is p k and for the (n 2 k) independent failures the probability is q n – k. Since both successes and failures are independent, we have Probability{k successes in a given sequence} ¼ pk qnk

(4:2:1)

In the second part we address the situation of k successes in any sequence in n trials. The number of ways k successes can be obtained from n trials is by sampling without replacement and without regard to order. This number from Eq. (3.3.2) is kn . These realizations are mutually exclusive since the occurrence of one realization precludes the occurrence of another. Hence the probabilities will add, giving rise to n k nk Probability{k successes in any sequence in n trials} ¼ b(k; n,p) ¼ pq (4:2:2) k This is called the binomial distribution, and b(k;n,p) is called a probability mass function. We now show that this is a probability distribution by summing over all k. Since n kwill nk p q is the kth term in the binomial expansion of ( p þ q)n, we can write k n n X X n k nk b(k; n,p) ¼ (4:2:3) p q ¼ ( p þ q)n ¼ 1 k k¼0 k¼0 The probability of getting utmost r successes is given by P{number of successes r} ¼

r X

b(k; n,p) ¼

k¼0

r X n k¼0

k

pk qnk

(4:2:4)

pk qnk

(4:2:5)

and the probability of getting at least r successes is given by P{number of successes r} ¼

n X k¼r

b(k; n,p) ¼

n X n k¼r

k

4.2 Binomial Distribution

39

Since n! grows very rapidly with n, computational difﬁculties in calculating b(k;n,p) can be avoided by using the recursive formula shown below with starting value b(0;n,p). n n k p n k nk kþ1 nk1 pq ¼ p q b(k þ 1; n,p) ¼ kþ1 q k kþ1 ¼ Example 4.2.1

nk p b(k; n,p) kþ1 q

(4:2:6)

A fair die is rolled 10 times, and we will calculate

1. The probability that an ace comes up exactly 4 times. Substituting p ¼ 16 and q ¼ 56, n ¼ 10, and k ¼ 4 in Eq. (4.2.3), we obtain 10! 1 4 5 6 ¼ 0:05427 P{6 aces in 10 trials} ¼ 6!4! 6 6 2. The probability that utmost 4 aces show up. In probability 1 (above) we wanted exactly 4 aces coming up. Now we are looking into the possibility of 4 or fewer aces showing up. This probability is obtained as follows: k 10k 10! 1 5 ¼ 0:98454 P{number of aces 4} ¼ k!(10 k)! 6 6 k¼0 4 X

3. The probability that the number of aces showing is between 2 and 6 inclusive. Here we have to sum b(k: 10, 16) between 2 and 6. Performing the indicated summation, we have k 10k 10! 1 5 ¼ 0:22451 P{2 number of aces 6} ¼ k!(10 k)! 6 6 k¼2 k 5 10k The graph of the probability mass function 10i ( 16 P ) ( 10 ) is i shown 10iin Fig. 4.2.1, is shown in and that of the cumulative distribution function ki¼0 10i 16 56 Fig. 4.2.2 for k ¼ 0, . . . , 10. 6 X

FIGURE 4.2.1

40

Discrete Distributions

FIGURE 4.2.2

Example 4.2.2 We shall use the recursive form [Eq. (4.2.6)] to compute the binomial coefﬁcients in Example 4.2.1 with initial condition b(0;n,p) ¼ 0.161506. These values are substituted in Eq. (4.2.4), and the ﬁrst four steps are shown: 1 b 0; 10, ¼ 0:161506 6 1 10 1 0:161506 ¼ 0:323011 b 1; 10, ¼ 6 1 5 1 9 1 b 2; 10, ¼ 0:323011 ¼ 0:29071 6 2 5 1 8 1 b 3; 10, ¼ 0:29071 ¼ 0:155045 6 3 5 The true values and the computed values are shown in Table 4.2.1, and they are the same. Example 4.2.3 Two experiments are performed. In the ﬁrst experiment 6 fair dice are tossed and the person wins if she gets 1 or more aces. In the second experiment 12 dice are tossed and the person wins if she gets 2 or more aces. We want to ﬁnd in which experiment the probability of winning is higher.

TABLE 4.2.1 b(k;n,p) k

Computed

Calculated

k

Computed

Calculated

0 1 2 3 4 5

0.161506 0.323012 0.290711 0.155046 0.054266 0.013024

0.161506 0.323011 0.29071 0.155045 0.054266 0.013024

6 7 8 9 10

0.002171 0.000248 0.000019 0.000001 0

0.002171 0.000248 0.000019 0.000001 0

4.3 Multinomial Distribution

41

The probability of getting one or more aces out of 6 tosses is the same as the complement of the probability of getting no aces in 6 throws. This is obtained from Eq. (4.2.5) with n ¼ 6, p ¼ 16 and q ¼ 56: 0 6 1 5 6 1 P{no aces in 6 throws} ¼ 1 ¼ 0:6651 0 6 6 The probability of getting 2 or more aces in 12 throws is the same as the complement of the sum of the probabilities of getting no aces and one ace in 12 throws. With n ¼ 12, we obtain from Eq. (4.2.5) 1 P{no aces in 12 throws} P{one ace in 12 throws} 1 11 0 12 12 12 1 5 1 5 1 ¼1 6 6 6 6 1 0 ¼ 1 0:1122 0:2692 ¼ 0:6186 Hence the probability of getting one or more aces in 6 throws is better than getting two or more aces in 12 throws. Example 4.2.4 An urn consists of 40 red balls and 60 green balls. What is the probability of getting exactly k red balls in a sample size of 10 if the sampling is done 1. With replacement? The meaning of “with replacement” is to pick a ball from the 100 in the urn note its color and replace it and pick again, and this process is continued for 10 times. Thus Pfred ballg ¼ p ¼ 25 and Pfgreen ballg ¼ q ¼ 35 and with n ¼ 10, the probability of k red balls in 10 trials is given by k 10k 2 2 3 10 b k; n, ¼ k 5 5 5 2. Without replacement? Here we sample the balls and do not replace it in the urn. 100 Thus, the number of ways of getting 10 balls from 100 balls 40 is 10 . Out of these 10 balls, the number of ways of getting kred balls is k and the number of ways of getting (10 2 k) blue balls is 1060 k . Since getting k red balls and 40 (10 602k) blue balls are functionally independent, the total number of ways are k 10 k using the product rule. Hence the probability P(k) of k red balls without replacement is given by 40 60 k 10 k P(k) ¼ 100 10

B 4.3

MULTINOMIAL DISTRIBUTION

The binomial distribution of the previous section can be generalized to n repeated independent trials where each trial has m outcomes fEi, i ¼ 1, . . . , mg with probabilities PfEig ¼ fpi, i ¼ 1, . . . , mg and p1 þ p 2 þ þ p m ¼ 1

(4:3:1)

42

Discrete Distributions

Each of the outcomes fEi, i ¼ 1, . . . , mg occurs fki, i ¼ 1, . . . , mg times with k1 þ k2 þ þ km ¼ n

(4:3:2)

The probability Pfk1, k2, . . . , kmg is given by the multinomial probability distribution, n P{k1 , k2 , . . . , km } ¼ pk1 pk2 pkmm k1 , k2 , . . . , km 1 2 n! pk1 pk2 pkmm k1 !k2 !; km ! 1 2

¼

(4:3:3)

If m ¼ 2, the multinomial distribution reduces to the binomial distribution. Example 4.3.1 A pair of dice is rolled 8 times. We will ﬁnd the probability that 7 or 11 occurs thrice; 2, 3, or 12 twice; any pair twice; and any other combination once. For each rolling of a pair there are four events E1, E2, E3, E4 and their probabilities are obtained from Table 2.2.1 and are shown below: E1 ¼ {7 or 11}

8 36 4 P{E2 } ¼ p2 ¼ 36 6 P{E3 } ¼ p3 ¼ 36 18 P{E4 } ¼ p4 ¼ 36 P{E1 } ¼ p1 ¼

E2 ¼ {2, 3 or 12} E3 ¼ {any pair} E4 ¼ {other combination}

In Eq. (4.3.3) we substitute k1 ¼ 3, k2 ¼ 2, k3 ¼ 2, k4¼1 with p1, p2, p3, p4 as given above, yielding 3 2 2 1 8! 8 4 6 18 ¼ 0:003161 P{3, 2, 2, 1} ¼ 3!2!2!1! 36 36 36 36

B 4.4

GEOMETRIC DISTRIBUTION

Closely allied to the binomial distribution is the geometric distribution. A sequence of independent Bernoulli trials with p as the probability of success can be modeled in two different ways. The distribution of the number of successes k is binomial. The waiting time until the ﬁrst success occurs is geometrically distributed. If the ﬁrst success is to occur at the kth trial, then we must have (k 2 1) failures preceding the ﬁrst success. Since the trials are independent, this probability is given by g(k,p) ¼ (1 p)k1 p, k ¼ 1,2, . . .

(4:4:1)

This is the geometric distribution with parameter p. Further, the waiting times between each success and the next are independent and are geometrically distributed. We can show that the sum to inﬁnity of g(k,p) is indeed 1 as shown below: 1 X k¼1

g(k,p) ¼ p

1 X k¼1

(1 p)k1 ¼

p ¼1 1 (1 p)

4.4 Geometric Distribution

43

Example 4.4.1 In an urn there are r red balls and b blue balls. Balls are selected at random with replacement so that the ﬁrst blue ball is obtained. We have to ﬁnd the probability that 1. Exactly k draws are needed for the ﬁrst blue ball. The probability of a blue ball is p ¼ b/(b þ r). Substituting this value p in Eq. (4.4.1), the required probability is b b k1 b ¼ 1 g k, bþr bþr bþr k1 r b br k1 ¼ ¼ bþr b þ r (b þ r)k 2. At least m draws needed for the ﬁrst blue ball. The ﬁrst success occurs at the mth draw after (m 2 1) failures. Since the failure probability is q ¼ [r/(b þ r)], we have from the deﬁnition of the geometric distribution this probability is [r/(b þ r)]m21. The same result can also be obtained from the cumulative summation as follows: k1 1 1 X b b X r g k, ¼ bþr b þ r k¼m b þ r k¼m # m1 " 2 b r r r ¼ 1þ þ þ bþr bþr bþr bþr 2 3 m1 m1 b r 1 r 6 7 ¼ ¼ 4 r 5 bþr bþr bþr 1 bþr This geometric distribution is shown in Fig. 4.4.1 for r ¼ 100, b ¼ 20, and k ¼ 1, . . . , 10 and the cumulative distribution in Fig. 4.4.2.

FIGURE 4.4.1

44

Discrete Distributions

FIGURE 4.4.2

B 4.5

NEGATIVE BINOMIAL DISTRIBUTION

The binomial distribution ﬁnds the probability of exactly k successes in n independent trials. The geometric distribution found the number of independent trials needed for the ﬁrst success. We can generalize these two results and ﬁnd the number of independent trials required for k successes. This distribution is known as the negative binomial or the Pascal distribution. Let us now deﬁne the following events to ﬁnd this distribution: E ¼ n trials for exactly k successes A ¼ exactly (k 1) successes occur in (n 1) trials B ¼ nth trial results in kth success Since E ¼ A > B and A and B are independent, we have P{E} ¼ P{A} P{B}

(4:5:1)

We will now compute PfAg. In a sequence of (n 2 1) trials we have exactly (k 2 1) successes and [n 2 1 2 (k 2 1)] ¼ (n 2 k) failures. Since these trials are independent, the probability of that particular sequence is p k21q n21. There are n1 k1 mutually exclusive sequences, and hence n 1 k1 nk (4:5:2) P{A} ¼ b(k 1; n 1, p) ¼ p q k1 Since the probability PfBg ¼ p, we can substitute this and Eq. (4.5.2) in Eq. (4.5.1) and write n 1 k nk n 1 k1 nk ¼ nb(k; n,p) ¼ P{E} ¼ p p q , n ¼ k, k þ 1, . . . (4:5:3) p q k1 k1 We have to show that the sum of nb(k;n,p) from n ¼ k to 1 equals 1. Or 1 X n¼k

nb(k; n,p) ¼

1 X n1 n¼k

k1

pk qnk ¼ 1

(4:5:4)

4.5

TABLE 4.5.1

Negative Binomial Distribution

45

Negative binomial

n

Probability

n

Probability

n

Probability

5 6 7 8

0.16807 0.252105 0.226894 0.158826

9 10 11 12

0.095296 0.05146 0.02573 0.01213

13 14 15

0.005458 0.002365 0.000993

By substituting (n 2 k) ¼ m, we can rewrite Eq. (4.5.4) as 1 1 1 X X kþm1 k m X kþm1 k m nb(k; k þ m,p) ¼ pq ¼ p q ¼1 k1 m m¼0 m¼0 m¼0

(4:5:5)

The Taylor series expansion of (1 2 x)2k yields (k)( k 1) (k)(k 1)(k 2) (x)2 þ (x)3 þ 2! 3! 1 X kþm1 m ¼ x m m¼0

(1 x)k ¼ 1 þ kx þ

(4:5:6)

Substituting x ¼ q in Eq. (4.5.6) and (1 2 q) ¼ p, we obtain 1 1 X X kþm1 m kþm1 k m k or 1 ¼ q , p q , which is Eq. (4:5:5) p ¼ m m m¼0 m¼0 In Eq. (4.5.3), if we substitute k ¼ 1, then we get the geometric distribution. Example 4.5.1 A computer software company wants to recruit 5 new engineers. The probability that a person interviewed is given a job offer is 0.7. We want to ﬁnd the distribution of the number of people interviewed for the 5 positions. We use Eq. (4.5.3) with p ¼ 0.7 and k ¼ 5. Thus n1 P{E ¼ n} ¼ (0:7)5 (0:3)n5 , n ¼ 5, 6; . . . 51 PfE ¼ ng is shown for n ¼ 5,6, . . . ,15 in Table 4.5.1, and the probability mass function is shown in Fig. 4.5.1.

FIGURE 4.5.1

46

Discrete Distributions

From the table and the ﬁgure we see that the maximum probability of 0.2521 occurs at k ¼ 6 and it is most likely that company will interview exactly 6 persons for the 5 jobs.

B 4.6

HYPERGEOMETRIC DISTRIBUTION

The binomial distribution is obtained while sampling with replacement. Let us select a sample of n microprocessor chips from a bin of N chips. It is known that K of these chips are defective. If we sample with replacement then the probability of any defective chip is K/N. The probability that k of them will be defective in a sample size of n is given by the binomial distribution since each drawing of a chip constitutes a Bernoulli trial. On the other hand, if we sample without replacement, then the probability for the ﬁrst chip is K/N, the probability for the second chip is (K 2 1)/(N 2 1), and for each succeeding sample the probability changes. Hence the drawings of the chips do not constitute Bernoulli trials and binomial distribution does not hold. Hence we have to use alternate methods to arrive at the probability of ﬁnding k defective chips in n trials. The number of sample points in this sample space is choosing n chips out of the total N chips without regard to order. This can be done in Nn ways. Out of this sample size of n, there are k defective chips and (n 2 k) good chips.The number of ways of picking k defective chips out of the total of K defective chips is Kk and the number of ways of drawing (n 2 k) good chips out of the remaining (N 2 K) good chips is NK nk . We denote the total number of ways of drawing a sample of n chips with k bad chips as the event E. Since these drawings are functionally independent the number of ways is Kk NK nk . Hence the probability of the event E is given by K NK k nk P{E ¼ k} ¼ (4:6:1) N n This is known as the hypergeometric distribution. We came across this distribution in Example 4.2.4. Since the summation over all k must be one, we must have the following in Eq. (4.6.1): n X K N K N ¼ (4:6:2) k n k n k¼0 This result follows directly from Vandermonde’s identity of Eq. (3.3.6) given by n X n m nþm ¼ (3:3:6) k rk r k¼0 with n ¼ K, m ¼ N 2 K, and r ¼ n. Example 4.6.1 (Mass Megabucks) Massachusetts megabucks lottery is played as follows. A player has to choose 6 numbers from a slate of 42 numbers. Twice a week the lottery commission draws 6 numbers. If all the 6 numbers of the player match all the 6 numbers drawn by the commission, the player wins megabucks. There are smaller prizes for matching 5, 4, or 3 numbers. We have to calculate the probability of winning megabucks or the smaller prizes.

4.6 Hypergeometric Distribution

TABLE 4.6.1

47

Mass megabucks

k

N(k)

C(k)

P(k)

0 1 2 3 4 5 6

1947792 2261952 883575 142800 9450 216 1

3 2 6 37 555 24286 5,245,786

0.371306035 0.431194105 0.168435197 0.02722185 0.001801446 0.000041176 0.000000191

Let k ¼ 0,1, . . . , 6. Substituting N ¼ 42, n ¼ 6, and K ¼ 6, we can ﬁnd from Eq. (4.6.1) PfE ¼ kg: 6 42 K k 6k N(k) ¼ P{E ¼ k} ¼ , k ¼ 0,1, . . . , 6 D(k) 42 6 The results for k ¼ 0, . . . , 6 in Table 4.6.1. In this table, the second column are tabulated is the number of ways k digits can be formed from 6 digits. In the third N(k) ¼ 6k 42K 6k column C(k) ¼ D(k)/N(k) represents the chance of winning with k digits rounded to the nearest digit and in the fourth column P(k) is the probability of winning. Example 4.6.2 Hypergeometric distribution is used to estimate an unknown population from the data obtained. To estimate the population N of tigers in a wildlife sanctuary, K tigers are caught, tagged, and released. After the lapse of a few months, a new batch of n tigers is caught, and k of them are found to be tagged. It is assumed that the population of tigers does not change between the two catches. The total number of tigers N in the sanctuary is to be estimated. In Eq. (4.6.1), the known quantities are K, k, and n. The total number N has to be estimated in the following equation: K NK k nk Pk {N} ¼ N n We will ﬁnd the value of N that maximizes the probability PkfNg. Such an estimate is called the maximum-likelihood estimate, which will be discussed in Chapter 18. Since the numbers are very large, we investigate the following ratio: NK , K N1K K nk nk k k Pk {N} ¼ N N1 Pk {N 1} n n N K N1 nk n Nn NK ¼ ¼ N N1K N NKnþk n nk

48

Discrete Distributions

The ratio (Pk {N})=(Pk {N 1}) is greater than 1 if (N 2 n)(N 2 K ) . N (N 2 K 2 n þ k), Nk , nK, or N , nK/k. Conclusion. If N is less than nK/k, then Pk(N) is increasing, and if N is greater than nK/k, then Pk(N ) is decreasing. Hence the maximum value of Pk(N ) occurs when N is equal to the nearest integer not exceeding nK/k. Thus, if K ¼ 100, n ¼ 50, and k ¼ 20, then the population of tigers is 100.50/20 ¼ 250. Approximation to Binomial Distribution

If N and K are very large with respect to n and k, then the hypergeometric distribution can be approximated by the binomial distribution as shown below: P{E ¼ k} ¼

K k

NK nk

N n

¼

k! (N K)! n!(N n)! k!(K k)! (n k)!(N K n þ k)! N!

¼

n! K! (N n)! (N K)! k!(n k)! (K k)! N! (N K n þ k)!

¼

n K(K 1) (K k þ 1) k N(N 1) (N k þ 1)

(N K)(N K 1) (N K n þ k þ 1) (N k)(N K 1) (N k n þ k þ 1)

(4:6:3)

Since K k and N n, Eq. (3.3.7) can be approximated as P{E ¼ k} ¼

n K K K (N K)(N K) (N K) N N N k N N N k times

(4:6:4)

Nk times

Approximating p ﬃ K/N and q ﬃ (N 2 K )/n in Eq. (3.3.8), we obtain n k nk P{E ¼ k} ﬃ pq k

(4:6:5)

which is a binomial distribution.

B 4.7

POISSON DISTRIBUTION

Perhaps three of the most important distributions in probability theory are (1) Gaussian, (2) Poisson, and (3) binomial. We have already discussed the binomial and other allied distributions that are characterized by discrete time (n) and discrete state, that is, the cumulative distribution function has jumps occurring only at discrete points. In the regular Poisson distribution the cumulative distribution has unit jumps at random times and not at discrete times. Thus, it is discrete in state but continuous in time.

4.7 Poisson Distribution

49

The Poisson distribution takes values 0,1, . . . k, . . . at random points with a parameter l and is given by p{k ; l} ¼ el

lk , k!

k0

(4:7:1)

We will show that this is a probability function by summing k from 0 to 1 as shown below: 1 X

p{k ; l} ¼

k¼0

1 X

el

k¼0

1 X lk lk ¼ el ¼ el el ¼ 1 k! k! k¼0

(4:7:2)

Similar to Eq. (4.2.6) we can derive a recursive form for evaluating the Poisson distribution as shown below: 1 X k¼0

p{k þ 1 ; l} ¼

1 X k¼0

el

1 lkþ1 l X lk l p{k ; l} ¼ el ¼ (k þ 1)! k þ 1 k¼0 k! k þ 1

(4:7:3)

This last equation is solved from the starting value p(0, l) ¼ e 2l. Example 4.7.1 During a certain experiment the number of a particles passing through a sensor in a 1-ms interval is 3.2. The Poisson distribution from Eq. (4.7.1) with l ¼ 3.2 can be written as p{k ; 3:2} ¼ e3:2

(3:2)k , k!

k0

This is graphed in Fig. 4.7.1. We want to ﬁnd the probability that 1. Six a particles pass through the sensor in a given millisecond. Substituting l ¼ 3.2 and k ¼ 6, we obtain p{6 ; 3:2} ¼ e3:2

ð3:2Þ6 ¼ 0:06079 6!

FIGURE 4.7.1

50

Discrete Distributions

2. No more than three a particles pass through the sensor. We have to ﬁnd the cumulative distribution given by 3 X

p{k ; 3:2} ¼

k¼0

3 X k¼0

e3:2

(3:2)k ¼ 0:6025 k!

Example 4.7.2 The average number of customers arriving at the service counter in a supermarket is l, and the probability of k customers arriving in the interval (0,t) minutes is given by the Poisson distribution pfk ; lg ¼ e 2l[(lt)k/k!]. In the nonoverlapping intervals (0,t1) and (t1,t), we can write the following Poisson law P{n1 customers in (0,t1 ) and n2 in (t1 ,t)} ¼ P{n1 customers in (0,t1 )} P{n2 customers in (t1 ,t)} and the two events are independent in disjoint intervals. We will ﬁnd the conditional probability, Pfn1 customers in (0,t1) j (n1 þ n2) customers in (0,t)g. From the deﬁnition of conditional probability, we can write P{n1 ; (0,t1 ) j (n1 þ n2 ); (0,t)} ¼ ¼

P{½n1 ; (0,t1 )½(n1 þ n2 ); (0,t)} P{(n1 þ n2 ); (0, t)} P{½n1 ; (0,t1 )½n2 ; (t1 ,t)} P{(n1 þ n2 ); (0, t)}

elt1 (lt1 )n1 el(tt1 ) ½l(t t1 )n2 ¼ n1 ! n2 ! ¼

(lt1 )n1 ½l(t t1 )n2 (n1 þ n2 )! (lt)n1 þn2 n1 !n2 !

¼

t1n1 (t t1 )n2 (n1 þ n2 )! tn1 þn2 n1 !n2 !

,

elt (lt)n1 þn2 (n1 þ n2 )!

which yields a surprising result independent of l! Poisson distribution has been used in a number of applications such as the 1. Number of telephone calls coming into an exchange 2. Number of cars entering a highway ramp 3. Number of packets transmitted along wireless lines 4. Number of customers at the checkout counter 5. Number of misprints in a book 6. Number of photons emitted Approximation to Binomial Distribution

Poisson distribution is also used for approximating a binomial distribution with parameters (n,p) under the condition that n is large compared to k and p is small enough that the product np ¼ l is of moderate magnitude. As a rule of thumb, l may

4.7 Poisson Distribution

51

be a single digit. We shall now derive this approximation: n k nk n! b(k;n,p) ¼ ¼ pk (1 p)nk pq k!(n k)! k n(n 1) (n k þ 1) l k l nk ¼ 1 k! n n ¼

n(n 1) (n k þ 1) lk (1 l=n)n k! (1 l=n)k nk

(4:7:4)

As n becomes very large and for moderate l, the quantity (1 l=n)n can be approximated by e 2l and the quantity (1 l=n)k can be approximated as 1. Finally, as k is much smaller than n, the quantity ½n(n 1) (n k þ 1)=nk can be approximated by 1. Thus the binomial distribution b(k;n,p) can be approximated as b(k; n,p)

lk l e , k!

k0

(4:7:5)

Example 4.7.3 A computer system consists of 1000 microchips, and the probability of failure of a microchip is 0.005. We have to ﬁnd the probability of k microchips failing using both the binomial distribution and its Poisson approximation for k ¼ 0,1, . . . ,15. We will also ﬁnd the percentage error between the two. Using binomial distribution, k chips failing in 1000 trials given by b(k; 1000,0.005) ¼ b(k) is b(k; 1000,0:005) ¼ b(k) ¼

1000! 0:005k 0:9951000k k!(1000 k)!

The value b(k) can be solved using the recursive formula developed in Eq. (4.2.6), and the steps are 1000 k 0:005 b(k) k þ 1 0:995 1000 k 1 b(k) ¼ k þ 1 188

b(k þ 1) ¼

Using the Poisson approximation we have np ¼ l ¼ 5, and substituting in the Poisson distribution, we have k 5 5 p(k ; l) ¼ p(k ; 5) ¼ p(k) ¼ e k! Both b(k) and p(k) are shown for k ¼ 0,1, . . . , 15 in Table 4.7.1 along with the percentage error. Table 4.7.1 is also graphed in Fig. 4.7.2 for the binomial distribution and the Poisson approximation. The ﬁgure shows that the approximation is very close. The error in the approximation is also graphed in Fig. 4.7.3. From the table and the graphs we see that the approximation becomes worse as k is increased and the approximation is excellent in the vicinity of k ¼ 5 ¼ l, about 0.05 –0.25%.

52

Discrete Distributions

TABLE 4.7.1

Comparison of binomial with Poisson approximation

k

Binomial

Poisson

%Error

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.006654 0.033437 0.083929 0.140303 0.175731 0.175908 0.14659 0.104602 0.065245 0.036138 0.017996 0.008139 0.003371 0.001287 0.000456 0.000151

0.006738 0.03369 0.084224 0.140374 0.175467 0.175467 0.146223 0.104445 0.065278 0.036266 0.018133 0.008242 0.003434 0.001321 0.000472 0.000157

21.26208 20.75577 20.352343 20.050683 0.150021 0.250272 0.250272 0.14992 20.051188 20.353762 20.758823 21.267706 21.88207 22.603907 23.435549 24.379687

FIGURE 4.7.2

FIGURE 4.7.3

4.7 Poisson Distribution

53

Example 4.7.4 (See Ref. 47) We will revisit the two or more birthdays matching problem posed in Example 3.2.4, and solve it using the Poisson approximation and compare the results. In addition, we will use the Poisson approximation to ﬁnd probability of matching three or more birthdays that would have been very difﬁcult using the exact method. Given the total numbers of persons n 365, we will ﬁnd, using the Poisson approximation, the probability of two or more birthdays matching. The probability of two birth1 , and this we will set equal to p. The number days matching is 365 n of ways we can obtain a sample of two without replacement and without ordering is 2 , each of which has the 1 same probability of 365 of matching. Hence the 2n ways constitute Bernoulli trials and 1 since p ¼ 365 is very small and 1 n(n 1) n ¼ ¼l np ¼ 2 365 730

is less than 10, all the conditions for approximation by a Poisson distribution are satisﬁed. Hence

P{2 or more people share the same birthday} ¼ 1 P{no birthdays match} n(n 1) l0 ¼ 1 p(0; l) ¼ 1 exp 730 0!

and [1 2 p(0 ; l)] is shown in Table 4.7.2 for the same values of n as given in Table 3.2.3 with the corresponding percentage error shown in Table 4.7.3. The percent error curve between the true value and the Poisson approximation for values of n ¼ 1, . . . ,100 is shown in Fig. 4.7.4. An examination of the error curve reveals that the Poisson approximation underestimates the true values, and for values of n higher than 82, the error is almost zero. Even then the maximum error is only about 1.46% occurring at n ¼ 26. As mentioned before, this approximation can be extended to the case of three or more , and matches. Approaching, as before, the number of 3-combinations is n3 ¼ n(n1)(n2) 6

TABLE 4.7.2

Poisson approximation for two matches

#

Probability

#

Probability

#

Probability

20 21 22 23 24 25 26

0.0405805 0.437488 0.468938 0.500002 0.530536 0.560412 0.589513

30 31 32 33 34 35 36

0.69632 0.720282 0.743058 0.764625 0.784972 0.804097 0.82201

44 45 46 47 48 49 50

0.925113 0.933618 0.941318 0.948266 0.954517 0.960121 0.965131

54

Discrete Distributions

TABLE 4.7.3

Error in Poisson approximation

#

%Error

#

%Error

#

%Error

20 21 22 23 24 25 26

1.369162 1.397497 1.420488 1.438108 1.450358 1.457273 1.458918

30 31 32 33 34 35 36

1.415262 1.392672 1.365868 1.335125 1.300742 1.263039 1.222352

44 45 46 47 48 49 50

0.833107 0.781948 0.731364 0.681649 0.633067 0.58586 0.540238

FIGURE 4.7.4

the probability p of three matches is

1 2 365 .

Substituting in the Poisson formula we obtain

P{3 or more share the same birthday} ¼ 1 P{no 3 birthdays match} n(n 1)(n 2) l0 ¼ 1 p(0; l) ¼ 1 exp 0! 6 3652 n(n 1)(n 2) ¼ 1 exp 799350

This probability is graphed in Fig. 4.7.5 for values of n ¼ 1, . . . , 150 and tabulated in Table 4.7.4 for values of n ¼ 80 – 86, 90– 96, and 104 –110. Examination of Fig. 4.7.3 and Table 4.7.4 reveals that for an even chance for at least three matches, there must be about 84 people and for a 70% chance there must be about 100 people compared to 23 and 30 for at least two matches. This approximation technique can be extended to any number of matches from 2 to 365, whereas exact computation becomes extremely cumbersome, if not impossible, for anything more than two matches.

4.8 Logarithmic Distribution (Benford’s Law)

55

FIGURE 4.7.5

TABLE 4.7.4

B 4.8 4.8.1

Poisson approximation for three matches

#

Probability

#

Probability

#

Probability

80 81 82 83 84 85 86

0.460278 0.472929 0.485593 0.498257 0.510911 0.523543 0.536141

90 91 92 93 94 95 96

0.58597 0.598231 0.610393 0.622444 0.634375 0.646176 0.657838

104 105 106 107 108 109 110

0.745102 0.755146 0.764978 0.774593 0.783987 0.793155 0.802095

LOGARITHMIC DISTRIBUTION Finite Law (Benford’s Law)

First Signiﬁcant Digit

Most physical data depend on units. Examples are rainfall in inches, land area in acres, blood pressure in millimeters of mercury, stock prices in dollars, distances in miles, and volume in cubic feet. If clusters of these quantities can be described by a probability distribution P(x), then such a probability distribution must be invariant under a scale change, so that P(k x) ¼ KP(x) (4:8:1) Ð Ð where k and K are scale factors. Since P(x)dx ¼ 1, we have P(k x)dx ¼ 1=k, and hence K ¼ 1=k is the normalization factor. Thus 1 P(kx) ¼ P(x) or kP(kx) ¼ P(x) k Differentiating Eq. (4.8.2) with respect to k, we obtain P(kx) þ k

d d(kx) P(kx) ¼0 d(kx) dk

(4:8:2)

56

Discrete Distributions

Substituting k ¼ 1 in the equation above, we have d x P(x) ¼ P(x) dx The solution for Eq. (4.8.3) is P(x) ¼

1 x

(4:8:4)

Equation (4.8.4) can be considered as a probability distribution only for ﬁnite values of x since the integral for inﬁnite values of x diverges. We shall assume that the data are deﬁned to some arbitrary base B rather than base 10. If X1 is the random variable corresponding to the occurrence of the ﬁrst digit d to the base B, then the probability of occurrence of PX1(d) is given by Ð dþ1 1 dx d ¼ Ð x ; d ¼ 1,2, . . . , B 1 PX1 (d) ¼ dÐ B B1 1 P(x)dx 1 x dx 1 ln 1 þ ln (d þ 1) ln (d) d ¼ ¼ ln (B) ln (1) ln (B) Ð dþ1

P(x)dx

(4:8:5)

We can show that indeed Eq. (4.8.5) is a probability function as follows: B1 X 1 1 1 1 ln 1 þ ¼ ln 1 þ þ ln 1 þ þ ln 1 þ d 1 2 B1 1 ¼ ln

B 1 Y 1

1 234 B 1þ ¼ ln ¼ ln (B) d 123 B 1

(4:8:6)

Thus PX1(d ) over all digits 1 to B21 sums to 1. If the base B is 10, the generic ln becomes log, and the probability of the ﬁrst signiﬁcant digit d from 1 to 9 is, 1 PX1 (d) ¼ log 1 þ ; d ¼ 1,2, . . . , 9 (4:8:7) d This equation is called Benford’s law, ﬁrst stated in 1938 by Benford [5]. It gives the probability of occurrence of the ﬁrst signiﬁcant digit in a cluster of distributions. Incidentally, 0 cannot be the ﬁrst signiﬁcant digit. It can be a signiﬁcant digit only after the ﬁrst one. Example 4.8.1 In a cluster of numbers from different scale factors, we want to ﬁnd the occurrence of 1,2, . . . , 9 as the ﬁrst signiﬁcant digit. Using Eq. (4.8.7), we have 1 ¼ 0:30103 PX1 (1) ¼ log 1 þ 1 The probability of 1 as the ﬁrst signiﬁcant digit is 30.1% and not a uniform distribution of 11.1% as one would expect! The probability of 2 is 17.6%. The probabilities of other signiﬁcant digits are tabulated in Table 4.8.1 and plotted in Fig. 4.8.1. They show that 9 as the ﬁrst signiﬁcant digit has the least probability of occurrence of 4.6%. Example 4.8.2 As a realistic example we have taken the values of physical constants (73 in number), in Chapter 17 of Ref. 18 and tabulated the signiﬁcant digits from 1 to 9 in

4.8 Logarithmic Distribution (Benford’s Law)

TABLE 4.8.1

57

Benford distribution

d

fD(d )

1 2 3 4 5 6 7 8 9

0.30103 0.176091 0.124939 0.09691 0.079181 0.066947 0.057992 0.051153 0.045757

FIGURE 4.8.1

Table 4.8.2. The physical constants have totally different units. The probabilities of the signiﬁcant digits are shown in Fig. 4.8.2 along with the graph of Benford’s law. The table and the graphs show that even for a small sample of 73, the probabilities follow the logarithmic distribution reasonably closely except for the number 3. In conclusion, in a mixed population of different scale factors, the probabilities of the digits 1 to 9 are logarithmically distributed and are not uniform—a surprising result!

TABLE 4.8.2 k

p(k)

1 2 3 4 5 6 7 8 9

0.342 0.205 0.055 0.096 0.096 0.055 0.027 0.041 0.082

58

Discrete Distributions

FIGURE 4.8.2

Second Signiﬁcant Digit

We shall now compute the occurrence of 0,1, . . . , B 2 1 as the second signiﬁcant digit. Note that 0 can occur as a subsequent signiﬁcant digit after the ﬁrst. We will only consider numbers of only ﬁrst-order magnitude since the pattern will be the same for any other order. With base B and the total number of digits being B 2 1, the occurrences of 2 as the second signiﬁcant digit is 1.2,2.2, . . . (B 2 1) . 2. Figure 4.8.3 shows in logarithmic scale the arrangement of numbers for base 10. Thus, the probability of 2 as the second signiﬁcant digit occurring after the ﬁrst digit 1 is given by an equation similar to Eq. (4.8.5):

13 ln ln (1:3) ln (1:2) 12 ¼ P{2 as second digit after first digit 1} ¼ ln (B) ln (B) ln (1)

(4:8:8)

If X2 is the random variable corresponding to the second digit, then the probability of 2 occurring after all the B 2 1 ﬁrst digits is given by 13 23 (B 1)3 ln þ ln þ þ ln 12 22 (B 1)2 PX2 (2) ¼ ln (B)

(4:8:9)

Summing Eq. (4.8.10) over all the ﬁrst digits d ¼ 0, . . . , B 2 1, we obtain the probability

FIGURE 4.8.3

4.8 Logarithmic Distribution (Benford’s Law)

of occurrence of the second signiﬁcant digit, PX2(d ), as B1 1 X 1 ln 1 þ PX2 (d) ¼ ; d ¼ 0,1, . . . ,B 1 ln (B) k¼1 kB þ d ! B 1 Y 1 1 1þ ln ¼ ln (B) kB þ d k¼1 It can be shown that " B1 B 1 Y X ln 1þ d¼0

k¼1

1 kB þ d

#

" ¼ ln

B 1 B 1 Y Y d¼0 k¼1

1 1þ kB þ d

59

(4:8:10)

# ¼ ln (B)

(4:8:11)

and hence, PX2(d ) is a probability function. We can extend the preceding analysis to obtain a general probability function for the (n þ 1)st signiﬁcant digit after the ﬁrst one digit n ¼ 0,1, . . . , B21 as shown below. The third digit after the ﬁrst one can be obtained from Eq. (4.8.10). The occurrence of 3 as the third signiﬁcant digit is 103, 113, . . . , 1(B 2 1)3; 203, 213, . . . , 2(B 2 1)3; . . . ; (B 2 1)03,(B 2 1)13, . . . , (B 2 1)(B 2 1)3. Thus, there are B(B 2 1) possible ways in which the digit 3 or any other digit d ¼ 0,1, . . . , B21 can occur as the third signiﬁcant number. An equation analogous to Eq. (4.8.9) can be written for the probability of 3 occurring as the third signiﬁcant digit: 2 1 1 BX 1 ln 1 þ (4:8:12) PX3 (3) ¼ ln (B) k¼B kB þ 3 Similar to Eq. (4.8.10), we can write the probability of any digit n occurring as the third signiﬁcant digit: 2 1 1 BX 1 ln 1 þ PX3 (d) ¼ ln (B) k¼B kB þ d ! 2 BY 1 1 1 ln ; 1þ ¼ ln (B) kB þ d k¼B

d ¼ 0, . . . , B 1

(4:8:13)

We can extend Eq. (4.8.13) to all (n þ 1) signiﬁcant digits after the ﬁrst with n ¼ 1,2, . . . , B 2 1: Bn1 1 X 1 PX(nþ1) (d) ¼ ln 1 þ ln (B) k¼Bn1 kB þ d n ¼ 1,2, . . . , B 1;

d ¼ 0, . . . , B 1

(4:8:14)

Example 4.8.3 Using Eq. (4.8.14) with the base B ¼ 10, we will ﬁnd the occurrence of the digits 0,1, . . . , 9 occurring as the ﬁrst, second, . . . , tenth signiﬁcant digit, noting that the probability of occurrence of 0 as the ﬁrst signiﬁcant digit is 0. Equation (4.8.14) becomes n 10 1 X 1 1 ln 1 þ PX(nþ1) (d) ¼ ln (10) k¼10n1 k 10 þ d n ¼ 0,1, . . . , 9;

d ¼ 0,1, . . . , 9

60

Discrete Distributions

TABLE 4.8.3

Benford distribution for 1 – 9 signiﬁcant digits

d

PX1(d )

PX2(d )

PX3(d)

PX4(d )

PX5(d )

PX6(d )

PX7(d )

0 1 2 3 4 5 6 7 8 9

0 0.30103 0.176091 0.124939 0.09691 0.079181 0.066947 0.057992 0.051153 0.045757

0.119679 0.11389 0.108821 0.10433 0.100308 0.096677 0.093375 0.090352 0.08757 0.084997

0.101784 0.101376 0.100972 0.100573 0.100178 0.099788 0.099401 0.099019 0.098641 0.098267

0.100176 0.100137 0.100098 0.100059 0.100019 0.09998 0.099941 0.099902 0.099863 0.099824

0.100018 0.100014 0.10001 0.100006 0.100002 0.099998 0.099994 0.09999 0.099986 0.099982

0.100002 0.100001 0.100001 0.100001 0.1 0.1 0.099999 0.099999 0.099999 0.099998

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

FIGURE 4.8.4

The values of PX (nþ1)(d) have been tabulated in Table 4.8.3 for values of d ¼ 0, . . . , 9 and n ¼ 1, . . . , 6. It is interesting to note that the ﬁrst signiﬁcant digit is considerably different from uniform distribution. As the level of signiﬁcance increases, the distribution tends to be more and more uniform and the signiﬁcant digit is practically uniform. In fact, the third signiﬁcant digit is almost a uniform distribution as shown in Fig. 4.8.4. Benford’s law is being used widely by the Internal Revenue Service for detecting fraudulent tax returns.

4.8.2

Inﬁnite Law

We will also present two versions of a logarithmic distribution where the number of outcomes are inﬁnite: Version 1. The ﬁrst version is deﬁned by p(k,u) ¼

uk ; k ln (1 u)

0 , u , 1, k 1

(4:8:15)

4.8 Logarithmic Distribution (Benford’s Law)

61

TABLE 4.8.4 k

P(u,k)

1 2 3 4 5 6 7 8 9 10

0.390865 0.175889 0.105534 0.071235 0.051289 0.038467 0.029675 0.023369 0.018695 0.015143

where u is the shape parameter and k are the inﬁnite outcomes. Here u is between 0 and 1, and k is greater than or equal to 1. We will show that this is a probability distribution by summing over all k: 1 X

p(k,u) ¼

k¼1

1 X

1 X uk 1 uk ln (1 u) ¼ ¼ ¼1 k ln (1 u) ln (1 u) k¼1 k ln (1 u) k¼1

(4:8:16)

In Eq. (4.8.15), as u tends to zero, p(k,u) tends to 1 as shown below: uk kuk1 (1 u) ¼ lim ¼1 u!0 k ln (1 u) u!0 k

lim p(k,u) ¼ lim

u!0

(4:8:17)

The probability mass function (pmf) given by Eq. (4.8.15) is tabulated for k ¼ 1, . . . ,10 and u ¼ 0.9 in Table 4.8.4 and shown in Fig. 4.8.5. This logarithmic distribution is used to model the number of items purchased by a customer in a supermarket in a given amount of time.

FIGURE 4.8.5

62

Discrete Distributions

Version 2. The second version of this logarithmic distribution is obtained by substituting l ¼ (1 2 u) in Eq. (4.8.15) to obtain p(k,l) ¼

(1 l)k : 0 , l , 1, k 1 k ln (l)

(4:8:18)

Here, as l tends to 1, p(k,l) tends to 1 as shown below: (1 l)k klk1 l ¼ lim ¼1 u!0 k ln (l) u!0 k

(4:8:19)

lim p(k,l) ¼ lim

u!0

B 4.9

SUMMARY OF DISCRETE DISTRIBUTIONS

Assumptions: Pfsuccessg ¼ p; Pffailureg ¼ q; p þ q ¼ 1. Distribution Type

Deﬁnition

Bernoulli

8 < p; b(p,q) ¼ q; : 0;

Binomial

8 < n k nk pq ; k b(k; n,p) ¼ : 0;

k¼1 k¼0 otherwise

Multinomial

p(k1 ,k2 , . . . , km ) ¼ m X

ki ¼ n;

i¼1

m X

k ¼ 0,1, . . . , n otherwise

n k1 ,k2 , . . . , km

pi ¼ 1

i¼1

qk1 ; 0;

k ¼ 1, 2, . . . k0

Geometric

g(k,p) ¼

Negative binomial: Pascal

8 < n 1 k nk pq ; k1 nb( p) ¼ : 0;

Hypergeometric

pk11 pk22 pkmm

8 K NK > > > k > > > > : n 0;

k ¼ k, k þ 1, . . . otherwise

k ¼ 0,1, . . . , n

9 > > > > =

otherwise

> > > > ;

(Continued)

4.9

Summary of Discrete Distributions

Distribution Type

Poisson

Logarithmic – Benford

63

Deﬁnition

p(k,l) ¼

8 < :

e

l

lk ; k!

k ¼ 0,1, . . .

9 =

; 0; otherwise 1 1 p(d) ¼ log 1 þ ; d ¼ 1, . . . , 10N 1 N d

Logarithmic 1—semiinﬁnite

p(k,u) ¼

uk ; k ln (1 u)

Logarithmic 2—semiinﬁnite

p(k,u) ¼

(1 u)k ; k ln (u)

0 , u , 1, k 1 0 , u , 1, k 1

CHAPTER

5

Random Variables

B 5.1

DEFINITION OF RANDOM VARIABLES

What is a random variable? An outlandish deﬁnition would be that it is neither random nor a variable! The real deﬁnition of a random variable is that it is a function X that assigns a rule of correspondence for every point j in the sample space S called the domain, a unique value X(j) on the real line R called the range. Let F be the ﬁeld associated with the sample space and FX be the ﬁeld associated with the real line. The random variable X induces a probability measure PX in R and hence X is a mapping of the probability space {S, F, P} to the probability space {R, FX , PX } as shown below: X: {S,F,P} ! {R, FX , PX }

(5:1:1)

Consider an event D , S [ F. The random variable X maps every point ji in the event D to points xi in the event Ix , called the image of D under X, where Ix , R [ FX . X can be a random variable only if the inverse image X 1 (Ix ) belongs to the ﬁeld F of subsets of S, and hence it must be an event. The mapping and the inverse mapping are shown in Fig. 5.1.1. With this restriction we should be able to ﬁnd the induced probability measure PX in terms of the probability measure P as follows: PX {Ix } ¼ P{X 1 (Ix )} ¼ P{D} ¼ P{j : X(j) [ Ix }

(5:1:2)

Since Ix belongs to the ﬁeld FX , on the real line it may consist of sets of the form {1 , h x}, {x , h , 1}, {x1 , h x2 }, {h ¼ x}. Out of these sets we deﬁne Ix by Ix ¼ {1 , h x}

(5:1:3)

All the other sets {x , h , 1}, {x1 , h x2 }, {h ¼ x} can be expressed in terms of the deﬁned Ix as follows: I x ¼ {x , h 1} Ix2 Ix1 ¼ {1 , h x2 } {1 , h x1 } ¼ {x1 , h x2 } lim {IxþDx Ix } ¼ {h ¼ x}

Dx!0þ

With the deﬁnition of Ix given by Eq. (5.1.3), we can write Eq. (5.1.2) as follows: PX {Ix } ¼ P{j : X(j) [ Ix } ¼ P{j : X(j) x}

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

64

(5:1:4)

5.1 Deﬁnition of Random Variables

65

FIGURE 5.1.1

We now deﬁne the quantity P{j : X(j) x} as the cumulative distribution function (cdf) FX (x), and rewrite Eq. (5.1.4) in abbreviated notation as P{j : X(j) [ Ix } ¼ P{X x} ¼ FX (x)

(5:1:5)

Equation (5.1.5) converts from a cumbersome set function P{X x} into a convenient point function FX(x). The value x scans the entire real line, that is, 21 , x , 1, and FX(x) must be deﬁned for all x on the real line. Note that {X x} in Eq. (5.1.5) is meaningless unless it is deﬁned carefully, because X is a function and a function without an argument cannot be less than a number! For example, we cannot have an exponential less than 2, unless we specify the argument of the exponential. The abbreviated notation in Eq. (5.1.5) is to be deﬁned as the probability of the set of all points j [ S such that the number X(j) is less than or equal to the number x. The random variable X should not confused with the value x that the random variable takes. The random variable X can take any value, for example, FX (y) ¼ P{X y}. Having deﬁned the cumulative distribution function (cdf), we now deﬁne the probability density function (pdf) as the derivative of the cdf FX(x) with respect to x: P(X x þ Dx) P(X x) Dx!1 Dx

fX ðxÞ ¼ lim

FX (x þ Dx) FX (x) d ¼ FX (x) Dx!1 Dx dx

¼ lim

(5:1:6)

The cdf FX(x) can be given in terms of the pdf fX(x) as follows: ðx FX (x) ¼

fX (j)dj

(5:1:7)

1

Whereas the distribution function FX(x) is a probability measure, the density function fX(x) is not a probability measure unless it is multiplied by the inﬁnitesimal Dx to yield fX (x)Dx ¼ P(x , X x þ Dx)

(5:1:8)

66

Random Variables

B 5.2 DETERMINATION OF DISTRIBUTION AND DENSITY FUNCTIONS Most of the difﬁculty in determining distribution functions for random variables can be avoided if the following step by step procedure is followed. Step 1. Since the random variable is a mapping from the sample space S to the real line, the mapping diagram is drawn connecting every point j in the sample space to the points on the real line R. Step 2. The regions for x on the entire real line from the mapped values are determined. The regions may correspond to the different subsets of S that map to the real line R. The set of points Ix will be equal to the right closed set ð1, x or, Ix ¼ {1 , h x}. Note that whereas Ix is a right closed subset, the region of deﬁnition of x is a subset of the real line R. Step 3. On the sample space S, the probability of the set of all points that map into Ix for every region of x in step 2 is determined. Step 4. The cdf FX (x) ¼ P{X x} is graphed for all the regions of x in step 2 on the entire real line. Step 5. The derivative of FX (x) determines the density function fX (x). Example 5.2.1 A die is tossed, and the random variable X is deﬁned by the amount won (þ) or lost (2) on the face of the die as shown in Table 5.2.1. We have to ﬁnd the cdf FX(x) and the pdf fX(x). We will follow the steps to ﬁnd the probability functions. Step 1. The mapping diagram (Fig. 5.2.1) is drawn with positive numbers indicating win and negative numbers indicating loss. Step 2. From the mapping diagram the regions of x are (a) x 29, (b) 29 , x 24, (c) 24 , x 5, (d) 5 , x 8, and (e) x . 8. The corresponding set Ix on the real line for all the 5 regions is Ix ¼ {1 , h x}. Step 3. We will ﬁnd the all the points in the sample space that map into Ix for every region of x: (a) x 29: Since no points in S map into Ix (Fig. 5.2.1a), we have FX(x) ¼ 0. (b) 29 , x 24: In this region (Fig. 5.2.1b), only one point f3g maps into Ix. Since P{3} ¼ 16 , FX (x) ¼ 16. (c) 24 , x 5: Here the points f3,2,6g from S map into Ix (Fig. 5.2.1c). Hence FX (x) ¼ 16 þ 16 þ 16 ¼ 12. (d) 5 , x 8: In this region (Fig. 5.2.1d) the points f3,2,6,5g in S map in to Ix. Hence FX (x) ¼ 12 þ 16 ¼ 23 TABLE 5.2.1 Pips on the Die 1,4 2,6 3 5

Win or Loss $ þ8 24 29 þ5

5.2 Determination of Distribution and Density Functions

67

FIGURE 5.2.1

(e) x . 8: In this region (Fig. 5.2.1e) all six points, that is, the entire sample space S, map into Ix. Hence FX (x) ¼ 1. In terms of unit step functions we can write the cdf FX(x) as follows: 1 1 1 1 FX (x) ¼ u(x þ 9) þ u(x þ 4) þ u(x 5) þ u(x 8) 6 3 6 3 Step 4. The cdf FX(x) can be graphed as shown in Fig. 5.2.2.

68

Random Variables

FIGURE 5.2.2

Step 5. We can now ﬁnd the density function fX(x) by differentiating the distribution function FX(x), bearing in mind that the differentiation of the unit step is the Dirac delta function. Performing the indicated differentiation, we obtain 1 1 1 1 fX (x) ¼ d(x þ 9) þ d(x þ 4) þ d(x 5) þ d(x 8) 6 3 6 3 Note that the sum of the strengths of the impulse function adds to 1. The pdf fX(x) is shown in Fig. 5.2.3. From Fig. 5.2.2 we can write the following probabilities: (a) P{X ¼ 4} ¼ FX (4þ) FX (4) ¼ 12 16 ¼ 13 1 1 (b) P{X 4} ¼ FX (4þ) FX (1) ¼ 2 0 ¼ 2

(c) P{5 , X 4} ¼ FX ( 4þ) FX (5) ¼ 12 16 ¼ 13 1 1 1 (d) P{4 , X 4} ¼ FX (4þ) FX (4) ¼ 2 6 ¼ 3

(e) P{5 X , 8} ¼ FX (8) FX (5þ)(Note closure on the left) ¼ 12=3 2=3 ¼ 0 The example above shows that the mapping is from a discrete sample space S to discrete points on the real line R.

FIGURE 5.2.3

5.2 Determination of Distribution and Density Functions

69

TABLE 5.2.2 Time Interval

Win or Loss ($)

6:00– 6:15 6:15– 6:45 6:45– 7:15 7:15– 8:00

þ2 24 24 þ6

Example 5.2.2 A telephone call occurs at random between 6:00 p.m. and 8:00 p.m. Depending on the interval of arrival, the win/loss table is shown in Table 5.2.2. We have to ﬁnd the cdf FX(x) and the pdf fX(x). Since the phone call is uniformly distributed in 120 min, the probability of a phone call in the region (0,x] is given by x/120. We follow the steps as outlined in Example 5.2.1: Step 1. The mapping diagram is shown in Fig. 5.2.4. Note that the mapping is from the real line R to the real line R. Step 2. From the mapping diagram the regions of x are (a) x 24, (b) 24 , x 2, (c) 2 , x 6, (d) x . 6. Step 3. (a) x 24: No points map FX(x) ¼ 0. 60 ¼ 12. (b) 24 , x 2: The interval that maps is 30 min; hence FX (x) ¼ 120 75 (c) 2 , x 6: The interval that maps is 45 min; hence FX (x) ¼ 120 ¼ 58. (d) x . 6: The entire sample space of 120 min maps; hence FX (x) ¼ 1. In terms of unit step functions, the cdf FX(x) is 1 1 3 FX (x) ¼ u(x þ 4) þ u(x 2) þ u(x 6) 2 8 8 Step 4. The cdf FX(x) is graphed in Fig. 5.2.5 Step 5. The pdf fX(x) is obtained by differentiating FX(x), and the result is 1 1 3 fX (x) ¼ d(x þ 4) þ d(x 2) þ d(x 6) 2 8 8 fX(x) is graphed in Fig. 5.2.6.

FIGURE 5.2.4

70

Random Variables

FIGURE 5.2.5

FIGURE 5.2.6

In the preceding example, the sample space is continuous whereas the range space is discrete. Since the random variable takes discrete values, we have a discrete random variable. Example 5.2.3 A telephone call occurs at random between 6:00 p.m. and 8:00 p.m. We want to ﬁnd the cdf FX(x) and the pdf fX(x). Here we have a continuous mapping from the real line to the real line and X is a continuous random variable. Step 1. See the mapping diagram in Fig. 5.2.7. Step 2. From the mapping diagram the regions of x are (a) x 0, (b) 0 , x 120, (c) x . 120, and Ix ¼ {1 , h x). Step 3 (a) x 0: No points in S map into Ix. Hence FX(x) ¼ 0. (b) 0 , x 120: The call is uniformly distributed in the interval (0,120], and since (0,x] is an interval mapping into Ix, we have FX(x) ¼ x/120. (c) x . 120: Here the entire sample space maps into Ix, and hence FX (x) ¼ 1. Step 4. The cdf FX (x) is graphed in Fig. 5.2.8. Step 5. The density function fX(x) is obtained by taking the derivative of FX (x), or 1 , 0 x 120. This is sketched in Fig. 5.2.9. fX (x) ¼ 120 We found the cdf and pdf for discrete random variables in Examples 5.2.1 and 5.2.2 and for continuous random variables in Example 5.2.3. We will now give an example of a mixed random variable that has both continuous and discrete components.

5.2 Determination of Distribution and Density Functions

71

FIGURE 5.2.7

FIGURE 5.2.8

FIGURE 5.2.9

Example 5.2.4 A telephone call occurs at random between 6:00 p.m. and 8:00 p.m. However, between 6:20 and to 6:40 p.m. calls accumulate and are released exactly at 6:40 p.m. with a similar situation between 7:00 and 7:20 p.m. and are released at exactly 7:20 p.m. We want to ﬁnd the cdf FX(x) and the pdf fX(x). Step 1. The mapping diagram shown in Fig. 5.2.10 is similar to Fig. 5.2.7 except that there are deadspots between 6:20 and 6:40 p.m. and 7:00 and 7:20 p.m. Step 2. From the mapping diagram the regions of x are (a) x 0, (b) 0 , x 20, (c) 20 , x , 40, (d) x ¼ 40, (e) 40 , x 60, (f) 60 , x , 80, (g) x ¼ 80, (h) 80 , x 120, (i) x . 120. Step 3. (a) x 0: No points in S map into Ix; hence FX(x) ¼ 0.

72

Random Variables

FIGURE 5.2.10

(b) 0 , x 20: The call is uniformly distributed in the interval (0,120]; hence FX (x) ¼ x/120. 20 ¼ 16. (c) 20 , x , 40: There is a deadspot in this region; hence FX (x) ¼ FX (20) ¼ 120 (d) x ¼ 40: There is jump at this point equal to P{X ¼ 40} ¼ 16; hence FX (40) ¼ 16 þ 16 ¼ 13. (e) 40 , x 60: Here also (0,x] is an interval in Ix and we have FX (x) ¼ x=120 and FX (40) , 13. (f) 60 , x , 80: Here again we have a deadspot; and hence FX (x) ¼ FX (60) ¼ 60 1 120 ¼ 2. (g) x ¼ 80: There is again a jump at this point equal to P{X ¼ 80} ¼ 16; hence FX (80) ¼ 12 þ 16 ¼ 23. (h) 80 , x 120: Here (0,x] is an interval in Ix and we have FX (x) ¼ x=120 and FX (80) ¼ 23. (i) x . 120: The entire sample space maps into Ix; hence FX(x) ¼ 1 Step 4. The cdf FX(x) is graphed in Fig. 5.2.11.

FIGURE 5.2.11

5.3 Properties of Distribution and Density Functions

73

FIGURE 5.2.12

Step 5. The pdf fX(x) is the derivative of FX(x) and is graphed in Fig. 5.2.12. From the graph of the cdf FX(x) of Fig. 5.2.11, we can obtain the following probabilities: 1. P{X ¼ 40} ¼ P{X ¼ 80} ¼ 16. 2. P{X , 80} ¼ 12. 3. P{X 80} ¼ 23 : 4. P{40 X 80} ¼ 13 : 5. P{X ¼ 100} ¼ 0: 5 5 6. P{50 , X 100} ¼ 56 12 ¼ 12 :

B 5.3 PROPERTIES OF DISTRIBUTION AND DENSITY FUNCTIONS Cumulative Distribution Function FX(x)

1. From the three examples given in Section 5.2 the cdf of a discrete random variable is a staircase function with jumps. 2. FX(x) 0, that is, nonnegative. 3. FX(21) ¼ 0, FX(1) ¼ 1. 4. If x1 , x2, then FX(x1) FX(x2), that is, FX(x) is a nondecreasing function (monotone increasing). For a continuous random variable, if x1 , x2, then FX(x1) , FX(x2), that is, FX(x) is an increasing function (strict monotone increasing). 5. FX(x) is right-continuous: FX (x) ¼ lim FX (x þ 1) 1!0

for 1 . 0

From this deﬁnition, the value that FX (x) takes at the point of a jump, is to the right of the jump, and if there is an event A in the interval {x1 , X x2 }, then the interval is closed on the righthand side so that we can write P{x1 , X x2 } ¼ P{X x2 } P{X x1 } ¼ FX (x2 ) FX (x1 ). This ensures that the point x1 is not counted twice. Figure 5.3.1 will clarify this point. ðx FX (x) ¼ fX (j)dj ¼ P{X x} (5:3:1) 6. 1

74

Random Variables

FIGURE 5.3.1

7. At the point x ¼ a of the jump ð aþ FX (x) ¼ fX (j)dj ¼ FX (aþ) FX (a) ¼ P{X ¼ a}

(5:3:2)

a

Probability Density Function fX(x)

1. fX (x) 0, nonnegative. 2. fX (x) consists of the sum of Dirac delta functions for a discrete random variable. 3. fX (x) is not a probability measure, but fX (x)Dx is a probability measure as shown: fX (x)Dx P{x , X x þ Dx}

4.

(5:3:3)

P{x , X x þ Dx} d ¼ FX (x) Dx dx

ð5:3:4Þ

fX (j)dj FX (b) FX (a) ¼ P{a , X b}

(5:3:5)

fX (x) lim

Dx!0

ðb 5. a

Probability Mass Functions (pmf’s)

We have already come across probability mass functions in connection with discrete random variables. We will now formalize this concept. The probability of a random variable equal to a number is called the probability mass function (pmf). In Eq. (5.3.2), P{X ¼ a} is the pmf. 1. P{X ¼ a} is the probability of the jump at the point x ¼ a in the cdf FX (x). Or, P{X ¼ a} ¼ FX (a) FX (a). 2. The pmf P{X ¼ a} is nonzero for discrete random variables at x ¼ a, where they jump. 3. P{X ¼ a} ¼ 0 for continuous random variables, since at any point of continuity a, FX (a) FX (a) ¼ 0. 4. The density function fX(x) for discrete random variables involves summation of the pmf’s with Dirac delta functions multiplying the pmf at every point in the range. For example, the pdf of the binomial distribution given by the

5.4 Distribution Functions from Density Functions

n

pk qnk is n X n k nk fX (x) ¼ p q d(x k) k k¼0

probability mass function

75

k

(5:3:6)

The pdf of the Poisson distribution given by the pmf el (lk =k!) is fX (x) ¼

1 X

el

k¼0

lk d(x k) k!

(5:3:7)

B 5.4 DISTRIBUTION FUNCTIONS FROM DENSITY FUNCTIONS Given any density function (pdf), the distribution function (cdf) can be found by integrating the pdf over all x on the real line as given in Eq. (5.3.1). Care must be taken in integrating over various regions along the x axis. Example 5.4.1 expressed as

We will start with a simple example. The density function fX (x) is

fX (x) ¼

8 k > > > :1, 2

2,x3

and is shown in Fig. 5.4.1. We have to ﬁrst ﬁnd k and then the distribution function FX (x) for all x. Integrating fX (x) over the range of deﬁnition of x, we have ð3 ð2 ð3 k 1 k 1 fX (x) ¼ dx þ dx ¼ þ ¼ 1, [ k ¼ 1 4 2 2 2 0 0 2 1 1 fX (x) ¼ u(x) þ u(x 2) 4 4

FIGURE 5.4.1

76

Random Variables

FIGURE 5.4.2

We note that there are four natural ranges for x given by (1) x 0, (2) 0 , x 2, (3) 2 , x 3, (4) x . 0. We will evaluate FX(x) in all these ranges: 1. x 0: Since fX (x) is zero, FX (x) ¼ 0. 2. 0 , x 2: Using Eq. (5.3.5), we can write ðx FX (x) ¼

1 x dj ¼ 4 04

3. 2 , x 3: Again using Eq. (5.3.5) and taking care to carry over FX (2), we have ð2 FX (x) ¼

1 dj þ 04

ðx

1 1 1 1 dj ¼ þ (x 2) ¼ (x 1) 2 2 2 2 2

4. x . 3: Since fX(x) is zero in this range, FX(x) ¼ FX(3) ¼ 1. FX(x) is graphed in Fig. 5.4.2.

FIGURE 5.4.3

5.4 Distribution Functions from Density Functions

77

Example 5.4.2 This is a more involved example that characterizes a mixed random variable. The density function is given by fX (x) ¼ K{ex ½u(x 1) u(x 5) þ ex ½u(x 1) u(x 5)g 1 þ ½d(x þ 2) þ d(x) þ d(x 2) 8 (see Fig. 5.4.3). We will ﬁnd K and graph the distribution function FX(x) for all x. We integrate fX(x) over the range (25,5] to evaluate the constant K. Thus ð 5 K

ex dx þ

5

1 1 ex dx þ þ ¼ 1 8 8 1

ð5

2K½e1 e5 þ

3 ¼1 8

[K¼

5 e5 )

16(e1

The ranges for x for integrating fX(x) are given by (1) x 25, (2) 25 , x 22, (3) 22 , x 21, (4) 21 , x 0, (5) 0 , x 1, (6) 1 , x 2, (7) 2 , x 5, (8) x . 5. We shall now evaluate FX(x) in each of these ranges: 1. x 25: Since fX(x) ¼ 0, we have FX(x) ¼ 0 2. 5 , x 2:

3. 2 , x 1:

FX (x) ¼

FX (x) ¼

4. 1 , x 0: FX (x) ¼

5. 0 , x 1:

FX (x) ¼

5 16(e1 e5 Þ

ðx

ej dj ¼

5

5(ex e5 ) 1 þ 1 5 16(e e Þ 8

5(e1 e5 ) 1 7 þ ¼ 1 5 16(e e Þ 8 16

7 1 9 þ ¼ 16 8 16

FIGURE 5.4.4

5(ex e5 ) 16(e1 e5 Þ

78

Random Variables

6. 1 , x 2:

FX (x) ¼

7. 2 , x 5: FX (x) ¼

8. x . 5: FX (x) ¼

9 þ 16

ðx

ej dj ¼

1

9 5(e1 ex ) þ 16 16(e1 e5 )

9 5(e1 ex ) 1 þ þ 16 16(e1 e5 ) 8

9 5(e1 e5 ) 1 þ ¼1 þ 16 16(e1 e5 ) 8

The graph of FX (x) is shown in Fig. 5.4.4. In the next chapter we will discuss the various continuous distributions.

CHAPTER

6

Continuous Random Variables and Basic Distributions

B 6.1

INTRODUCTION

In the last chapter we deﬁned a continuous random variable as the mapping from a continuous sample space S to continuous points on the real line R. We also discussed some of the properties of continuous pdf fX(x) and the continuous cdf, FX(x). In this chapter we will discuss three of the basic continuous distributions: uniform, exponential, and Gaussian. Other distributions will be discussed in the next chapter.

B 6.2

UNIFORM DISTRIBUTION

The basic continuous distributions are uniform, exponential, and Gaussian. We have already seen (in Example 5.2.3) the distribution of a telephone call occurring at random in a given interval. The pdf fX(x) of a random variable uniformly distributed in the interval (a, b], (notation X U(a, b]), is given by 8 0 xa > > > < 1 (6:2:1) fX (x) ¼ a,xb > b a > > : 0 x.b and the corresponding distribution function FX(x) is 8 0 xa > > > > > : 1 x.b

(6:2:2)

Such distributions occur in transmission of sinusoids where the phase angle will be uniformly distributed in (0,2p). The density function is shown in Fig. 6.2.1, and the distribution function is shown in Fig. 6.2.2. Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

79

80

Continuous Random Variables and Basic Distributions

FIGURE 6.2.1

FIGURE 6.2.2

Example 6.2.1 A sinusoidal signal with random phase angle Q, uniformly distributed between (0,2p] is sent along a communications channel. We want to ﬁnd the probability that the phase angle is between (p/4, 3p/4]. Since it is uniformly distributed, the pdf is 1/2p in (0,2p], and hence the Pfp/4, 3p/ 4g is given by p 3p 1 3p p 1 ,u ¼ ¼ P 4 4 2p 4 4

B 6.3

EXPONENTIAL DISTRIBUTION

Exponential distributions ﬁnd wide applicability in queuing theory, reliability theory, and communication theory. It is the continuous analog of the geometric distribution discussed in Chapter 4. The pdf of an exponential distribution is given by 0 x0 (6:3:1) fX (x) ¼ lelx x . 0 where the parameter l is the average rate at which events occur. The corresponding cdf FX(x) is given by 0 x0 (6:3:2) FX (x) ¼ 1 elx x . 0 The pdf is shown in Fig. 6.3.1, and the cdf is shown in Fig. 6.3.2 for values of l ¼ 0.5,1,2,3,4. The exponential distribution belongs to a class of distributions possessing memoryless (Markov) properties that we will presently show. A random variable X is called memoryless if it satisﬁes the conditional probability P{X . x þ x0 j X . x0 } ¼ P{X . x}

(6:3:3)

6.3 Exponential Distribution

81

FIGURE 6.3.1

FIGURE 6.3.2

The condition given by Eq. (6.3.3) is equivalent to P{X . x þ x0 , X . x0 } ¼ P{X . x} P{X . x0 }

(6:3:4)

or P{X . x þ x0 } ¼ P{X . x}P{X . x0 } To demonstrate the memoryless property, let X be a random variable representing the time to failure of a computer with cdf given by Eq. (6.3.2). We are given that the computer has lasted x0 hours. We want to ﬁnd the conditional probability that the computer will fail at time x beyond x0 given that it has lasted for x0 hours. We proceed as follows: P{X x þ x0 j X . xo } ¼ FX (x þ x0 j X . x0 ) ¼

P{X x þ x0 , X . x0 } P{x0 , X x þ x0 } ¼ P{X . x0 } 1 FX (x0 )

¼

FX (x þ x0 ) FX (x0 ) 1 el(xþx0 ) 1 þ elx0 ¼ 1 1 þ elx0 1 FX (x0 )

¼

elx0 (1 elx ) ¼ 1 elx , x . x0 elx0

(6:3:5)

82

Continuous Random Variables and Basic Distributions

The conditional distribution FX(x þ x0 j X . x0) is the same as FX(x)! The computer “forgets” that it has been operating for x0 hours and has no memory of previous hours of operation. Hence the exponential distribution is called a memoryless distribution.

Poisson Arrival Process

The number k of events occurring in an interval (0,t] is Poisson-distributed with p(k,l) ¼ e 2lt[(lt)k/k!], and the waiting times for the occurrence from one event to the next are independent and are exponentially distributed as e 2lt. Example 6.3.1 A computer company leases its computers. The lifetime of one of the 1 computers is exponentially distributed with l ¼ 50;000 h. If a person leases the computer for 10,000 h, we have to ﬁnd the probability that the computer company will have to replace the computer before the lease runs out. By the memoryless property derived in Eq. (6.3.5), it is immaterial how long it has been leased before, and hence we have P{computer fails in t 10,000} ¼ 1 e10000l ¼ 1 e(1=5) ¼ 0:1813 and the probability that the computer will not be replaced before the lease runs out is P{lifetime . 10,000} ¼ 1 0:1813 ¼ 0:8187 However, if the failure time FX(x) is not exponentially distributed, then the probability of replacement has to be computed, taking into account the prior use of x0 hours of the computer. This is given by the conditional probability P{lifetime . x0 þ 10,000 j lifetime . 10000} ¼

1 FX (x0 þ 1000) 1 FX (x0 )

which means that we have to know x0, the number of hours the computer was operational before the lease. Hazard Rate

Given that an item has not failed upto time t, the conditional probability that it will fail in the next small interval of time Dt is determined as follows P{t , X t þ Dt j X . t} ¼ ¼

P{t , X t þ Dt, X . t} P{X . t} P{t , X t þ Dt} fX (t)Dt ¼ b(t)Dt ¼ P{X . t} 1 FX (t)

(6:3:6)

where FX(t) ¼ PfX tg represents the failure distribution and the quantity b(t) ¼ fx(t)/ [1 2 FX(t)] is called the instantaneous hazard rate. It is not a probability function as determined by the following analysis: ðt ðt ðt fX (j) dfX (x) dj ¼ ¼ ln½1 FX (t) b(j)dj ¼ (j) 1 F 1 FX (j) X 0 0 0

6.3 Exponential Distribution

83

Ð Since t0b(j)dj ! 1 as t ! 1, we conclude that b(t) is not a probability function. However, solving for FX(t), we obtain ðt ln½1 FX (t) ¼ b(j)dj (6:3:7) 0

or ðt 1 FX (t) ¼ exp b(j)dj ¼ 0 as t ! 1 0

From Eq. (6.3.7) we can obtain the cdf FX(t), given by ðt FX (t) ¼ 1 exp b(j)dj

(6:3:8)

0

Thus, b(t) uniquely determines the failure probability distribution function FX(t). While b(t) is not a probability density function, it is to be noted that the function b(x), given by 8 < fX (x) , x . t (6:3:9) b(x) ¼ 1 FX (t) : 0, xt is the conditional failure density conditioned on x . t and b(t) is the value of b(x) at x ¼ t. Example 6.3.2 Physicians are of the opinion that the death rate of a person E exercising regularly is half that of a person S who has sedentary habits. We want to compare the probabilities of survival of E and S given that the death rate of an exercising person E is bE(t) and the rate for a sedentary person S is bS(t) with bE(t) ¼ 12bS(t). Since the rates are conditional failure rates, we will assume that both persons have lived upto the age T1, and we want to compare the probabilities of their reaching the age T2 with T2 . T1. Thus we need to ﬁnd P{Xi . T2 j Xi . T1 } ¼

P{Xi . T2 } 1 FXi (T2 ) ¼ P{Xi . T1 } 1 FXi (T1 )

where Xi ¼ E or S. Substituting from Eq. (6.3.8), we have ÐT ð T2 1 FXi (T2 ) exp 0 2 bXi (j)dj ¼ ¼ exp bXi (j)dj , Ð 1 FXi (T1 ) exp T1 bX (j)dj T1 0

i ¼ E or S

i

Simplifying, we obtain ð 1 T2 P{E . T2 j E . T1 } ¼ exp bS (j)dj 2 T1 ð T2 P{S . T2 j S . T1 } ¼ exp bS (j)dj T1

Thus, the probability of an exercising person who has lived upto T1 years living to T2 years is the square root of the probability of the sedentary person. If the death rate of a sedentary person, bS(t) ¼ 0.001t, T1 ¼ 60 years and T2 ¼ 70 years, then the following probabilities

84

Continuous Random Variables and Basic Distributions

can be evaluated: ð 70 P{S . 70 j S . 60} ¼ exp 0:001j dj ¼ 0:522

60

1 P{E . 70 j E . 60} ¼ exp 2

ð 70

0:001j dj ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:522 ¼ 0:7225

60

The probability of a 60 year old sedentary person reaching 70 years is 0.522, whereas the same person exercising regularly will reach 70 years with a probability of 0.7225.

B 6.4

NORMAL OR GAUSSIAN DISTRIBUTION

The most important distribution in probability theory is the Gaussian distribution. Under some general conditions, a random variable X consisting of a number of components, each with a general distribution tends to a normal distribution as the number becomes very large. The probability density fX(x) of a Gaussian distribution is given by 2 1 fX (x) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ eð1=2Þ½(xm)=s 2ps

(6:4:1)

where m is called the mean and s2 is called the variance. These quantities will be deﬁned later. The cumulative distribution function FX(x) is given by ðx 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ e(1=2)½(jm)=s dj FX (x) ¼ (6:4:2) 2ps 1 The Gaussian density function fX(x) given by Eq. (6.4.1) is shown in Fig. 6.4.1 for m ¼ 3 and s2 ¼ 4. It also shows that the area under the Gaussian pdf between (m 2 s, m þ s] is 68.3%, between (m 2 2s, m þ 2s] is 95.5%, and between (m 2 3s, m þ 3s] is 99.7%. Thus, almost the entire pdf (99.7%) is between the m + 3s points of the Gaussian curve. The Gaussian distribution function, FX(x) is shown in Fig. 6.4.2 along with (m 2 s, m þ s], (m 2 2s, m þ 2s], and (m – 3s, m þ 3s] points.

FIGURE 6.4.1

6.4 Normal or Gaussian Distribution

85

FIGURE 6.4.2

A standard Gaussian is deﬁned as the random variable with m ¼ 0 and s2 ¼ 1, given by 1 2 fX (x) ¼ pﬃﬃﬃﬃﬃﬃ e(1=2)x 2p

(6:4:3)

and the corresponding cumulative distribution function is denoted by FX(x), given by ðx FX (x) ¼

2 1 pﬃﬃﬃﬃﬃﬃ e(1=2)j dj 2p 1

(6:4:4)

There is no closed-form solution for Eq. (6.4.4). Hence, FX(x) is tabulated for values of x between 25 and 5 in Appendix B. The standard Gaussian pdf fX(x) is shown in Fig. 6.4.3 with the 68.3% points between (21,1], 95.5% points between (22,2], and the 99.7% points between (23,3].

FIGURE 6.4.3

86

Continuous Random Variables and Basic Distributions

Given a random variable X with any m and s2, the random variable Z¼

X mX sX

is a standard Gaussian. Hence, the distribution function FX(x) can be given in terms of the standard Gaussian FZ (x) as follows: FX (x0 ) ¼ P{X x0 ¼} ¼ P

X mX x0 mX x0 mX ¼ FX sX sX sX

(6:4:5)

We shall now prove that the fX(x) is a probability density function by showing that if ð1 I¼

1 2 pﬃﬃﬃﬃﬃﬃ e(1=2)x dx 2p 1

then I ¼ 1. Direct evaluation of the integral is difﬁcult. Hence we square I and write ð1 1 (1=2)x2 1 2 pﬃﬃﬃﬃﬃﬃ e(1=2)y dy e dx 2p 2p 1 1 ð ð 1 1 1 ð1=2Þ(x2 þy2 ) e dx dy ¼ 2p 1 1

I2 ¼

ð1

(6:4:6)

Substituting x ¼ r cos(u) and y ¼ r sin(u) in Eq. (6.4.5) we obtain, I2 ¼

1 2p

ð 2p ð 1 0

2

re(1=2)r dr du ¼ 1

(6:4:7)

0

and the result is proven. Example 6.4.1 The number of malfunctioning computers is Gaussian-distributed with m ¼ 4 and s ¼ 3. We want to ﬁnd the probability that the number of bad computers is between 3 and 5. Denoting the malfunctioning computers by the random variable X, we have to ﬁnd Pf3 , X 5g. Using Eq. (6.4.5) and referring to the Gaussian tables, we have 34 X4 54 1 1 , P{3 , X 5} ¼ P ¼P ,Z 3 3 3 3 3 1 1 FX ¼ 0:2611 ¼ FX 3 3 Example 6.4.2 We shall study what grading on the curve in examinations means. If the average of the class in a particular examination is m, then one method of grading is to assign A to those receiving over m þ 2s, AB to those receiving between m þ s and m þ 2s, B to those receiving between m and m þ s, C to those receiving between m 2 s and m, D to those receiving between m 2 s and m 2 2s, and F to those receiving less than m 2 2s. We want to calculate the percentage of students receiving the grades A, AB, B, C, D, and F.

6.4 Normal or Gaussian Distribution

We can calculate these probabilities from Eq. (6.4.5) as follows: Xm A: P{X . m þ s} ¼ P . 2 ¼ 1 F(2) ¼ 0:0228 ¼ 2:28% s Xm AB: P{m þ s , X m þ 2s} ¼ P 1 , 2 ¼ F(2) F(1) s ¼ 0:1359 ¼ 13:59% B:

Xm P{m , X m þ s} ¼ P 0 , 1 ¼ F(1) F(0) s ¼ 0:3413 ¼ 34:13%

C:

Xm P{m s , X m} ¼ P 1 , 0 ¼ F(0) F(1) s ¼ 0:3413 ¼ 34:13%

D:

F:

Xm P{m 2s , X m s} ¼ P 2 , 1 ¼ F(1) F(2) s ¼ 0:1359 ¼ 13:59% Xm P{X m 2s} ¼ P 2 ¼ F(2) ¼ 0:0228 ¼ 2:2:8% s

These grades are shown in Fig. 6.4.4. Properties of Standard Gaussian

Density and Distribution 1. The pdf fX(x) is an even function.

pﬃﬃﬃﬃﬃﬃ 2. fX(x) attains its maximum value of 1= 2p at x ¼ 0. Ð 3. The cdf FX(x) ¼ x21fX(j)dj.

FIGURE 6.4.4

87

88

Continuous Random Variables and Basic Distributions

4. FX (0) ¼ 1 : 2 5. The error function erf(x) is deﬁned as 2 erf(x) ¼ pﬃﬃﬃﬃ p

ðx

2

ej dj

(6:4:8)

0

6. The complement of the error function erfc(x) is deﬁned as ð 2 1 j2 e dj ¼ 1 erf(x) erfc(x) ¼ pﬃﬃﬃﬃ p x

(6:4:9)

7. The tails of the distribution Q(x) is used in describing the probability of error in transmission of communication signals. It is deﬁned as ð 1 1 (j2 =2) e dj (6:4:10) Q(x) ¼ pﬃﬃﬃﬃﬃﬃ 2p x We will discuss this function in greater detail in the next section. 8. Q(x) in terms of error function complement erfc(x) is 1 x Q(x) ¼ erfc pﬃﬃﬃ 2 2

(6:4:11)

9. The general distribution function FX(x) in terms of erf(x) and erfc(x) is pﬃﬃﬃﬃ ðx ð 1 1 2 ðxmÞ= 2s j2 (1=2)½(jm)=s2 FX (x) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ e dj ¼ pﬃﬃﬃﬃ e dj 2 p 1 2ps 1 1 1 xm 1 xm (6:4:12) ¼ þ erf pﬃﬃﬃﬃﬃﬃ ¼ 1 erfc pﬃﬃﬃﬃﬃﬃ 2 2 2 2s 2s 1 10. (6:4:13) FX (m) ¼ 2

Gaussian Tails Q(x)

As mentioned in property 7 [Eq. (6.4.10)] Q(x) is used to express the bit error probability in transmission of communication signals. Since Q(x) does not have a closed-form solution, bounds can be established. The distribution Q(x) can expanded into an asymptotic series [1] given by 2 ex 1 1 1:3 n 1:3 (2n 1) (6:4:14) Q(x) ¼ pﬃﬃﬃﬃﬃﬃ 1 2 þ 4 þ þ (1) x x x2n 2p x and for large x, Q(x) can be approximated by 2

ex 1 Q(x) pﬃﬃﬃﬃﬃﬃ 2p x

(6:4:15)

For small x, Q(x) diverges, and hence this is not a very good bound. A better bound is 2

ex 1 Q(x) QA(x) ¼ pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2p 1 þ x2

(6:4:16)

6.4 Normal or Gaussian Distribution

89

Even this bound, although better than Eq. (6.4.15), is not very good for small values of x. However, a tighter bound has been established by Bo¨rjesson and Sundberg [8] by ﬁrst integrating Q(x) by parts as shown below: ð pﬃﬃﬃﬃﬃﬃ ð 1 1 (j2 =2) 1 1 (j2 =2) p ﬃﬃﬃﬃﬃﬃ je e dj ¼ 1 2p dj Q(x) ¼ 2p x x j ð1 1 1 1 (j2 =2) 2 e dj ¼ pﬃﬃﬃﬃﬃﬃ e(x =2) 2 2p x x j and the expression for Q(x) can be manipulated to obtain Q(x) ¼

1

1 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ e(x =2) 2 (1 a)x þ a x þ b 2p

(6:4:17)

Substituting the values a ¼ 1 and b ¼ 1 in Eq. (6.4.17), we obtain Eq. (6.4.16). Better values for a and b can be obtained by minimizing the absolute relative error function 1(x), given by

Q0 (x) Q(x)

(6:4:18) 1(x) ¼

Q0 (x) and constraining 1(0) ¼ 0. In Eq. (6.4.18) Q0(x) is the true value given by Eq. (6.4.10). As a consequence, the upper bound QU(x) is obtained for values of a ¼ 0.33919 and b ¼ 5.533425, and the lower bound QL(x) is obtained for values of a ¼ 1/p and b ¼ 2p. Thus the upper bound is QU(x) ¼

1 1 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ e(x =2) 2 0:66081x þ 0:33919 x þ 5:33425 2p

(6:4:19)

and the lower bound is, QL(x) ¼

1 1 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ eðx =2Þ (1 1=p) þ 1=p x2 þ 2p 2p

(6:4:20)

The functions Q(x), QU(x), QL(x), and QA(x) are shown in Fig. 6.4.5 and tabulated in Table 6.4.1 showing the upper and lower bounds are for all x . 0. The bound QA(x) is not good for low values of x, whereas QU(x) and QL(x) practically lie on Q(x).

FIGURE 6.4.5

90

Continuous Random Variables and Basic Distributions

TABLE 6.4.1 x 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Lower Bound QL(x)

True Value Q(x)

Upper Bound QU(x)

Approx. Value QA(x)

0.5 0.30496416 0.15704991 0.06633866 0.0226461 0.00619131 0.00134729 0.00023233 0.00003164 0.0000034 0.00000029

0.5 0.30853754 0.15865525 0.0668072 0.02275013 0.00620967 0.0013499 0.00023263 0.00003167 0.0000034 0.00000029

0.5 0.3071816 0.15837866 0.06684731 0.02279147 0.00622374 0.00135302 0.00023314 0.00003173 0.0000034 0.00000029

0.39894228 0.3148968 0.17109914 0.07184344 0.02414549 0.00650985 0.00140147 0.00023974 0.00003246 0.00000347 0.00000029

The absolute percentage errors for the upper QU(x) and lower bounds QL(x) are shown in Fig. 6.4.6. Table 6.4.2 shows in addition to the upper- and lower-bound errors the absolute percentage error for the approximate value QA(x). Since these errors are large, they are not shown in Fig. 6.4.6. The errors for the approximate value are high (20.2%). The upper-bound errors are better for smaller values of x, whereas the lower-bound is better for higher values of x. Even then, these bounds clearly illustrate that the approximation for all x is excellent with the maximum error of about 1.16% occurring at low values of x. This is much better than the usual approximation QA(x), where the maximum error is 20.2% for low values. For high values of x, QU(x), QL(x), and QA(x) provide reasonably good approximation.

Properties of Q Function

1. The limit as x ! 1 is zero: lim Q(x) ¼ 0

x!1

FIGURE 6.4.6

(6:4:21)

6.4 Normal or Gaussian Distribution

TABLE 6.4.2 x 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

91

Error percentages Lower Bound

Upper Bound

Approximate Value

0 1.15816594 1.01184414 0.70133395 0.45727327 0.29559943 0.19344425 0.12922798 0.08836646 0.06185728 0.04427826

0.00000176 0.43947331 0.17433835 0.06003046 0.1816945 0.226622 0.23152986 0.21848937 0.19884525 0.1779578 0.15815582

20.21154392 2.06109849 7.8433496 7.53846047 6.13342513 4.83410996 3.82069533 3.05864395 2.48602699 2.05102577 1.71571131

2. The limit as x ! 21 is 1: lim Q(x) ¼ 1

(6:4:22)

x!1

3. The value of Q(x) at x ¼ 0 is half: Q(0) ¼

1 2

(6:4:23)

4. Q(x) is even: Q(x) ¼ Q(x) 5. Q(x) can be expressed in terms of error function as 1 x 1 x Q(x) ¼ 1 erf pﬃﬃﬃ ¼ erfc pﬃﬃﬃ , 2 2 2 2

(6:4:24)

x0

(6:4:25)

6. If F(x) is a standard Gaussian, then Q(x) ¼ 1 F(x)

(6:4:26)

7. For any Gaussian distributed random variable X with mean m and variance s2 a m (6:4:27) P{X . a} ¼ 1 FX (a) ¼ Q s

Gaussian Approximation to Binomial Distribution

In Example 4.7.3 we saw that the binomial distribution can be approximated by a Poisson distribution under the condition that n is large compared to k and p is small enough that the product np ¼ l is of moderate magnitude. On the other hand, if npq is quite large, the Poisson approximation does not fare very well. In this case we use the Gaussian approximation, which is called the DeMoivre –Laplace limit theorem, which states that, if Sn is the total number of successes in n Bernoulli trials, each with probability of success p and failure q with ( p þ q ¼ 1) with pmf b(k;n,p), then, as n ! 1, the pﬃﬃﬃﬃﬃﬃﬃﬃ standardized random variable Z ¼ (Sn 2 np)/ npq converges to a standard normal distribution function where m ¼ np is the mean and s2 ¼ npq is the variance. Or, if

92

Continuous Random Variables and Basic Distributions

n1 , n2, then Sn np lim Z ¼ P n1 , pﬃﬃﬃﬃﬃﬃﬃﬃ n2 ! F(n2 ) F(n1 ) n!1 npq

(6:4:28)

and the probability mass function b(k;n,p) converges to the density function fZ (z). Or lim

2 n! 1 pk qnk ! pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e(1=2)½(xnp)=npq k)! 2pnpq

n!1 k!(n

(6:4:29)

The Demoivre –Laplace limit theorem is a special case of the central limit theorem presented in a later chapter.

Example 6.4.3 (Gaussian and Poisson Approximation) We will ﬁnd the Gaussian approximation for the binomial distribution given in Example 4.7.3, where the Poisson approximation has already been found. In the binomial distribution b(k;n,p) of that example, n ¼ 1000, p ¼ 0.005 with np ¼ 5, and npq ¼ 4.975. The approximate Gaussian pdf will be given by 2 1 fX (k) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ eð1=2Þ½(k5) =4:975 b(k; 1000,0:005) 2p:4:975

For the Poisson approximation np ¼ l ¼ 5, and this is given in Example 4.7.3 as k 5 pX (k;l) ¼ pX (k;5) ¼ pX (k) ¼ e5 k! Since l ¼ 5 is a single digit and npq ¼ 4.975 is not very large, we would reasonably infer that the Poisson approximation will be better than the Gaussian. The binomial distribution with the Poisson and Gaussian approximations are shown in Fig. 6.4.7 for k ¼ 0, . . . , 15. The true value of b(5; 1000, 0.005) ¼ 0.17591, the Gaussian approximation fX(5) ¼ 0.17886 with an absolute error of 1.678% and the Poisson approximation pX(5; 5) ¼ 0.17547 with an error of 0.25%, conﬁrming our inference that the Poisson approximation is better. Figure 6.4.7 clearly shows that the Poisson approximation follows the binomial more closely than the Gaussian.

FIGURE 6.4.7

6.4 Normal or Gaussian Distribution

93

FIGURE 6.4.8

Example 6.4.4 We shall now modify Example 6.4.3 by keeping n ¼ 1000 and changing p ¼ 0.5, yielding np ¼ l ¼ 500, and npq ¼ 250. Here both l and npq are quite large, so we can infer that the Gaussian will be a better approximation than the Poisson. The approximate Gaussian pdf is given by 2 1 fX (k) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e(1=2)½(k500) =250 b(k; 1000,0:5) 2p:250

and the corresponding Poisson approximation is PX (k;l) ¼ PX (k;500) ¼ PX (k) ¼ e500

500k k!

The binomial distribution with the Poisson and Gaussian approximations for this case are shown in Fig. 6.4.8. TABLE 6.4.3

Comparison of binomial with Gaussian and Poisson approximations

k

Binomial

Gaussian

Error 1BG

Error 1BP

2 3 4 5 6 7 8

0.083929 0.140303 0.175731 0.175908 0.14659 0.104602 0.035245

n ¼ 1000, p ¼ 0.005, np ¼ 5, npq ¼ 4.975 0.072391 0.084224 13.74699 0.119653 0.140374 14.718176 0.161758 0.175467 7.951344 0.17886 0.175467 1.678441 0.161758 0.146223 10.347485 0.119653 0.104445 14.38896 0.072391 0.065278 10.953113

0.352343 0.50683 0.150021 0.250272 0.250272 0.14992 0.051188

497 498 499 500 501 502 503

0.024775 0.025024 0.25175 0.25225 0.025175 0.025024 0.024775

n ¼ 1000, p ¼ 0.5, np ¼ 500, npq ¼ 250 0.024781 0.017731 0.023389 0.02503 0.017803 0.02438 0.025181 0.017838 0.024978 0.025231 0.017838 0.025177 0.025181 0.017803 0.024978 0.02503 0.017732 0.02438 0.024781 0.017626 0.023389

28.431591 28.85844 29.141872 29.283306 29.283306 29.141589 28.857017

Poisson

94

Continuous Random Variables and Basic Distributions

The true value for k ¼ 500 for binomial b(500,1000,0.5) ¼ 0.025225, the Gaussian approximation fX(k) ¼ 0.025231 with an absolute error of 0.025% and the Poisson approximation pX(k) ¼ 0.017838 with an error of 29.28%. In this case, the Gaussian approximation, as expected, fares very much better than the Poisson one. The results graphed in Fig. 6.4.8 for k ¼ 400, . . . , 600 clearly show that the Gaussian approximation follows the binomial much more closely than does the Poisson, unlike the scenario described in Example 6.4.3. The percentage absolute errors for the Gaussian and Poisson approximations to the binomial are deﬁned by

b(k;n,p) fX (k;np)

100 Gaussian: 1BG ¼

b(k;n,p)

b(k;n,p) pX (k;l)

100 Poisson: 1BP ¼

b(k;n,p) These errors are tabulated for p ¼ 0.005 and 0.5 in Table 6.4.3 in the neighborhoods of np ¼ l ¼ 5 and 500, respectively. The absolute error percentages also show that for large n and for values of moderate np and npq, the Poisson approximation is better and for large values of np and npq, the Gaussian approximation is better.

CHAPTER

7

Other Continuous Distributions

B 7.1

INTRODUCTION

We will now discuss some of the other important continuous distributions that occur in communications, signal processing, queuing, statistical inference, and other probabilistic situations. This list is by no means exhaustive. For a more complete list of distributions, refer to Web references W1 and W2 (last two entries in References list at end of the book). Most of these distributions are derivable from the three basic distributions discussed in Chapter 6.

B 7.2

TRIANGULAR DISTRIBUTION

If two independent random variables X and Y are uniformly distributed in the interval (a, b], then Z ¼ X þ Y has a triangular density deﬁned by 8 za > a,zb > > (b a)2 > < (7:2:1) fZ (z) ¼ (z 2b þ a) b , z 2b a > 2 > > (b a) > : 0 otherwise The density fZ (z) is shown in Fig. 7.2.1 with the corresponding uniform densities of X and Y for a ¼ 1 and b ¼ 5. The distribution function FZ (z) is given by 8 0 za > > > > 1 z a 2 > > a,zb > > 1 b , z 2b a > > 2 ba > : 1 z . 2b a The distribution function FZ (z) is shown for a ¼ 1 and b ¼ 5 in Fig. 7.2.2.

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

95

96

Other Continuous Distributions

FIGURE 7.2.1

FIGURE 7.2.2

B 7.3

LAPLACE DISTRIBUTION

The Laplace density, also called a double-exponential density, is used to model navigation and pilot errors, speech, and image processing. As the term suggests, double-exponential density is deﬁned by fX (x) ¼

l ljxj e 2

l.0

(7:3:1)

The cdf FX(x) is given by 8 1 > < elx FX (x) ¼ 2 > : 1 1 elx 2

x0 (7:3:2) x.0

It is an even density function, as shown for l ¼ 0.5 in Fig. 7.3.1. The cdf FX(x) is shown in Fig. 7.3.2. An alternate form of Laplace density in terms of the variance s2 is deﬁned by pﬃﬃ 1 fX (x) ¼ pﬃﬃﬃ e 2(jxj=s) 2s

(7:3:3)

7.4 Erlang Distribution

97

FIGURE 7.3.1

FIGURE 7.3.2

with the corresponding cdf FX(x) given by 8 pﬃﬃ 1 > < e 2(x=s) FX (x) 2 pﬃﬃ > : 1 1 e 2(x=s) 2

x0 (7:3:4) x.0

where s2 is the variance of the Laplace distribution.

B 7.4

ERLANG DISTRIBUTION

We already saw that the waiting time between successive events of a Poisson arrival process is exponentially distributed with parameter l. The distribution of the waiting time for n successive independent events of a Poisson arrival process is given by the Erlang distribution with n degrees of freedom and is deﬁned for positive integral values of n by fX (x) ¼

l(lx)n1 elx , (n 1)!

where n is an integer

(7:4:1)

98

Other Continuous Distributions

The Erlang distribution is obtained by the n-fold convolution of n independent exponential distributions. The cumulative distribution corresponding to Eq. (7.4.1) can be derived as follows. Let Xn denote the time at which the nth event occurs. Let N(x) be the number of events in the interval (0,x]. Hence the event fXn x) can occur only if the number of events occurring in (0,x] is at least n. Since N(x) is Poisson-distributed, the cdf FXn(x) can be given by P{Xn x} ¼ FXn (x) ¼ P{N(x) . n} ¼

1 X

el x

k¼n

n1 X (l x)k (l x)k ¼1 el x k! k! k¼0

(7:4:2)

We can obtain the pdf fXn(x) by differentiating Eq. (7.4.2) with respect to x. Performing the indicated differentiation, we can write ( ) n1 k X d d (l x) {FXn (x)} ¼ 1 el x k! dx dx k¼0 ¼ el x

n1 X k(l x)k1 l k¼0

( ¼ le

l x

k!

n1 X (l x)k

n1 X (l x)k k¼0

n1 X (l x)k1

k!

)

(k 1)! ( ) n1 n2 X (l x)k X (l x)k l x ¼ le k! k! k¼0 k¼0 k¼0

k!

þ lel x

k¼1

or d lel x (l x)n1 {FXn (x)} ¼ fXn (x) ¼ (n 1)! dx

(7:4:3)

and Eq. (7.4.3) is the Erlang distribution. This distribution is shown in Fig. 7.4.1 for n ¼ 10 and l ¼ 2,2.5,3,3.5,4,5. Another family of Erlang distributions is shown for l ¼ 4 and n ¼ 2,3,4,5,6 in Fig. 7.4.2.

FIGURE 7.4.1

7.5 Gamma Distribution

99

FIGURE 7.4.2

In both these cases the Erlang distributions reduce to an exponential distribution for n ¼ 1. In cases where the data may not be modeled as a simple exponential, the Erlang family provides greater ﬂexibility.

B 7.5

GAMMA DISTRIBUTION

A gamma distribution is a generalization of the Erlang distribution where n ¼ a, which may not be an integer. It ﬁnds wide use in statistics. Gamma density is a two-parameter family with parameters a . 0, termed the shape parameter, and l . 0, called the scale parameter, and is deﬁned as fX (x) ¼

l(lx)a1 elx , G(a)

a, l, x . 0

(7:5:1)

where G(a) is the gamma function deﬁned by the integral ð1 G(a) ¼ xa1 ex dx, a . 0

(7:5:2)

0

Integrating Eq. (7.5.2) by parts with u ¼ x a21 and dv ¼ e 2x dx, we have 1 ð1 ð1 a1 x a1 x G(a) ¼ x e dx, a . 0 ¼ x e þ (a 1) xa2 ex dx 0

0

¼ (a 1)G(a 1)

0

(7:5:3)

If a is an integer n, then G(n) ¼ (n –1)G(n –1) and continued expansion yields G(n) ¼ (n– 1)! Hence the gamma function can be regarded as the generalization of the factorial function for all positive real numbers. The most useful fractional argument for the gamma function is 12 with pﬃﬃﬃﬃ 1 G ¼ p (7:5:4) 2

100

Other Continuous Distributions

The incomplete gamma function Gx(a) is deﬁned by ðx Gx (a) ¼ ja1 ej dj

(7:5:5)

0

We will use Eq. (7.5.5) later for the x2 distribution function FX(x) in Section 7.7. We shall now show that the density function given by Eq. (7.5.1) indeed integrates to 1. If we deﬁne the integral Ix by ðx l(lh)a1 elh dh (7:5:6) Ix ¼ G(a) 0 Then, substituting lh ¼ j in Eq. (7.5.6), we can rewrite ð 1 lx a1 j Ix ¼ j e dj G(a) 0

(7:5:7)

Applying Eq. (7.5.5) to Eq. (7.5.7), we have Ix ¼

Glx (a) G(a)

or

lim Ix ¼

x!1

G(a) ¼1 G(a)

thus showing the validity of Eq. (7.5.1) as a proper density function. If a is an integer, then we have the Erlang distribution. The gamma density family is shown in Fig. 7.5.1 with the parameter l ¼ 2 and a ¼ 0.5,1,1.5,2.5,3.5,4.5,5.5. The cumulative distribution function FX(x) is given by ð 1 x Gx (a) l(lj)a1 elj dj ¼ FX (x) ¼ (7:5:8) G(a) 0 G(a) This integral in Eq. (7.5.8) has no closed-form solution unless a is an integer, in which case it is the Erlang distribution given by Eq. (7.4.2). However, the integral given by Eq. (7.5.8) has been extensively tabulated. Reproductive Property of Gamma Distribution

If X1, X2, . . . , Xn are n independent gamma-distributed random variables with shape parameters a1, a2, . . . , an and having the same scale parameter l, then random variable

FIGURE 7.5.1

7.6

Weibull Distribution

101

Z ¼ X1 þ X2 þ þ Xn is also gamma-distributed with shape parameter b ¼ a1 þ a2 þ þ an. As a consequence of this property, since an exponential random variable is gamma-distributed with parameters (1,l), the sum of n exponential random variables is gamma-distributed with parameters (n,l), which is an Erlang distribution given by Eq. (7.4.1).

B 7.6

WEIBULL DISTRIBUTION

We will deﬁne the Weibull distribution as a three-parameter family of distribution functions, in which the parameter l . 0 is the scale parameter, a . 0 is the shape parameter, and m is the location parameter. It is deﬁned by a a a l (x m)a1 e½l(xm) x . m fX (x) ¼ (7:6:1) 0 xm We will now ﬁnd the cdf FX (x) of the Weibull distribution. We can write ðx a FX (x) ¼ a la (j m)a1 e½l(jm) dj

(7:6:2)

m

Substituting h ¼ ½l(j m)a and hence dh ¼ ½l(j m)a1 al dh, we can rewrite Eq. (7.6.2) as follows: ½l(xm)a ð ½l(xm) a eh dh ¼ eh , x.m (7:6:3) FX (x) ¼ 0

0

Substitution of the limits in Eq. (7.6.3) yields a 1 e½l(xm) x . m FX (x) ¼ 0 xm

and

FX (1) ¼ 1

(7:6:4)

Two families of Weibull distributions are shown for m ¼ 0 and a ¼ 2, l ¼ 0.5,1,2,3,4 (Fig. 7.6.1) and for l ¼ 2 and a ¼ 0.5,1,2,3,4,5 (Fig. 7.6.2). When m ¼ 0 and l ¼ 1 in Eqs. (7.6.1) and (7.6.4), then the resulting distribution is called the standard Weibull.

FIGURE 7.6.1

102

Other Continuous Distributions

FIGURE 7.6.2

The Weibull distribution was originally used for modeling fatigue data and is at present used extensively in engineering problems for modeling the distribution of the lifetime of an object that consists of several parts and that fails if any component part fails.

B 7.7

CHI-SQUARE DISTRIBUTION

The Laplace, Erlang, gamma, and Weibull distributions are all based on exponential distributions. The chi-square distribution straddles both the exponential and Gaussian distributions. The general gamma distribution has parameters (a, l). In the gamma distribution, if we substitute a ¼ n/2 and l ¼ 12, we obtain the standard chi-square distribution, deﬁned by 8 (n=2)1 (x=2) x e > < , x.0 n=2 2 G(n=2) (7:7:1) fX (x) ¼ > : 0, x0 where n is the degrees of freedom for the x2 distribution. We can introduce another parameter s2 and rewrite Eq. (7.7.1) as 8 (n=2)1 (x=2s2 ) > e > > : 0 , x0 If we substitute x ¼ x2 in Eq. (7.7.2), we obtain the familiar form of the chi-square distribution, given by 2

fX (x2 ) ¼

2

x2½(n=2)1 e(x =2s ) (2s2 )n=2 G(n=2)

(7:7:3)

where G(x) is the gamma function deﬁned in Eq. (7.5.2). If n is even in Eq. (7.7.2), then G(n=2) ¼ (n=2 1)!, and if n is odd, then G(n=2) in Eqs. (7.7.1) pﬃﬃﬃﬃ –(7.7.3) is obtained by iterating (n=2 1)G(n=2 1) with the ﬁnal iterant Gð12Þ ¼ p. . . . , Xn are Equation (7.7.3) can also be obtained from the following. If X1, X2, P n i.i.d. Gaussian random variables with zero mean and variance s2, then X ¼ nk¼1 X k2 is x2 -distributed with n degrees of freedom.

7.7 Chi-Square Distribution

103

The cumulative chi-square distribution function FX(x) is given by integrating Eq. (7.7.2): 8 ð x n=21 (j=2s2 ) e < j dj x . 0 n=2 2 FX (x) ¼ (7:7:4) : 0 (2s ) G(n=2) 0 x0 This equation can be given in terms of the incomplete gamma function Gx(a) given by Eq. (7.5.5). Substituting h ¼ j/2s2 and dj ¼ 2s2 dh in Eq. (7.7.4), we obtain ðx FX (x) ¼

2

jn=21 e(j=2s ) 2s2 dj 2 n=21 G(n=2) 0 (2s )

ð x=2s2 ¼ 0

Gx=2s2 (n=2) hn=21 e(h) dh ¼ (7:7:5) G(n=2) G(n=2)

When x ! 1 then Gx=2s2 (n=2) ! G(n=2) and hence FX(x) ¼ 1. The cdf FX(x) in Eq. (7.7.5) has been extensively tabulated. The chi-square distribution is shown in Fig. (7.7.1) for the number of degrees of freedom n ¼ 2,3,4,5,6,10. The chi-square distribution is used directly or indirectly in many tests of hypotheses. The most common use of the chi-square distribution is to test the goodness of a hypothesized ﬁt. The goodness-of-ﬁt test is performed to determine whether an observed value of a statistic differs enough from a hypothesized value of a distribution to draw the inference whether the hypothesized quantity is the true distribution. Its utility is shown in Chapter 15. Example 7.7.1 The density function of fZ (z) of Z ¼ X 2 þ Y 2, where X and Y are independent random variables distributed as N(0, s2), can be found by substituting n ¼ 2 in Eq. (7.7.2), yielding 8 2 < e(z=2s ) z.0 FZ (z) ¼ 2 : 2s 0 z0

FIGURE 7.7.1

104

Other Continuous Distributions

which is a x2 density with 2 degrees of freedom. The corresponding cdf, FZ (z), is given by ( 2 1 e(z=2s ) z . 0 FZ (z) ¼ 0 z0

B 7.8

CHI AND OTHER ALLIED DISTRIBUTIONS

The Rayleigh and Maxwell distributions are special cases of the chi distribution. Just as the sum of the squares of n zero mean Gaussian random variables with variance s2 is chisquare-distributed with n degrees of freedom, we can consider that the sum of the square roots of the sum of the squares of n zero mean Gaussian random variables with variance s2 is chi-distributed with n degrees of freedom. Accordingly, let Zn be the random variable given by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (7:8:1) Zn ¼ X1 þ X2 þ þ Xn2 where fXi . i ¼ 1, . . . , ng are zero mean Gaussian random variables with variance s2. Then the x-density fZ (z) is given by 8 9 1 zn1 (1=2)(z=s)2 < = e z.0 (7:8:2) fZ (z) ¼ 2n=21 G(n=2) sn ; : 0 z0 where G(n/2) is the gamma function deﬁned in Eq. (7.5.2). A family of chi distributions is shown in Fig. 7.8.1 for s ¼ 1.5 and n ¼ 1,2,3,5,7,9, 11,13,15. The distribution function FZ (z) is given by integrating Eq. (7.8.2): ðz 1 z n1 (1=2)(z=s)2 Fz (z) ¼ e dz z . 0 (7:8:3) n=21 G(n=2) sn 02 There is no closed-form solution for Eq. (7.8.3), but it can be expressed in terms of the incomplete gamma function GZ (z) deﬁned in Eq. (7.5.5). By substituting y ¼ z2 =2s2 and

FIGURE 7.8.1

7.8 Chi and Other Allied Distributions

105

dz ¼ (s2 dy)=z in the in Eq. (7.8.3), we can write 2n=21 FZ (z) ¼ n=21 2 G(n=2) ¼

ð z2 =2s2

y(n=21) ey dy

0

n G 2 2 (n=2) 2n=21 (z =2s ) 2 =2s2 ) ¼ G (z 2 2n=21 G(n=2) G(n=2)

(7:8:4)

where G(z2 =2s2 ) (n=2) is the incomplete gamma function deﬁned in Eq. (7.5.5). Clearly, as z ! 1, G(z2 =2s2 ) (n=2) ! G(n=2), thus showing that FZ (1) ¼ 1. There are two very important special cases of the chi distribution when n ¼ 2 and n ¼ 3. Rayleigh Distribution (n 5 2)

When n ¼ 2 in the chi distribution of Eq. (7.8.2), we have ( z ) (1=2)(z=s)2 e z . 0 fZ (z) ¼ s2 0 z0

(7:8:5)

This is called the Rayleigh density and ﬁnds wide application in communications fading channels. If we have two independent zero mean Gaussian random variables X and Y with variance s2, and if we make the transformation X ¼ Z cos(Q) and Y ¼ Z sin(Q), pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 then Z ¼ X þ Y 2 is Rayleigh-distributed, which is a chi distribution with 2 degrees of freedom. Q is uniformly p distributed in the interval (0,2p]. The Rayleigh density is ﬃﬃﬃﬃﬃ shown in Fig. 7.8.2 for s ¼ 10,4,5. The cdf corresponding to the Rayleigh density can be expressed in a closed form as follows: ðz j j2 =2s2 e dj or, FZ (z) ¼ 2 0s ( 2 2 1 ez =2s z . 0 (7:8:6) ¼ 0 z0

FIGURE 7.8.2

106

Other Continuous Distributions

Maxwell Density (n 5 3)

This is one of the earliest distributions derived by Maxwell and Boltzmann from statistical mechanics considerations. They derived that the velocity vi of gas molecules in any direction i, based on the assumptions that the molecules are identical and distinguishable, and can occupy any energy state, follow the probability density function fV (vi ) ¼

e(1i =kT) Z

(7:8:7)

where 1i is the energy function given by 1i ¼ mv2i =2 with m as the mass of the particle, k is the Boltzmann constant, and T is the absolute temperature. Z is the normalization constant found from integrating Eq. (7.8.7) over all vi, or rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð 1 1 (mv2i =2kT) 2pkT e dvi ¼ 1 and Z ¼ Z 1 m so that the velocity density function along any direction follows a Gaussian distribution given by rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m (mv2i =2kT) , i ¼ x,y,z (7:8:8) fV (vi ) ¼ e 2pkT From Eq. (7.8.8) the velocity vector V ¼ fvx,vy,vzg will be distributed with density function, " # m 3=2 m(v2x þ v2y þ v2z ) (7:8:9) exp fV (vx ,vy ,vz ) ¼ 2kT 2pkT Eq. (7.8.9) is a chi-square density with 3 degrees of freedom. The speed x of gas molecules will be deﬁned by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x ¼ v2x þ v2y þ v2z Hence x is a distributed as a chi-density with three degrees of freedom. Its probability density can be derived as h m i3=2 2 f (x) ¼ 4p x2 emx =2kT , x . 0 (7:8:10) 2pkT If we substitute kT=m ¼ s2 in the Maxwell – Boltzmann distribution of Eq. (7.8.10), we obtain rﬃﬃﬃﬃ 2 x2 x2 =2s2 e , x.0 (7:8:11) fX (x) ¼ p s3 Equation (7.8.11) is a special case of the chi density of Eq. (7.8.2) with n ¼ 3 degrees of for the Maxwell and Rayleigh densities are compared in freedom. The pdf values fX(x) pﬃﬃﬃﬃﬃ Fig. 7.8.3 for values of s ¼ 10, 4, and 5. The cdf FX(x) corresponding to the Maxwell density fX(x) is obtained by integrating Eq. (7.8.11) and is given by rﬃﬃﬃﬃ ð x 2 2 j j2 =2s2 e dj (7:8:12) FX (x) ¼ p 0 s3

7.8 Chi and Other Allied Distributions

107

FIGURE 7.8.3

There is no closed form solution for Eq. (7.8.12). However, integrating Eq. (7.8.12) by parts, it can be expressed as rﬃﬃﬃﬃ pﬃﬃﬃ ð x 2 x x2 =2s2 2 j2 =2s2 FX (x) ¼ pﬃﬃﬃﬃ e dj e ps s p 0 rﬃﬃﬃﬃ x 2 x x2 =2s2 e ¼ erf pﬃﬃﬃ x.0 (7:8:13) ps 2s where erf(x) is as deﬁned in Eq. (6.4.6). Rice’s Density

In the Rayleigh distribution discussed earlier, the random variables X and Y must be zero mean with the same variances s2. In many communication problems involving fading channels, X and Y will not be zero mean p and will have mean values equal to mX and mY. If ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ the random variable Z is given by Z p ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X 2 þﬃ Y 2 , then fZ (z) has a Rice density [49,50] with a noncentrality parameter m ¼ m2X þ m2Y , given by 2 z (z þ m2 ) mz fZ (z) ¼ 2 exp (7:8:14) I0 2 s 2s2 s where I0(z) is the zeroth-order modiﬁed Bessel function of the ﬁrst kind. The general nth-order modiﬁed Bessel function is given by the series In (z) ¼

1 z n X

2

(z2 =4)k k! G(n þ k þ 1) k¼0

If n is an integer, then In(z) can also be given by ð 1 p z cos u In (z) ¼ e cos (nu) du p 0 and I0(z) the zeroth-order modiﬁed Bessel function by ð 1 X 1 p z cos u (z2 =4)k I0 (z) ¼ e du ¼ 2 p 0 k¼0 (k!) and I0(z), I1(z), I2(z), I3(z) are shown in Fig. 7.8.4.

(7:8:15)

(7:8:16)

(7:8:17)

108

Other Continuous Distributions

FIGURE 7.8.4

From Fig. 7.8.4 we see that I0(0) ¼ 1, and hence when m ¼ 0 (zero mean), then fZ (z) is Rayleigh-distributed. Rice’s density is shown in Fig. 7.8.5 for m ¼ 0,2,4,6,8,10,12,14, and m ¼ 0 corresponds to the Rayleigh density. The distribution function FZ(z) corresponding to Eq. (7.8.14) is given by integrating fZ (z): ðz j (j2 þ m2 ) mj exp FZ (z) ¼ I0 2 dj (7:8:18) 2 2 2s s 0s There is no closed-form solution for this integral.

Nakagami Density

The Nakagami distribution [49], like the Rayleigh density and unlike Rice’s distribution, also belongs to the class of central chi-square distributions. It is used for modeling data from multipath fading channels and has been shown to ﬁt empirical results more generally than the Rayleigh distributions. Sometimes, the pdf of the amplitude of a mobile signal can also be described by the Nakagami m distribution.

FIGURE 7.8.5

7.8 Chi and Other Allied Distributions

109

The Nakagami distribution may be a reasonable means to characterize the back scattered echo from breast tissues to automate a scheme for separating benign and malignant breast masses. The shape parameter m has been shown to be useful in tissue characterization. Chi-square tests showed that this distribution is a better ﬁt to the envelope than the Rayleigh distribution. Two parameters, m (effective number) and V (effective cross section), associated with the Nakagami distribution are used for the effective classiﬁcation of breast masses. The Nakagami m density is deﬁned by fX (x) ¼

2 h m im 2m1 mx2 =V x e G(m) V

(7:8:19)

where m is the shape parameter and V controls the spread of the distribution. The interesting feature of this density is that, unlike the Rice distribution, it is not dependent on the modiﬁed Bessel function. The Nakagami family is shown in Fig. 7.8.6 for V ¼ 1 and for m ¼ 3,2,1.5,1,0.75,0.5. The cdf FX(x) is given by ðx 2 h m im 2m1 mj2 =V FX (x) j e dj (7:8:20) 0 G(m) V This has no closed form solution. However, it can be given in terms of the incomplete gamma function by substituting y ¼ mj2/V and dj ¼ (V=2mj) dy in Eq. (7.8.20). This results in m 2 mj2 1 mj2 =V dj FX (x) ¼ e V j 0 G(m) m ð mx2 =V 2 2 mj2 1 dy emj =V ¼ G(m) V 2(mj2 =V) 0 ð mx2 =V 2 m y 1 y e dy ¼ G(m) 2y 0 ð mx2 =V Gmx2 =V (m) 1 m1 y y e dy ¼ ¼ G(m) G(m) 0 ðx

Since Gmx2 =V (m) ! G(m) as x ! 1 we have FX (1) ¼ 1:

FIGURE 7.8.6

(7:8:21)

110

Other Continuous Distributions

B 7.9

STUDENT-t DENSITY

The t distribution ﬁrst developed by William Gosset under the pseudonym “Student” is used to estimate the conﬁdence intervals for the unknown population mean where the variance is also unknown. In this case the statistic used for determining the conﬁdence interval is given by T¼

m^ X mX pﬃﬃﬃ s= ^ n

(7:9:1)

and the distribution of T is governed by the Student-t distribution with density given by

nþ1 ½(nþ1)=2 G t2 2 1þ fT (t) ¼ pﬃﬃﬃﬃﬃﬃ n n np G 2

1 , t , 1

(7:9:2)

where n is the number of degrees of freedom of T. It will be shown in Example 13.4.5 that if Z is a standard Gaussian random variable, Wn is a chi-square random of freedom, and Z and Wn are independent, then the variable with n degreespﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ random variable T ¼ Z= Wn =n is t-distributed with n degrees of freedom. The Student-t distribution with n ¼ 2 is compared to a standard Gaussian in Fig. 7.9.1. After integrating the density function, we can write the distribution function FT (t) as 1 ½(nþ1)=2 G nþ t2 2 1þ dt FT (t) ¼ n n 1 pﬃﬃﬃﬃﬃﬃ np G 2 ðt

This is shown in Fig. 7.9.2.

FIGURE 7.9.1

(7:9:3)

7.10

Snedecor F Distribution

111

FIGURE 7.9.2 Limit of Student-t as n ! 1

We shall ﬁnd the limit as n ! 1 in Eq. 7.9.2. We have ½(nþ1)=2 t2 2 lim 1 þ ¼ et =2 n!1 n 1 G nþ 1 2 ¼ pﬃﬃﬃﬃﬃﬃ lim n!1 pﬃﬃﬃﬃﬃﬃ n 2p np G 2

(7:9:4)

and hence ½vþ(1=2) G( nþ1 ) t2 1 2 lim pﬃﬃﬃﬃﬃﬃ 2 n 1 þ ¼ pﬃﬃﬃﬃﬃﬃ et =2 n!1 np G( ) n 2p 2

(7:9:5)

that is, as the number of degrees of freedom n goes to inﬁnity, the Student-t tends to a standard Gaussian. Gaussian random variables If X1, X2, . . . , Xn are n independent identically distributed P with mean mX and standard deviation sX, then m^ X ¼ (1=n) ni¼1 Xi is also Gaussiandistributed with mean mX and variance s2X =n.PSince we do not know s2X , we use the sample variance given by s^ 2X ¼ 1=(n 1) ni¼1 (Xi m^ X )2 and s^ 2X is chi-squaredistributed with n – 1 degrees of freedom. Under these conditions the random variable T¼

(m^ X mX ) pﬃﬃﬃ s^ X = n

(7:9:6)

has a t distribution with n 2 1 degrees of freedom.

B 7.10

SNEDECOR F DISTRIBUTION

If X1 and X2 are two independent chi-squared-distributed random variables with n1 and n2 degrees of freedom, then the density of the random variable F ¼ (X1 =n1 )=(X2 =n2 ) is n þ n n n1 =2 1 2 1 G 2 n2 (n 2)=2 , x.0 (7:10:1) fn1 ,n2 (x) ¼ (n1 þn2 )=2 x 1 n1 n2 1 þ n1 G G x 2 2 n2

112

Other Continuous Distributions

FIGURE 7.10.1

FIGURE 7.10.2

This density is shown in Fig. 7.10.1 for three cases: n1 ¼ 15, n2 ¼ 25: n1 ¼ 5, n2 ¼ 25: n1 ¼ 10, n2 ¼ 5. Note that in general fn1 ,n2 (x) is not equal to fn2 ,n1 (x). The corresponding distribution function is given by n þ n n n1 =2 1 2 1 ðx G 2 n2 (n1 2)=2 Fn2 , n1 (x) ¼ dj (7:10:2) n n 1 þ n (n1 þn2 )=2) j 0 1 2 1 G G j 2 2 n2 Equation (7.10.2) is shown in Fig. 7.10.2 for the same values of n1 and n2 as in Fig. 7.10.1. As n1 and n2 tend to inﬁnity, then the density function becomes more sharply spiked around 1. The F distribution is used to draw inferences about the population variances in the case of two-sample situations.

B 7.11

LOGNORMAL DISTRIBUTION

A random variable X has lognormal density with parameters a, m, and s2 if the transformed random variable Y ¼ ln(X ) has a normal distribution with mean m and variance

7.11

Lognormal Distribution

113

FIGURE 7.11.1

s2. This density, deﬁned only for positive values of x, is given by ( ) 1 1 ln (x a) m 2 fX (x) ¼ pﬃﬃﬃﬃﬃﬃ exp , 2 s 2p s (x a)

x . a; m, s . 0

(7:11:1)

where a is the location parameter, m is the scale parameter, and s is the shape parameter. When a and m are zero, fX(x) is called the standard lognormal distribution with parameter s. The case where a ¼ 0 and m ¼ 1 is shown in Fig. 7.11.1 for s ¼ 0.4,0.5,1.0,1.5,2 for values of x . 0.1. The cumulative distribution function FX(x) is obtained by integrating Eq. (7.11.1) and is given by ðx

2 1 2 pﬃﬃﬃﬃﬃﬃ e( ln (ja)m) =2s dj 2p s(j a) 0 ð ln (xa)m s 1 1 ln (x a) m 2 pﬃﬃﬃﬃﬃﬃ eh =2 dh ¼ erf pﬃﬃﬃ ¼ 2 2p 2s 0

FX (x) ¼

(7:11:2)

where 1 1 x 2 pﬃﬃﬃ eh =2 dh ¼ erf pﬃﬃﬃ 2 2p 2 0

ðx

The cdf FX(x) is shown in Fig. 7.11.2 for a ¼ 0, m ¼ 1 and for the same values of s as the density function. In ﬁnance, prices for primary instruments such as stocks and bonds are often assumed to be lognormally distributed. A standard assumption for the evolution of a stock price is that in an interval Dt the fractional increase in the stock price, DS/S, is normally distributed with mean mDt and variance s2Dt. The lognormal distribution is also used to model the lifetime of units whose failures are of a fatigue– stress nature. Consequently, the lognormal distribution is an adjunct to the Weibull distribution when attempting to model these types of units.

114

Other Continuous Distributions

FIGURE 7.11.2

B 7.12

BETA DISTRIBUTION

The beta distribution is a two-parameter a and b family of density functions fX(x) for 0 x 1. It is often used to represent proportions and percentages. It has the following probability density function: fX (x) ¼

G(a þ b) a1 x (1 x)b1 , G(a)G(b)

0x1

(7:12:1)

The choices of the two parameters a and b makes it amenable to a variety of shapes. For example, for a ¼ b ¼ 1, fX(x) becomes the rectangular distribution. The triangular distribution arises when one parameter has the value 2 and the other is unity. If both parameters are less than unity and positive, the density becomes U-shaped, and if only one of the parameters is less than unity, it becomes J-shaped. The distribution is unimodal when both parameters are positive, and symmetric when they are equal. The pdf curves are shown in Fig. 7.12.1 for the cases described above. The beta distribution is related to the Student-t distribution [Eq. (7.9.2)] by the following transformation. If Z is a t-distributed

FIGURE 7.12.1

7.13

Cauchy Distribution

115

FIGURE 7.12.2

random variable with n degrees of freedom, then the transformation 1 Z X ¼ þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 n þ Z2 is beta-distributed with a ¼ n/2 and b ¼ n/2. Example 7.12.1 We will give an example of how the beta distribution is used in stockmarket analysis. The analyst from his model of stock performances predicts before the start of each day that the proportion of the listed stocks has a beta distribution with parameters a ¼ 7.3 and b ¼ 5.2. The average value is a=(a þ b) ¼ 0:584. We want to ﬁnd the probability that more than 70% of the stocks increase in value and this is given by ð 0:7 fX (x) dx ¼ 1 0:794 ¼ 0:206 P{X . 0:7} ¼ 1 0

This probability is 20.6%, as illustrated in Fig. 7.12.2.

B 7.13

CAUCHY DISTRIBUTION

Cauchy distribution resembles a normal distribution in appearance but has much heavier tails. However, in studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of how sensitive the tests are to heavy-tail departures from normality. The general Cauchy density is given by fX (x) ¼

b p½(x a)2 þ b2

(7:13:1)

where a is the location parameter and b is the shape parameter. It is an example of a pathological case in the sense the moments do not exist even though the ﬁrst moment can be evaluated to be equal to 0 using the Cauchy principle value method. The Cauchy density is plotted for a ¼ 5 and b ¼ 2,3,4,5 in Fig. 7.13.1. Since the moments do not exist for the Cauchy distribution, it is characterized by the width at half the maximum value equal to 2b, called full width half maximum (FWHM). For b ¼ 2 the maximum is 0.159154 and the half-maximum is 0.079577. This is shown in Fig. 7.13.2. For comparison, a Gaussian distribution is superposed on the same ﬁgure for the same FWHM of 4 that is, a Gaussian with mean 5 and s ¼ 4/2.35482 ¼ 1.69864.

116

Other Continuous Distributions

FIGURE 7.13.1

FIGURE 7.13.2

FIGURE 7.13.3

7.14

Pareto Distribution

The distribution function FX(x) can be determined as follows: ðx 2 1 2 e( ln (ja)m) =2s dj FX (x) ¼ pﬃﬃﬃﬃﬃﬃ 2p s(j a) 0 Substituting (j a)=b ¼ tan(u) in Eq. (7.13.2), we have ð tan1 {(xa)=b} sec2 (u)du FX (x) ¼ p sec2 (u) p=2 tan1 {(xa)=b} u 1 xa 1 ¼ ¼ tan1 þ p p=2 p b 2

117

(7:13:2)

(7:13:3)

The term FX(x) is shown in Fig. 7.13.3 for the same values of a and b as in Fig. 7.13.1.

B 7.14

PARETO DISTRIBUTION

Many synthetic and naturally occurring phenomena, including city sizes, incomes, word frequencies, and earthquake magnitudes, are distributed according to a power-law distribution. A power law implies that small occurrences are extremely common, whereas large instances are extremely rare. These can be modeled as a Pareto distribution. In addition, trafﬁc measurements reveal features such as nonstationarity, long-range dependence, and signiﬁcant ﬂuctuations in different timescales (e.g., self-similarity). The essential features of the burstiness observed in measurements can be modeled as a heavy-tailed on-time and off-time distributions such as the Pareto distribution. The Pareto density is given by a b aþ1 , x.0 (7:14:1) fX (x) ¼ b xþb and the distribution function can be obtained from integrating Eq. (7.14.1): ðx a b aþ1 b a dj ¼ 1 FX (x) ¼ xþb 0b jþb

FIGURE 7.14.1

(7:14:2)

118

Other Continuous Distributions

The Pareto density and the distribution functions are plotted in Fig. 7.14.1 for a ¼ 5 and b ¼ 10,15,20. Note that the tails are long compared to other distributions. One of the uses of the Pareto distribution is to model the high end of the distribution of incomes.

B 7.15

GIBBS DISTRIBUTION

The Gibbs distribution has been widely used in image processing and pattern recognition problems. An example using this distribution in tomographic imaging is presented in Chapter 23. It is a generalization of the Maxwell – Boltzmann distribution and has origins in statistical mechanics. It is deﬁned on a random ﬁeld N as P(v) ¼

where,

1 ð1=TÞU(v) e Z

(7:15:1)

P wjk V(vj vk ) U(v) ¼ j,keN V(v)

Total energy function Potential function, where v is some vector ﬁeld

wjk j, k

Weighting functions Pixels in neighboring ﬁeld N

T

Absolute temperature that controls sharpness of distribution P Normalization constant given by Z ¼ v e(1=T)U(v) , also called the partition function

Z

The potential function V(v) depends on the local conﬁguration of the ﬁeld. Some of the properties of potential functions [38] are given below: 1. V(0) ¼ 0 2. V(v) 0: nonnegative 3. V(v) ¼ V(v): even function d V(v) . 0 for v . 0 4. dv 5. lim V(v) ! 1 v!1

6.

sup 0v,1 2

7.

B 7.16

d dv V(v)

,1

d V(v) . 1 dv2

MIXED DISTRIBUTIONS

There are any number of derived and mixed distributions, depending on the application, and new distributions are also constantly being formulated. For example, the doubledouble-exponential or double-Laplace distribution, given by

p ðjxjÞ 1 p jxjb (7:16:1) e a þ e f (x) ¼ 2a 2b

7.17

Summary of Distributions of Continuous Random Variables

119

FIGURE 7.16.1

with f1 (x) ¼

p ðjxjÞ 1p e a : f2 (x) ¼ e 2a 2b

jxj b

is shown in Fig. 7.16.1 with p ¼ 0.3, a ¼ 1, b ¼ 4. This distribution has been specially constructed for modeling navigation errors over transatlantic ﬂights consisting of both the pilot and instrument errors [51]. The pilot errors are modeled by f2(x) and the instrument errors, by f1(x). The instrument errors have a sharper peak than do pilot errors, which extend over a greater distance. Whereas the instrument are easier to eliminate, the pilot errors are more difﬁcult. The composite error curve is f(x) shown in Fig. 7.16.1. The distributions given here are by no means exhaustive, and only some of the more important distributions are discussed. More comprehensive sets of distributions can be obtained from Web references W1 and W2 (see References at end of book). A complete list of all the densities and distributions discussed in this chapter is tabulated in Section 7.17.

B 7.17 SUMMARY OF DISTRIBUTIONS OF CONTINUOUS RANDOM VARIABLES Density Function

Name Uniform

Triangular

1 ba xa ; a,xb (b a)2 (x 2b þ a) (b a)2 b , x 2b a

Closed-Form Distribution Function xa ba 1 x a 2

a,xb 2 b a 1 x þ a 2b 2 1 2 ba b , x 2b a (continued)

120

Other Continuous Distributions

Density Function

Name Exponential Laplace

Closed-Form Distribution Function

1 lx e u(x), l . 0 l l ljxj e , l0 2

(1 elx )u(x), 1 lx e , 2

l.0

x0

1 12 elx , x . 0 Erlang

l(lx)n1 elx , (n 1)!

1

n integer

n integer; x . 0

l(lx)a1 elx , G(a)

a,l,x . 0

ala (x m)a1 e½l(xm) x.m 1 ðx2 =2Þ Standard Gaussian pﬃﬃﬃﬃﬃﬃ e 2p Weibull

Chi-square

k!

k¼0

x.0 Gamma

n1 X (lx)k elx

2

(x2 )(n=2)1 ex =2s n (2s2 )n=2 G 2

Gx (a) G(a) Incomplete gamma

a a

1 e½l(xm) , 1 1 x þ erf pﬃﬃﬃ 2 2 2 n Gx2 =2s2 n 2 , G 2

2

x.m

x ¼ x2

Incomplete gamma 1

Chi

2(n=2)1 G

n

xn1 (1=2)(x=s)2 e s2

2

x.0

n Gx2 =2s2 n2 G 2 Incomplete gamma

Rayleigh

z (z2 =s2 ) e , s2

Maxwell

rﬃﬃﬃﬃ 2 x3 ðx2 =2s2 Þ e , p s3

Rice

x ½(x2 þm2 )=2s2 mx e I0 2 , s2 s

2

2

1 eðz =2s Þ , z . 0

z.0

x pﬃﬃﬃ s 2 erf 2s2

x.0

rﬃﬃﬃﬃ 2 ðx2 =2s2 Þ e p

x.0 x.0

No closed form (continued)

7.17

Summary of Distributions of Continuous Random Variables

Density Function

Name Nakagami

Student-t(n)

Snedecor F(n1 ,n2 )

Closed-Form Distribution Function

2 m m 2m1 mx2 =V x e G(m) V

Gmx2 =V (m) G(m)

x.0

Incomplete gamma

vþ1 ½(vþ1)=2 G t2 2 1 þ pﬃﬃﬃﬃﬃﬃ v v vp G 2 n þ n n n1 =2 1 2 1 x(n1 2)=2 2 n2 n n n1 (n1 þn2 )=2 1 2 G 1þ x G 2 2 n2 G

121

No closed form

No closed form

x.0 Lognormal

Beta

2 1 2 pﬃﬃﬃﬃﬃﬃ e½ln (xa)m =2s 2p s(x a)

G(a þ b) a1 x (1 x)b1 G(a)G(b)

1 ln (x a) m pﬃﬃﬃ erf 2 s 2 No closed form

0x1 Cauchy

b p½(x a)2 þ b2

1 1 1 x a þ tan 2 p b

Pareto

aþ1 a b , x.0 b xþb

1

Deﬁnitions

Zero-order modiﬁed Bessel function ð 1 p x cos (u) I0 (x) ¼ e du p 0

b xþb

a ,

x.0

Error function ð 2 2 x erf (x) ¼ pﬃﬃﬃﬃ ej dj p 0 Incomplete gamma function ðx Gx (a) ¼ ja1 ej dj 0

CHAPTER

8

Conditional Densities and Distributions

B 8.1 CONDITIONAL DISTRIBUTION AND DENSITY FOR P{A} = 0 In Section 2.2 we deﬁned conditional probability for events A and B that are subsets of the sample space S as 9 P{A > B} = P{B j A} ¼ if P{A} = 0 (8:1:1) P{A} ; P{A > B} ¼ P{B j A}P{A} We will extend this to distributions FX(x) and densities fX(x). The conditional distribution FX(x j A), given that an event A has occurred, is deﬁned as FX (x j A) ¼ P{X x j A} ¼

P{X x, A} P{A}

if

P{A} = 0

(8:1:2)

and if the event fX xg is independent of A, then PfX x j Ag ¼ PfX xg. The corresponding conditional density fX(x j A) is deﬁned by dFX (x j A) dx P{x , X x þ Dx j A} ¼ lim Dx!0 Dx

fX (x j A) ¼

Properties of Conditional Distributions 1. FX (1 j A) ¼ 1; FX (1 j A) ¼ 0 2. FX (x j A) is right-continuous 3. FX (x2 j A) FX (x1 j A) ¼ P{x1 , X x2 j A} Ðx 4. FX (x j A) ¼ 1 fX (j j A) dj

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

122

(8:1:3)

8.1 Conditional Distribution and Density for P{A}= 0

123

Example 8.1.1 A die is tossed and, corresponding to the faces of the die, a random variable is deﬁned by X ¼ 10k, k ¼ 1,2,3,4,5,6. The conditioning event A is given by A ¼ fthe toss resulted in an even numberg. We will ﬁrst ﬁnd the unconditional distribution FX(x) and then the conditional distribution FX(x j A). Using the same mapping techniques discussed in Chapter 2, the unconditional cdf is given by FX (x) ¼

6 1X u(x 10k) 6 k¼1

Since the sample space has been reduced to 3 points by the conditioning event A, the conditional cdf FX(x j A) is given by Fx (x j A) ¼ ¼

P{X 10k, X even} FX (10k), k ¼ 2,4,6 ¼ P{X even} 1=2 3 1X u(x 20k) 3 k¼1

Both these distributions are shown in Fig. 8.1.1. Conditioning Event A Depends on x. We shall now consider the case when the event A is dependent on x. Three cases arise: 1. A ¼ {X x1 }: Eq. (8.1.2) takes the form: FX (x j A) ¼

P{X x, X x1 } P{X x1 }

(8:1:4)

Two situations arise: (a) x . x1 : Since x . x1, the joint event fX x, X x1g ¼ fX x1g, and hence Eq. (8.1.4) can be rewritten as FX (x j A) ¼

P{X x1 } ¼1 P{X x1 }

(8:1:5)

(b) x x1: In this case the joint event fX x, X x1g ¼ fX xg, and hence Eq. (8.1.4) becomes FX (x j A) ¼

P{X x} FX (x) ¼ P{X x1 } FX (x1 )

FIGURE 8.1.1

(8:1:6)

124

Conditional Densities and Distributions

The corresponding density function is given by 8 < fX (x) fX (x j A) ¼ FX (x1 ) : 0

x x1

(8:1:7)

x . x1

2. A ¼ fx1 , X x2g with x1 , x2 Analogous to Eq. (8.1.4), we have the following: FX (x j A) ¼

P{X x, x1 , X x2 } P{x1 , X x2 }

(8:1:8)

Here three situations arise: (a) x . x2: FX (x j A) ¼

P{x1 , X x2 } ¼1 P{x1 , X x2 }

(8:1:9)

(b) x1 . x x2: FX (x j A) ¼

P{x1 , X x} FX (x) FX (x1 ) ¼ P{x1 , X x2 } FX (x2 ) FX (x1 )

(8:1:10)

FX (x j A) ¼

P{1} ¼0 P{x1 , X x2 }

(8:1:11)

(c) x x1:

The corresponding density function is given by 8 0 x x1 > > < fX (x) x1 , x x2 fX (x j A) ¼ > F (x ) FX (x1 ) > : X 2 0 x . x1

Example 8.1.2

(8:1:9)

A random variable has a Gaussian distribution, given by ( " #) 1 1 x4 2 fX (x) ¼ pﬃﬃﬃﬃﬃﬃ exp 2 2 2 2p

We will ﬁnd the conditional distribution FX(x j A) and the conditional density fX(x j A) given that (1) A ¼ fX 4.5g and (2) A ¼ f1.5 , X 4.5g. Using Eq. (8.1.6), we obtain 1. A ¼ fX 4.5g:

FX (x) 1 1 x4 p ﬃﬃ ﬃ 1 þ erf FX (x j A) ¼ ¼ 9 9 2 2 2 FX FX 2 2

and using Eq. (8.1.10), we obtain 2. A ¼ f 1.5 , X 4.5 g:

1 x 4) 3 p ﬃﬃ ﬃ 1 þ erf FX FX (x) FX (x1 ) 2 2 (2 2 ¼ FX (x j A) ¼ 9 3 FX (x2 ) FX (x1 ) FX FX 2 2

8.1 Conditional Distribution and Density for P{A}= 0

125

FIGURE 8.1.2

where 2 erf(x) ¼ pﬃﬃﬃﬃ p

ðx

2

ej dj

0

The corresponding density functions are as follows for (1) ( " 2 #) 1 1 x4 pﬃﬃﬃﬃﬃﬃ exp 2 2 2 2p fX (x j A) ¼ 9 FX 2 and for (2)

( " #) 1 1 x4 2 pﬃﬃﬃﬃﬃﬃ exp 2 2 2 2p fX (x j A) ¼ 9 3 FX FX 2 2

The conditional distribution and densities FX(x j A) and fX(x j A) along with the unconditional distributions and densities FX(x) and fX(x) are shown in Fig. 8.1.2 for (1) A ¼ fX 4.5g and in Fig. 8.1.3 for (2) A ¼ f1.5 , X 4.5g.

FIGURE 8.1.3

126

Conditional Densities and Distributions

B 8.2 CONDITIONAL DISTRIBUTION AND DENSITY FOR P{A} 5 0 In Section 8.1 we assumed that the probability of the conditioning event A is not zero. We will now determine the conditional probabilities when PfAg ¼ 0. We need to ﬁnd PfB j X ¼ x1g. Since PfX ¼ x1g ¼ 0 for a continuous random variable, we have to use limiting forms to determine this probability. We will write PfB j X ¼ x1g as P{B j X ¼ x1 } ¼ lim P{B j x1 , X x1 þ Dx} Dx!0

(8:2:1)

The conditional probability PfB j x1 , x x1 þ Dxg in Eq. (8.2.1) can be written as follows: P{B j x1 , X x1 þ Dx} ¼

P{B, x1 , X x1 þ Dx} P{x1 , X x1 þ Dx}

¼

P{x1 , X x1 þ Dx j B}P{B} FX (x1 þ Dx) FX (x1 )

¼

FX {x1 þ Dx j B} FX {x1 j B}P{B} FX (x1 þ Dx) FX (x1 )

(8:2:2)

Dividing numerator and denominator of Eq. (8.2.2) by Dx and taking the limit as Dx ! 1 results in P{B j X ¼ x1 } ¼ lim

Dx!0

¼

{FX {x1 þ Dx j B} FX {x1 j B}P{B}g=Dx {FX (x1 þ Dx) FX (x1 )}=Dx

fX (x1 j B)P{B} fX (x1 )

(8:2:3)

Case of B: fY 5 yg

In an analogous manner we can ﬁnd an expression for the conditional probability PfB j X ¼ xg when B is the event B:fY ¼ y). In this case we can write P{Y ¼ y j X ¼ x} ¼ lim P{y , Y y þ Dy j x , X x þ Dx} Dx!0 Dy!0

¼ lim

Dx!0 Dy!0

P{y , Y y þ Dy, x , X x þ Dx} P{x , X x þ Dx}

Dividing the numerator and denominator of Eq. (8.2.4) by Dx Dy, we have, lim P{y , Y y þ Dy j x , X x þ Dx}

Dx!0 Dy!0

½P{y , Y y þ Dy j x , X x þ Dx}=DxDy Dx!0 ½P{x , X x þ Dx=DxDy

¼ lim

Dy!0

¼

fXY (x,y) Dy fX (x)

(8:2:4)

8.2 Conditional Distribution and Density for P{A} 5 0

127

Hence

lim

Dx!0 Dy!0

P{y , Y y þ Dy j x , X x þ Dx} ¼ fYjX ( y j X ¼ x) Dy ¼ fYjx ( y j x) ¼

fXY (x,y) fX (x)

(8:2:5)

The conditional distribution FYjX( y j X xg ¼ PfY y j X xg can be obtained from the basic deﬁnition of conditional probability as follows:

FYjX ( y j X x) ¼ P{Y y j X x} ¼

P{Y y, X xg FXY (x, y) ¼ P{X x} FX (x)

(8:2:6)

Even though Eq. (8.2.5) is of exactly the same form as Eq. (8.2.6), it cannot be obtained from the deﬁnition of conditional probability but must be obtained from limiting considerations.

Point of Clarity

Since fYjX( y j x) has been deﬁned as fYjX( y j X ¼ x), it is not the derivative with respect to y of the distribution FYjX ( y j x), which has been deﬁned as FYjX( y j X x). Bearing this in mind, we can write the following equations for clarity: dFYjX ( y j X x) dFyjX ( y j x) ¼ ¼ fYjX ( y j X x) dy dy (8:2:7) dFYjX ( y j X ¼ x) ¼ fYjX ( y j X ¼ x) ¼ fYjX ( y j x) dy Example 8.2.1 The probability that a machine fails in x months is given by (1 2 e 2x/d) u(x). If the machine failed sometime before c months, it is desired to ﬁnd the conditional distribution FX(x j X c) and the corresponding conditional density. Here we can use the deﬁnition of conditional distribution and write for FX(x j X c): FX (x j X c) ¼ P{X x j X c} ¼

¼

P{X x, X c} P{X c}

8 1 ex=d > > > < 1 ec=d

0,xc

> > > :

x.c x0

1 0

The corresponding density function for the event fX cg is dex=d fX (x j X c) ¼ 1 ec=d 0

0,xc otherwise

128

Conditional Densities and Distributions

FIGURE 8.2.1

We can now ﬁnd the conditional distribution FX(x j X ¼ c) if it failed at x ¼ c months. Since PfX ¼ cg ¼ 0, this has to be solved using limiting considerations: FX (x j X ¼ c) ¼ lim P{X x j c , X c þ Dc} Dc!0

¼ lim

Dc!0

P{X x, c , X c þ Dc} P{c , X c þ Dc}

P{c , X c þ Dc j X x}P{X x}=Dc Dc!0 P{c , X c þ D c}=Dc

¼ lim

¼ fX (c j X x)

FX (x) fX (c, X x) FX (x) ¼ fX (c) FX (x) fX (c)

Hence FX (x j X ¼ c) ¼

1 x.c 0 xc

and the corresponding density function for the event fX ¼ cg is given by fX (x j X ¼ c) ¼ d(x c) These distribution and density functions are shown in Fig. 8.2.1 for c ¼ 2, d ¼ 2. Example 8.2.2

The joint density function of random variables X and Y is (1 fXY (x, y) ¼

p 0

x2 þ y2 1 otherwise

The marginal density function fX(x) is obtained by integrating fXY(x,y) over all y: ﬃﬃﬃﬃﬃﬃﬃ2ﬃ ð p1x

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dy 2 1 x2 fX (x) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ ¼ p 1x2 p

8.2 Conditional Distribution and Density for P{A} 5 0

129

FIGURE 8.2.2

From Eq. (8.2.5), the conditional density fYjX (y j X ¼ x) is given by 8 1 < pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fXY (x, y) 1=p ﬃ x2 þ y2 1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 2 1 x2 fYjX ( y j X ¼ x) ¼ fX (x) 2 1 x2 =p : 0 otherwise Note that for each value of x the pdf fYjX (y j X ¼ x) is a constant and not a function of y. The corresponding conditional pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ distribution function FYjX (y j X ¼ x) is obtained by integrating the pdf from 1 x2 to y as shown below: ðy 1 y 1 FYjX (y j X ¼ x) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dh ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ 2 2 2 2 2 1x 1x 2 1 x In Fig. 8.2.2, the conditional pdf fYjX (y j X ¼ x) is shown as a function of x and the conditional cdf FYjX (y j X ¼ x) is shown as a function of y for x = 0.5. Example 8.2.3 This is an interesting example that again illustrates ﬁnding the conditional probability when the probability of the conditioning event is zero. Two friends A and B plan to meet between 7:00 and 8:00 p.m. However, they agree that each of them will wait only 15 min for the other. Each of them arrives independently at a random time between 7 and 8 p.m. We have to compute the probability that they will meet given that A arrives at 7:20 p.m. Let us ﬁrst deﬁne these random variables. X ¼ {time of arrival of A after 7:00 p.m. in minutes} Y ¼ {time of arrival of B after 7:00 p.m. in minutes} The pdf fX(x) and fY( y) are both uniformly distributed in (0, 60] as shown in Fig. 8.2.3. Since X and Y are independent, the joint density fXY(x,y) ¼ fX(x) . fY( y) and is given by 1 1 1 60 ¼ 3600 : The 60 60 Cartesian product space is shown in Fig. 8.2.4. fXY(x,y) ¼ 60

FIGURE 8.2.3

130

Conditional Densities and Distributions

FIGURE 8.2.4

If A comes at time x minutes after 7:00 p.m., then for the meeting to take place, we have to ﬁnd the probability of the event j Y 2 X j 15 under the condition X ¼ x. Hence, we can write PfY 2 X j 15 j X ¼ xg ¼ Pf215 þ X Y 15 þ X j X ¼ xg, and we have to ﬁnd the area under the shaded portion in Fig. 8.2.4. In essence, we have to integrate fY( y) in three different regions: (1) 0 , y x þ 15, 0 , x ¼ 15, (2) x 2 15 , y x þ 15, 15 , x 45, and (3) x þ 15 , y 60, 45 , x 60. Performing the indicated integration, we obtain 8 ð 15þx > dx > > 0 , x , 15 > > 60 > 0 > > ð < 15þx dx P{A and B meeting j X ¼ x} ¼ 15 , x 45 > 60 > > ð15þx > 60 > > dx > > 45 , x 60 : 60 15þx 8 15 þ x > > 0 , x 15 > > 60 > > < 1 ¼ 15 , x 45 > 2 > > > > > : 75 x 45 , x 60 60 The probability that A and B meet if A arrives exactly x minutes after 7:00 p.m. is shown in Fig. 8.2.5. From the curve, the Pfmeet j X ¼ 20g ¼ 0.5. The total probability of A and B meeting is given by ð 60 P{meet} ¼

P{meet j x ¼ x}fX (x) 0

¼

1 60 30 15 105 7 þ þ ¼ ¼ ¼ 0:4375 60 4 4 4 4:60 16

8.3 Total Probability and Bayes’ Theorem for Densities

131

FIGURE 8.2.5

B 8.3 TOTAL PROBABILITY AND BAYES’ THEOREM FOR DENSITIES We will use Eq. (8.2.3), shown in Eq. (8.3.1) as the starting point to derive the total probability law for density functions: P{B j X ¼ x1 } ¼

fX (x1 j B)P{B} fX (x1 )

(8:3:1)

Cross-multiplying and integrating both sides of the resulting equation, we obtain ð1

ð1 P{B j X ¼ x1 }fX (x1 )dx1 ¼

1

fX (x1 j B) P{B}dx1

(8:3:2)

1

In Eq. (8.3.2) we obtain the term ð1

ð1 fX (x1 j B)P{B}dx1 ¼ P{B}

1

fX (x1 j B)dx1 ¼ P{B} 1

and hence Eq. (8.3.2) becomes ð1 P{B j X ¼ x}fX (x)dx ¼ P{B}

(8:3:3)

1

where the subscript on x has been omitted without loss of generality. Equation (8.3.3) is the total probability law for density functions. We can now formulate Bayes’ theorem for density functions. From Eq. (8.3.1) we can write fX (x j B) ¼

P{B j X ¼ x}fX (x) P{B} P{B j X ¼ x}fX (x) 1 P{B j X ¼ x}fX (x)dx

¼ Ð1

(8:3:4)

132

Conditional Densities and Distributions

where fX(x j B) is the a posteriori pdf after observing the event B given the a priori pdf fX(x). If B is the event B ¼ {Y ¼ y}, then Eq. (8.3.4) can be written as fYjX (y j x)fX (x) 1 fYjX (y j x)fX (x)dx

fXjY (x j y) ¼ Ð 1

(8:3:5)

Equation (8.3.5) is Bayes’ theorem for connecting the a priori density fX(x) with the a posteriori density fXj Y (x j y) after observing the event fY ¼ yg.

Example 8.3.1 Let X be the random variable that a navigation system fails before x hours. Three suppliers A, B, and C have supplied the system to airlines with probabilities PfAg ¼ 0.35, PfBg ¼ 0.25, PfCg ¼ 0.4. The conditional probabilities of failure for the suppliers A, B, C are as follows: FX (x j A) ¼ (1 eax )u(x), a ¼ 1:625 104 FX (x j B) ¼ (1 ebx )u(x), b ¼ 5:108 104 FX (x j C) ¼ (1 ecx )u(x),

c ¼ 2:877 104

1. From the conditional probabilities of failure, we can ﬁnd the probability density fX(x) as follows: fX (x j A) ¼ aeax u(x),

a ¼ 1:625 104 : P{A} ¼ 0:35

fX (x j B) ¼ bebx u(x),

b ¼ 5:108 104 : P{B} ¼ 0:25

fX (x j C) ¼ cebx u(x), c ¼ 2:877 104 : P{C} ¼ 0:40 and fX (x) ¼ fX (x j A)P{A} þ fX (x j B)P{B} þ fX (x j C)P{C} ¼ (0:56875eax þ 1:277ebx þ 1:1508ecx )104 u(x) 2. If the navigation system fails after x hours, we will ﬁnd the probabilities PfA j X . xg, PfB j X . xg, PfC j X . xg. P{X . x} is given by P{X . x} ¼ (1 FX (x j A)ÞP{A} þ (1 FX (x j B))P{B} þ (1 FX (x j C))P{C} ¼ eax 0:35 þ ebx 0:25 þ ecx 0:40 The probability PfA j X . xg can be found from the equation

P{A j X . x} ¼

P{X . x j A}P{A} (1 FX (x j A))P{A} ¼ P{X . x} P{X . x}

8.3 Total Probability and Bayes’ Theorem for Densities

133

with similar equations for PfB j X . xg and PfC j X . xg. Hence these probabilities can be given by P{A j X . x} ¼

P{B j X . x} ¼

P{C j X . x} ¼

eax 0:35

eax 0:35 þ ebx 0:25 þ ecx 0:40

eax 0:35

ebx 0:25 þ ebx 0:25 þ ecx 0:40

ecx 0:40 eax 0:35 þ ebx 0:25 þ ecx 0:40

3. If the navigation system fails exactly at time X ¼ x, we will ﬁnd PfA j X ¼ xg, P{B j X ¼ x}; and P{C j X ¼ x}. From Eq. (8.2.3) we have P{A j X ¼ x} ¼

fX (x j A)P{A} 0:56875 104 eax ¼ fX (x) fX (x)

P{B j X ¼ x} ¼

fX (x j A)P{B} 1:277 104 ebx ¼ fX (x) fX (x)

P{C j X ¼ x} ¼

fX (x j A)P{C} 1:1508 104 ecx ¼ fX (x) fX (x)

4. We will ﬁnd the most likely supplier in part (3) if x ranges from 1000 to 10,000 and the total probability of a wrong decision Pe. The most likely supplier if the system failed at exactly x hours can be obtained by drawing the graphs of the probabilities PfA j X ¼ xg, PfB j X ¼ xg, and PfC j X ¼ xg as shown in Fig. 8.3.1. From the graph in Fig. 8.3.1 we can see that if the system fails between x ¼ 1000 and x ¼ 5629.17 h, the most likely supplier is C, and if it fails between x ¼ 5629.17 and

FIGURE 8.3.1

134

Conditional Densities and Distributions

10,000 h, the most likely supplier is A. The conditional probability PfB j X ¼ xg is smaller than the other two conditional probabilities in the two ranges. The probability of wrong decision Pe is given by Pe ¼ P{X 5629:17 j A}P{A} þ P{X . 5629:17 j C}P{C} þ P{B} Since supplier B does not ﬁgure in the wrong decision, PfBg appears without any multiplier. Hence P{wrong decision} ¼ Pe ¼ ½1 e5629:17a 0:35 þ ½1 (1 e5629:17c ) 0:4 þ 0:25 ¼ 0:539

CHAPTER

9

Joint Densities and Distributions

B 9.1

JOINT DISCRETE DISTRIBUTION FUNCTIONS

If the random variables fX ¼ X1, X2, . . . , Xng, are discrete, then we can deﬁne a joint probability mass function as pX1 Xn (x1 , . . . , xn ) ¼ P{X1 ¼ x1 , . . . , Xn ¼ xn }

(9:1:1)

and the total sum n X m

n X n X j

pX1 Xn (xi , . . . , xm ) ¼ 1

i

In addition, the marginal distributions are deﬁned by n X

pX1 Xn (x1 , . . . , xk ) ¼ pX1 Xn1 (x1 , . . . , xn1 )

(9:1:2)

k

Example 9.1.1 (Discrete) An urn contains 4 red, 5 green, and 6 blue balls, and 3 balls are taken at random. Given the number of red balls represented by the random variable X and the number of green balls by the random variable Y, we want to ﬁnd the joint probability pXY (k,m) of PfX ¼ k, Y ¼ mg, where k ranges from 0 to 4 and m ranges from 0 to 5. The total number of points in the sample space is n! 15! n C(n,r) ¼ ¼ ¼ ¼ 455 r r!(n r)! 3!12! The number of ways in which 3 balls can be picked from 15 balls is to choose k balls from 4 red balls, m balls from 5 green balls, and 3 2 k – m balls from 6 blue balls. Hence, the probability pXY (k,m) is given by C(6, 3 k m)C(4, k)C(5, m) k þ m 3 pXY (k,m) ¼ 0 otherwise

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

135

136

Joint Densities and Distributions

FIGURE 9.1.1

These probabilities are shown in Fig. 9.1.1 2 p(0,0) 1 6 p(1,0) 6 PXY ¼ C(15,3) 4 p(2,0) p(3,0)

for all values of k and m. 3 p(0,1) p(0,2) p(0,3) p(1,1) p(1,2) p(1,3) 7 7 p(2,1) p(2,2) p(2,3) 5 p(3,1) p(3,2) p(3,3)

As the probability mass functions PfX ¼ k) and PfY ¼ m) appear as sums in the margin, they are referred to as marginal probability mass functions. Thus, the probability of 2 red balls PfX ¼ 2g ¼ 66/455 and the probability of 3 green balls PfY ¼ 3g ¼ 10/455.

B 9.2

JOINT CONTINUOUS DISTRIBUTION FUNCTIONS

The joint distribution of n random variables fX1, X2, . . . , Xng, is the probability of the joint event fX1 x1g, . . . , Xn xn and is given by FX1 Xn (x1 , . . . , xn ) ¼ P{X1 x1 , . . . , Xn xn }

(9:2:1)

In particular, if X and Y are two random variables, the joint distribution function FXY (x,y) is given by FXY (x,y) ¼ P{X x, Y y}

(9:2:2)

The corresponding density function is given by fXY (x,y) ¼

@2 FXY (x,y) @x@y

(9:2:3)

Using the deﬁnition of the second derivative, Eq. (9.2.3) can also be given in terms of the distribution function as follows: FXY (x þ D x, y þ Dy) þ FXY (x,y) FXY (x þ D x,y) þ FXY (x,y þ Dy) lim fXY (x,y) ¼ D x!0 Dy!0 D xDy

(9:2:4)

Similarly, the distribution function FXY (x,y) can also be given in terms of the density function by ðx ðy fXY (j,h)dhdj (9:2:5) FXY (x,y) ¼ 1 1

9.2

Joint Continuous Distribution Functions

137

The marginal density functions fX(x) and fY( y) are obtained by integrating the joint density function FXY (x,y) over all y and x, respectively, giving ð1 fX (x) ¼

ð1 fXY (x,y)dy;

fY ( y) ¼

1

fXY (x,y)dx

(9:2:6)

1

Properties of Joint Densities and Distributions 1. FXY (1, 1) ¼ 0; FXY (1,1) ¼ 1 2. FXY (1,y) ¼ 0; FXY (x,1) ¼ 0 3. FXY (x,1) ¼ FX (x); FXY (1,y) ¼ FY (y) 4. P{x1 , X x2 , Y y} ¼ FXY (x2 ,y) FXY (x1 ,y) P{X x, y1 , Y y2 } ¼ FXY (x,y2 ) FXY (x,y1 ) 5. P{x1 , X x2 , y1 , Y y2 } ¼ FXY (x2 ,y2 ) FXY (x2 ,y1 ) FXY (x1 ,y2 ) þ FXY (x1 ,y1 ) 6. @FXY (x,y) ¼ @x @FXY (x,y) ¼ @y

(9:2:7)

(9:2:8)

ðy fXY (x,h)dh 1 ðx

(9:2:9) fXY (j,y)dj

1

In many problems we are given fXY (x,y) in the region {(x1,x2], ( y1,y2]} and the distribution function FXY (x,y) is desired in the entire XY plane. In these cases, we can visualize the XY plane and delineate the various regions as shown in Fig. 9.2.1 for which FXY (x,y) can be determined. Given FXY (x,y) in region I, we ﬁnd that using the properties enunciated earlier: FXY (x,y) ¼ 0 in region V, FXY (x,y) ¼ FX (x) in region II, FXY (x,y) ¼ FY ( y) in region III, and FXY (x,y) ¼ 1 in region IV. We shall give three different examples to illustrate how the joint distribution function can be determined.

FIGURE 9.2.1

138

Joint Densities and Distributions

Example 9.2.1 (Regions Bounded by Constants) The joint pdf fXY (x,y) is given by K(8 x y) 1 , x 3, 1 , y 2 fXY (x,y) ¼ 0 otherwise The constant K can be found by integrating over the range, yielding ð3 ð2 1 K(8 x y)dy dx ¼ 9K ¼ 1 or K ¼ 9 1 1 We plot the regions in Fig. 9.2.2 as shown in Fig. 9.2.1.

Region V: x 1 or y 1 FXY (x,y) ¼ 0 Region IV: 2x . 3, y . 2 FXY (x,y) ¼ 1 Region I: 1 , x 3, 1 , y 2 y ðx 1 1 h2 (8 j h)dh dj ¼ 8h jh 2 1 1 19 19 ðx 1 y2 15 8y jy þ j ¼ dj 2 2 19 x 1 j2 y jy2 j2 15j þ 8yj ¼ 2 2 9 2 2 1 1 j2 y jy2 j2 15j þ ¼ 8yj 2 2 9 2 2 2 2 2 2 1 x y xy x y 15x 15y ¼ þ þ 8xy þ7 2 2 2 9 2 2 2 ðx ðy

FIGURE 9.2.2

9.2

Joint Continuous Distribution Functions

139

Region II: 1 , x 3, y . 2 [marginal distribution FX(x)] 1 x2 13x 6 þ FXY (x,2) ¼ FX (x) ¼ 9 2 2 Region III: x . 3, 1 , y 2 [marginal distribution FY ( y)] FXY (3,y) ¼ FY ( y) ¼

1 2 y þ 12y 11 9

By differentiating the distribution function, we can write the density function for the various regions as follows: 8 0 > > > >1 > > > > 9 (8 x y) > > > > > 1 13 > > x > > > 9 2 > : 0

x1

or y 1

1 , x 3, 1 y 2 x . 3, 1 y 2 1 , x 3, y . 2 x . 3, y . 2

We can also conclude that since fXY (x,y) = fX (x) fY ( y), the random variables X and Y are not independent. Having determined the joint probability FXY (x,y), we can ﬁnd the probability for other regions such as f1.5 , X 2.5, 1 , Y 1.5g as shown in Fig. 9.2.3. ð 2:5 ð 1:5 P{1:5 , X 2:5, 1 , Y 1:5} ¼ 1:5

1

1 (8 x y)dy dx 9

¼ FXY (2:5, 1:5) FXY (1:5, 1:5) FXY (2:5, 1) þ FXY (1:5, 1) ¼ 0:4167 0:1528 0 0 ¼ 0:2639

FIGURE 9.2.3

140

Joint Densities and Distributions

We can also ﬁnd the conditional probability of Y y given that X x as follows FXY (x,y) FX (x) 1 x2 y xy2 x2 y2 15x 15y þ þ 8xy þ7 2 2 2 9 2 2 2 ¼ 2 1 x 13x þ 6 9 2 2

FY j X (y j X x) ¼

¼

y2 þ xy x 15y þ 14 x 12

At the endpoints x ¼ 3 and y ¼ 2, we have FY j X ( y j X 3) ¼ FY ( y) and FY j X (2 j X 3) ¼ FY ( y) ¼ 1. The corresponding conditional density function fY j X( y j X x) is given by fY j X (y j X x) ¼

x þ 2y 15 , x 12

1,x3 1,y2

We will now ﬁnd the conditional density fY j X ( y j X ¼ x) from Eq. (8.2.5) 1 fXY (x,y) 9 (8 x y) 2(8 x y) ¼ ¼ fY j X (y j X ¼ x) ¼ 1 13 fX (x) 13 2x x 9 2 in the region 1 , x 3, 1 , y 2. The corresponding distribution function FY j X ( y j X ¼ x) in the region 1 , x 3, 1 , y 2 is obtained by integrating the density function and is given by ðy 2(8 x h) dh FY j X (y j X ¼ x) ¼ 13 2x 1 ¼

y2 þ 2xy 2x 16y þ 15 2x 13

We see clearly from this example that FY j X ( y j X x) is different from FY j X ( y j X ¼ x) and the corresponding conditional density functions fY j X ( y j X x) and fY j X ( y j X ¼x) are also different. Note that fY j X ( y j X ¼ x) has been deﬁned as fY j X ( y j x) in Chapter 8. Example 9.2.2 (One Region Bounded by a Function) In this example one of the bounds on the region is a functional inequality. The joint pdf fXY (x,y) of two random variables X and Y is given by (8 (1 þ 3xy) 0 , x y 1 fXY (x,y) ¼ 7 0 otherwise We have to ﬁnd the joint distribution function FXY (x,y) in the entire xy plane. Integrating ﬁrst along the h axis and then along the j axis (Fig. 9.2.4), we have ðx ðy 8 (1 þ 3jh)dh dj FXY (x,y) ¼ 0 j7 ¼

1 3x4 þ 6x2 y2 4x2 þ 8xy , 7

0,xy1

9.2

Joint Continuous Distribution Functions

141

FIGURE 9.2.4

Integrating along the j axis ﬁrst and then along the h axis (Fig. 9.2.4), we have ðx ðh 8 FXY (x,y) ¼ (1 þ 3jh)dj dh 0 0 7 I |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ ﬄ} ðy ðx 8 þ (1 þ 3jh)dj dh x 07 II |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} 1 1 ¼ ð3x4 þ 4x2 Þ þ ð6x4 þ 6x2 y2 8x2 þ 8xyÞ 7 7 1 ¼ ð3x4 þ 6x2 y2 4x2 þ 8xyÞ; 0 , x y 1 7 Example 9.2.3 (Both Regions Bounded by Functions) The joint probability density function of two random variables X and Y is given by 8 2 > 0,x1 > > < 6 k(1 þ 2xy) 4 1 fXY (x,y) ¼ 0 , x2 , y x þ 1 > > 2 > : 0 otherwise The two boundaries are shown in Fig. 9.2.5.

FIGURE 9.2.5

142

Joint Densities and Distributions

The constant k is found by integrating fXY (x,y) over the ranges of deﬁnition as shown below: ð1 ð1 ð 1=2 ð xþ1=2 k(1 þ 2xy)dy dx þ k(1 þ 2xy)dy dx ¼ 1 0

x2

1=2 x2

Solving the above equation we obtain, k

161 ¼1 192

or k ¼

192 161

Method 1: Marginal Distributions by Integrating fXY (x,y). We will now ﬁnd the marginal density fX (x) by integrating fXY (x,y) over all y. In the region 0 , x 12, we have ð xþ1=2 192 (1 þ 2xy)dy fX (x) ¼ 161 x2 48 1 ¼ 2 þ 5x þ 4x3 4x 5 , 0 , x 161 2 In the region 12 , x 1, we have ð1

192 (1 þ 2xy)dy 161 192 ¼ 1 þ x x2 x5 , 161

fX (x) ¼

x2

1 ,x1 2

By integrating fX (x) in these two regions, we can write the distribution function FX(x) as 8 8 1 > > (12x þ 15x2 þ 16x4 4x6 ) 0,x > > 161 2 > < 31 32 1 2 3 6 FX (x) ¼ þ (6x þ 3x 2x x ) ,x1 > 161 161 2 > > > x.1 > :1 0 otherwise Similarly, we integrate fXY (x,y) over all x to obtain the marginal density fY ( y) as shown below: 8 ð pyﬃﬃ 192 1 > > > (1 þ 2xy)dx, 0,y < 161 2 fY ( y) ¼ ð 0pyﬃﬃ > 192 1 > > (1 þ 2xy)dx, ,y1 : 161 2 y1=2 or 8 192 pﬃﬃﬃ > > y y2 > < 161 pﬃﬃﬃ fY ( y) ¼ 16 6 þ 12 y 15y þ 24y2 12y3 > > > 161 : 0

0,y

1 2

1 ,y1 2 otherwise

9.2

Joint Continuous Distribution Functions

143

Integrating fY ( y) in the above regions we obtain the distribution function FY ( y) as shown below: 8 64 3=2 > > 2y þ y3 > > 161 > < 1 16 FY ( y) ¼ þ (6y þ 8y3=2 15y2 þ 8y3 3y4 ) > 7 161 > > > > :1 0

0,y

1 2

1 ,y1 2 y.1 otherwise

fX(x), FX (x), fY ( y) and FY ( y) are shown in Fig. 9.2.6. Method 2: Marginal Distributions from FXY (x,y). We will now ﬁnd the marginal distributions from the joint distribution FXY (x,y) obtained by integrating the joint density fXY (x,y) over the regions of deﬁnition. The joint distribution FXY (x,y) in the region 0 , x 12, 0 , y 12 is obtained from the following integration: ðx ðy FXY (x,y) ¼ 0

x2

192 (1 þ 2jh)dh dj 161

32 ¼ 6xy 2x3 þ 3x2 y2 x6 , 161

0,x1 0 , y 1=2

and in the region 0 , x 1, 12 , y 1, we have ð y(1=2) ð jþ(1=2) FXY (x,y) ¼ 0

ðx

j2

ðy

þ y(1=2) j

2

192 (1 þ 2jh)dh dj 161

192 (1 þ 2jh)dh dj 161

or FXY (x,y) ¼

1 8 þ 12y 15y2 þ 8y3 6y4 7 161 2 2

3

þ 24xy þ 12x y 8x 4x

FIGURE 9.2.6

6

(

0,x,1 1 ,y1 2

144

Joint Densities and Distributions

The marginal distributions FX (x) and FY ( y) can be obtained by substituting suitable boundary values in the joint distribution FXY (x,y) as shown below:

FX (x) ¼

8 1 > > F x,x þ , > XY > > 2 > > > > > < FXY (x,1), > > > FXY (1,1), > > > > > > > : FXY (0,0),

1 1 0,x , y. 2 2 1 1 , x 1, y . 2 2 1 x . 1, y . 2 1 x 0, y . 2

Similarly 8 pﬃﬃﬃ > > FXY ( y,y), > > > < pﬃﬃﬃ FY (y) ¼ FXY ( y,y), > > > > > : FXY (1,1), FXY (0,0),

1 0,y , 0,x1 2 1 , x 1, 0 , x 1 2 y . 1, x . 1 y 0, x . 1

and these are exactly the same as derived previously.

B 9.3

BIVARIATE GAUSSIAN DISTRIBUTIONS

If X and Y are jointly distributed random variables with mean value mX and mY and variances s2X and s2Y, then the joint density function fXY (x,y) is given by ( " 1 1 x mX 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp fXY (x,y) ¼ sX 2(1 r2 ) 2psX sY 1 r2 #) 2r(x mX )(y mY ) y mY 2 (9:3:1) þ sY sX s Y where r is the correlation coefﬁcient that determines the degree of similarity between X and Y, given by sXY (9:3:2) r¼ sX sY We have to show that ðð 1 FXY (x,y)dy dx ¼ 1

I¼

(9:3:3)

1

Substituting j ¼ (x 2 mX)/sx and h ¼ ( y 2 mY)/sy in Eq. (9.3.1), we can write the integral I as follows: 1 1 2 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp ½j 2rjh þ h dh dj 2(1 r2 ) 1 2p 1 r2

ðð 1 I¼

(9:3:4)

145

9.3 Bivariate Gaussian Distributions

Completing the squares in j in the integral (9.3.4) above, we have

ðð 1 1 1 (j rh)2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp I¼ þ h dh dj 1 r2 2 1 2p 1 r2

(9:3:5)

Substituting j rh w ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 r2 and dj dw ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 r2 the integral (9.3.5) can be written as follows: 2 2 ð ð1 1 w 1 1 h pﬃﬃﬃﬃﬃﬃ exp I¼ exp dw dh ¼ 1 2 2 2p 1 2p 1

(9:3:6)

If r ¼ 0 in Eq. (9.3.3), then X and Y are independent because fXY (x,y) ¼ fX (x) fY ( y). Thus, if Gaussian-distributed random variables X and Y are uncorrelated, they are also independent. We shall ﬁnd the conditional pdf fY j X (X ¼ x) given X ¼ x. From Eq. (8.2.5) we have fXY (x,y) fX (x) 8 9 1 1 x mX > > > > p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp > > > > 2) 2 > > s 2(1 r X 2ps 1 r s > > X Y > > > >

> > 2 > > > > (2r(x m )( y m ) y m X Y Y > > > > þ = < sX s Y sY ¼ 2 > > 1 1 x mX ) > > > > pﬃﬃﬃﬃﬃﬃ exp > > > > > > 2 s 2p s X > > X > > > > > > > > > > > > ; :

fY j X (y j X ¼ x) ¼

( " 2 ) 1 1 rsY ¼ pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp (x mX Þ y mY sX 2s2Y (1 r2 ) 2psY 1 r2 Thus the conditional density has a conditional mean value of rsY (x mX ) mY j X ¼ mY þ sX and a conditional variance of s2Y j X ¼ s2Y (1 r2 ):

CHAPTER

10

Moments and Conditional Moments

B 10.1

EXPECTATIONS

The expected value or the mean value of any random variable X is deﬁned by ð1 E ½X ¼ mX ¼ x fX (x) dx (10:1:1) 1

This deﬁnition is valid regardless of whether the random variable is continuous or discrete. Since the pdf of a discrete random variable is always given by a sum of delta functions, we have ) ð 1 (X 1 x P{X ¼ xi }d(x xi ) dx E½X ¼ mX ¼ 1

¼

1 X

i¼0

xi P{X ¼ xi }

(10:1:2)

i¼0

The expected value is the ﬁrst moment of the random variable X about the origin. Example 10.1.1

For a random variable uniformly distributed in the interval (a,b], we have fX (x) ¼

1 ba

and hence the mean value is given by

E½X ¼

1 ba

ðb x dx ¼ a

b 1 x2 1 b2 a2 b þ a ¼ ¼ b a 2 a 2 b a 2

Example 10.1.2 On a toss of a die, a person wins $10 if {1} or {3} results, loses $5 if {4} or {6} results, wins $5 if {2} results, and loses $10 if {5} results. From Eq. (10.1.2), Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

146

10.1 Expectations

147

the expected value of the winnings is 6 X

E{X} ¼ mX ¼

xi P{X ¼ xi }

i¼1

1 5 ¼ {10 þ 10 5 5 þ 5 10} ¼ 6 6 The expected value of any random variable is a constant. In other words, it smooths the variations of a random variable to a constant. It is a one-way operator, in the sense that once the random variable X is smoothed, there is no way of getting back the density function. Thus, the expected value of a constant a is the same constant a.

Properties of Expectations 1. It is a linear operator irrespective of the nature of random variables, in the sense that the expected value of the sum equals the sum of expected values E½X þ Y ¼ E½X þ E½Y

(10:1:3)

as shown below: ð1 ð1 (x þ y)fXY (xy)dx dy

E½X þ Y ¼ 1 1 ð1 ð1

ð1 ð1 xfXY (xy)dx dy þ

¼ 1 1 ð1

ð1

xfX (x)dx þ

¼ 1

yfXY (xy)dx dy 1 1

yfY (y)dy ¼ E{X} þ E{Y} 1

2. On the contrary, the expected value of the product given by ð1 ð1 xyfXY (xy)dx dy E½XY ¼

(10:1:4)

1 1

is not, in general, equal to the product of expected values. However, Equation (10.1.4) can be split only if X and Y are independent, in which case we have ð1 ð1 xy fXY (xy)dx dy E½XY ¼ 1 1

ð1

ð1

xfX (x)dx

¼ 1

yfY (y)dy 1

¼ E½X E½Y and the expected value of the product is equal to the product of expected values. The converse is not true. If E[XY ] ¼ E[X ] E[Y ], then X and Y are uncorrelated and need not be independent. Analogously, if E[XY ] ¼ 0, then X and Y are orthogonal. We can summarize the following deﬁnitions: Uncorrelated: E[XY ] ¼ E[X ] E[Y ] Orthogonal: E[XY ] ¼ 0 Independence: fXY (xy) ¼ fX (x) fY (y)

148

Moments and Conditional Moments

Conditional Expectation

Given that the random variable X has occurred, the expected value of Y is deﬁned by ð1 yfYjX ( y j x)dy (10:1:5) E½Y j X ¼ 1

The conditional expectation operator does not completely smooth the random variable Y to a constant but smooths it to random variable dependent on X. Thus, if we take the expectation of the conditional expectation, we get the expected value of Y: ð 1 ð 1 y fY j X ( y j x) dy fX (x)dx E{E½Y j X} ¼ 1

1

ð1

y fXY (xy) dy dx

¼ 1 ð1

y fY ( y)dy ¼ E½Y

¼

(10:1:6)

1

If X and Y are independent, then E[Y j X ] ¼ E[Y ] since fY j X ( y j x) ¼ fY ( y).

Example 10.1.3 A random variable Y is uniformly distributed in the interval (0,5}. Hence the distribution function FY( y) is given by y/5 for 0 , y 5. We will ﬁnd the conditional expectation of Y given that Y . x and 0 , x 5. We ﬁrst ﬁnd the conditional distribution FY ( y j Y . x): FY ( y j Y . x) ¼

P(x , Y y) FY ( y) FY (x) ¼ P(Y . x) 1 FY (x)

Differentiating this equation with respect to y, the conditional density fY( y j Y . x) is obtained as follows: fY (y) ¼

fY ( y) 1 ¼ 1 FY (x) 5 x

Hence the conditional expectation E[Y j Y . x] is given by 5 y y2 dy ¼ E½Y j Y . x ¼ 2(5 x)x x5 x ð5

¼

25 x2 5þx ¼ 2(5 x) 2

Joint Moment

We can now deﬁne the joint moment between two random variables X and Y as follows: ð1 ð1 E½XY ¼ mXY ¼ xy fXY (x,y) dx dy (10:1:7) 1

1

The joint moments determine the correlation between X and Y.

10.2

B 10.2

Variance

149

VARIANCE

The variance is the second moment about the mean value also known as the second central moment. It is deﬁned by ð1 (x mX )2 fX (x) dx var½X ¼ E½X mX 2 ¼ s2X ¼ 1 2

¼ E½X 2mX E½X þ

m2X

¼ E½X 2 m2X

(10:2:1)

The variance expresses the variability of the random variable about the mean value. If this variability is zero, then the random variable will be no longer random but a constant. This will be discussed further in Chapter 14. The preceding deﬁnition applies for any random variable. However, for a discrete random variable, analogous to Eq. (10.1.2), we have E½X mX 2 ¼ sX2 ¼

1 X

(xi mX )2 P{X ¼ xi )

(10:2:2)

i¼0

Example 10.2.1 For a random variable, X uniformly distributed between (a,a, the mean value mX ¼ 0. Hence the variance is given by ð 1 a 2 a2 sX2 ¼ E½X 2 ¼ x dx ¼ 3 2a a Example 10.2.2 From Eq. (10.2.2), the variance of the random variable given in Example 10.1.2 is 6 X 5 21 xi s2X ¼ 6 6 i¼1 2 1 55 552 252 252 252 552 10,950 þ 2 þ 2 þ 2 þ 2 þ 2 ¼ ¼ 6 6 6 6 6 6 62 216 Properties of Variance 1.

var½aX ¼ a2 var½X ¼ a2 s2X

2.

2

var½X + Y ¼ E½X + Y {E½X þ E½Y}

(10:2:3) 2

¼ E½X 2 {E½X}2 þ E½Y 2 {E½Y}2 + 2{E½XY E½XE½Y} ¼ sX2 þ s2Y + 2{E½XY E½XE½Y}

(10:2:4)

3. If X and Y are independent, then var½X + Y ¼ s2X þ s2Y

(10:2:5)

Conditional Variance. We can also deﬁne a conditional variance of Y given X has occurred as var½Y j X ¼ E½(Y mYjX )2 j X

(10:2:6)

150

Moments and Conditional Moments

Covariance. The covariance between two random variables X and Y is deﬁned by cov½XY ¼ sXY ¼ E ½X mX E ½Y mY ¼ E ½XY mX mY

(10:2:7)

If X and Y are independent, then cov[XY ] ¼ 0. The converse is not true. If cov[XY ] ¼ 0, then X and Y are uncorrelated but not necessarily independent. Correlation Coefﬁcient. The correlation coefﬁcient is the normalized covariance and is deﬁned by cov½XY sXY rXY ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ var½X var½Y sX sY

(10:2:8)

The correlation coefﬁcient satisﬁes 1 rXY 1 and shows the degree of selfsimilarity between X and Y.

B 10.3 MEANS AND VARIANCES OF SOME DISTRIBUTIONS Discrete Random Variables Binomial Distribution

Mean: n k nk b(k; n, p) ¼ pq k E½X ¼

n X k¼0

¼

k

(10:3:1)

n! pk qnk k!(n k)!

n X

n(n 1)! p pk1 qnk (k 1)!(n k)! k¼0

¼ np

n X

(n 1)! pk1 qnk (k 1)!(n k)! k¼1

¼ np( p þ q)n1 ¼ np

(10:3:2)

Variance: var½X ¼ E½X 2 m2X E½X 2 ¼

n X k¼0

k2

n! pk qnk k!(n k)!

(10:3:3) (10:3:4)

Substituting k 2 ¼ k (k 2 1) þ k in Eq. (10.3.4), we have E½X 2 ¼

n X k¼0

k(k 1)

n X n! n! pk qnk þ pk qnk k k!(n k)! k!(n k)! k¼0

(10:3:5)

10.3

Means and Variances of Some Distributions

151

The second summation in Eq. (10.3.5) is the mean value, already calculated as np in Eq. (10.3.2). The ﬁrst summation can be written as follows: n X

k(k 1)

k¼0

n X n! (n 2)! pk qnk ¼ n(n 1) p2 pk2 qnk k!(n k)! (k 2)!(n k)! k¼2

¼ n(n 1) p2 (p þ q)n2 ¼ n(n 1)p2

(10:3:6)

Thus E½X 2 ¼ n (n 1) p2 þ np

(10:3:7)

and the variance from Eq. (10.3.3) is given by var½X ¼ n (n 1) p2 þ np n2 p2 ¼ npq

(10:3:8)

Poisson Distribution

Mean: P(k; l) ¼ el E½X ¼ el

lk k! 1 X klk k¼0

k!

(10:3:9) ¼ lel

1 X lk 1 ¼ l el el ¼ l (k 1)! k¼1

(10:3:10)

Variance. We use the same technique as in the binomial distribution by substituting k 2 ¼ k(k 2 1) þ k. Hence E½X 2 ¼ el

1 X k(k 1)lk k¼0

¼ l2 el

k!

þ el

1 X klk k¼0

k!

1 X lk 2 þl (k 2)! k¼2

¼ l2 el el þ l ¼ l2 þ l

(10:3:11)

Substituting in var½X ¼ E½X 2 m2X from Eqs. (10.3.11) and (10.3.10), we have var ½X ¼ l2 þ l l ¼ l

(10:3:12)

For a Poisson distribution, the mean and the variance are the same and equal to l. Continuous Random Variables Exponential Distribution

Mean: fX (x) ¼ l elx u(x) ð1 E½X ¼ l xelx dx 0

(10:3:13) (10:3:14)

152

Moments and Conditional Moments

Integrating Eq. (10.3.14) by parts, we obtain ð1 ð1 delx ¼ x x delx E½X ¼ l l 0 0 1 ð 1 1 lx ¼ xe þ delx ¼ l 0 0 Variance. We ﬁrst calculate the second moment: ð1 E½X 2 ¼ l x2 elx dx

(10:3:15)

(10:3:16)

0

Integrating by parts [Eq. (10.3.16)] twice, we get the result as ð1 l

x2 elx dx ¼

0

1 elx 2 2 ¼ 2 (l x þ 2lx þ 2) 2 l l2 0

(10:3:17)

2 1 1 2¼ 2 2 l l l

(10:3:18)

and the variance is given by var½X ¼

Gaussian Distribution

Mean: 1 1 (x mX )2 pﬃﬃﬃﬃﬃﬃ exp 2 s2X sX 2p ð1 1 1 (x mX )2 x exp dx E½X ¼ pﬃﬃﬃﬃﬃﬃ 2 s2X sX 2p 1 ð1 1 1 (x mX )2 ¼ pﬃﬃﬃﬃﬃﬃ (x mX ) exp dx 2 s2X sX 2p 1 ð1 m 1 (x mX )2 þ pXﬃﬃﬃﬃﬃﬃ exp dx 2 s2X sX 2p 1

fX (x) ¼

(10:3:19)

(10:3:20)

In Eq. (10.3.20), the ﬁrst integral is zero because it is an odd function and the second integral equals mX. Hence E½X ¼ mX

(10:3:21)

Variance: var½X ¼

1 pﬃﬃﬃﬃﬃﬃ sX 2p

1 (x mX )2 (x mX )2 exp dx 2 s2X 1

ð1

Substituting y ¼ (x mX )=sX in Eq. (10.3.22), we have ð 1 1 2 2 (1=2)y2 sX y e dy ¼ s2X var½X ¼ pﬃﬃﬃﬃﬃﬃ 2p 1 pﬃﬃﬃﬃﬃﬃ Ð 1 2 where it can be shown that 1= 2p 1 y2 e(1=2)y dy ¼ 1

(10:3:22)

(10:3:23)

10.4

B 10.4

Higher-Order Moments

153

HIGHER-ORDER MOMENTS

We can now deﬁne higher-order moments. The kth-order moment is deﬁned by E½X k ¼ mk ¼

ð1

xk fX (x) dx

(10:4:1)

1

Clearly, m0 ¼ 1 and m1 ¼ mX. Similarly, the kth-order central moments are deﬁned by E½X mkX ¼ vk ¼

ð1

(x mkX ) fX (x) dx

(10:4:2)

1

Here v0 ¼ 1, v1¼0, and v2 ¼ s2X. We can ﬁnd relationships between the kth-order moments and the kth-order central moments. We can write these relationships as follows: v0 ¼ 1 v1 ¼ 0 v 2 ¼ s2 v3 ¼ m3 3m1 m2 þ 2m31

m0 ¼ 1 m1 ¼ m m2 ¼ v2 þ m2 m3 ¼ v3 þ 3m1 v2 þ m31

(10:4:3)

The moments of random variables are not arbitrary numbers but must satisfy certain inequalities. For example, since E½X mX 2 must be nonnegative, we have s2 ¼ m2 m21 0

and

m2k m2k 0

for all k

(10:4:4)

A random variable is completely determined if the density function is known. However, knowledge of only two moments gives only a limited information of a random variable. But under certain conditions, knowledge of moments mk for all k will uniquely determine the random variable. However, because only two of the moments of a Gaussian random variable are independent, it can be completely determined with two moments. All other moments are dependent on two moments. We will show that for a Gaussian random variable with density function 1 2 2 fX (x) ¼ pﬃﬃﬃﬃﬃﬃ ex =2s s 2p satisﬁes the following moment equations: 0 k E½X ¼ 1 3 5 (n 1)sn

n ¼ 2k þ 1 for all k n ¼ 2k for all k

(10:4:5)

(10:4:6)

Clearly the odd moments are 0 because fXÐ(x) is an even p function. To show the result for ﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 the even moments, we take the equation 1 ebx dx ¼ p=b and differentiate it k time with respect to b, yielding rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 1 3 5 (2k 1) p 2k bx2 x e dx ¼ (10:4:7) k 2kþ1 2 b 1 and the result is obtained by substituting b ¼ 1/2s2 in Eq. (10.4.7). Thus, the Gaussian random variable is completely determined if only two moments are known.

154

Moments and Conditional Moments

Higher-Order Cross-Moments

The concept of joint moments can be extended to higher-order cases. The k, jth higher-order cross-moment is deﬁned by, ð1 ð1 k j xk y j fXY (x,y) dx dy (10:4:8) E½X , Y ¼ mkj ¼ 1

1

and the k, jth higher order central moment is deﬁned by ð1 ð1 E½(X mX )k , (Y mY ) j ¼ hkj ¼ (x mX )k ( y mY ) j fXY (x,y) dx dy 1

(10:4:9)

1

and m11 ¼ E [XY ] the joint moment and h11 ¼ cov[XY ] ¼ sXY is the covariance.

B 10.5

BIVARIATE GAUSSIAN

Two random variables are jointly Gaussian-distributed if they satisfy the following density function with r as the correlation coefﬁcient: ( " 1 1 x mX 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp fXY (x,y) ¼ sX 2(1 r2 ) 2psX sY 1 r2 #) (x mX )(y my ) y mY 2 þ 2r (10:5:1) sY s X sY Ordinarily, uncorrelatedness does not imply independence, even though independence implies uncorrelatedness. However, in the case of Gaussian-distributed random variables if r ¼ 0, then Eq. (10.5.1) becomes ( " #) 1 1 x mX 2 y mY 2 exp þ (10:5:2) fXY (x,y) ¼ 2psX sY 2 sX sY and clearly fXY (x,y) ¼ fX (x) fY (y) ( ( ) ) 1 1 x mX 2 1 1 x mY 2 pﬃﬃﬃﬃﬃﬃ exp ¼ pﬃﬃﬃﬃﬃﬃ exp sX sY 2 2 2psX 2psY and hence X and Y are independent.

(10:5:3)

CHAPTER

11

Characteristic Functions and Generating Functions

B 11.1

CHARACTERISTIC FUNCTIONS

The characteristic function (CF) of any random variable X (discrete or continuous) is deﬁned by ð1 FX (v) ¼ E{e jvX} ¼ e jvx fX (x) dx (11:1:1) 1

From this deﬁnition it is seen that the characteristic function is the Fourier transform of the pdf fX (x) where the sign of v is positive instead of negative. However, all the properties of the Fourier transform hold good. From the deﬁnition of FX (v), we can see that FX (0) ¼ 1

(11:1:2)

This equation is used to check the correctness of the CF. Existence of CF

The CF is bounded as shown below. Applying the fact that the absolute value of an integral is less than or equal to the integral of its absolute value, we have ð1 ð 1 jvx e fX (x) dx je jvx jj fX (x)j dx 1

and since je

jvx

1

j ¼ 1 we have ð1 ð 1 jvx e f (x) dx je jvx jj fX (x)j dx X 1 1 ð1 j fX (x)jdx ¼ 1 ¼ 1

Thus, the CF always exists.

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

155

156

Characteristic Functions and Generating Functions

We can also obtain the pdf fX(x) from the CF from the inversion formula for the Fourier transform given by ð 1 1 FX (v)ejvx dv (11:1:3) fX (x) ¼ 2p 1 [Comment: Since FX(v) is the Fourier transformation of the pdf fX(x), the random variable X is also completely determined if FX(v) is known. A Fourier transform of a signal characterizes the frequency content of the signal. No such characterization for the CF can be given; it is just a convenient form of representing the pdf and has other very useful properties.] Example 11.1.1

We will ﬁnd the CF of an exponential distribution given by fX (x) ¼ lelx u(x)

Using Eq. (11.1.1), we have ð1 FX (v) ¼

lelx e jvx dx ¼

0

l l jv

Moment Generating Properties of CF

In Chapter 10 we found the moments of a random variable using integration techniques. Once the CF is known, the moments can be obtained by differentiation, which is a simpler process than integration. Expanding the CF as a Taylor series about the origin, we can write ð1 jvx ( jvx)2 (jvx)3 ( jvx)k þ þ þ þ dv (11:1:4) fX (x) 1 þ þ FX (v) ¼ 2! 3! k! 1! 1 Substituting ð1 mk ¼

xk fX (x) dx,

k ¼ 0,1; . . .

(11:1:5)

1

in Eq. (11.1.2), we have jvm1 ( jv)2 m2 ( jv)3 m3 ( jv)k mk þ þ þ þ þ FX (v) ¼ 1 þ 1! 2! 3! k! We now differentiate FX(v) term by term to yield d jm1 2( j)2 vm2 3( j)3 v2 m3 k( j)k vk1 mk þ þ þ þ þ FX (v) ¼ 1! 2! 3! k! dv

(11:1:6)

Substituting v ¼ 0 in Eq. (11.1.6), we get F0X (v)v¼0 ¼ jm1

and

m1 ¼

F0X ð0Þ j

(11:1:7)

Differentiating FX(v) successively k times and substituting in the resulting equations v ¼ 0 we can obtain all the moments of the random variables as shown below: (k) 2 k (11:1:8) F(2) X (v) v¼0 ¼ ( j) m2 , . . . , FX (v) v¼0 ¼ ( j) mk , . . . , We will give more examples in a later section.

11.2

Examples of Characteristic Functions

157

Joint Characteristic Functions

We can also deﬁne a joint characteristic function for two random variables X and Y as FXY (v1 , v2 ) ¼ E e j(v1 Xþv2 Y) ð1 ð1 ¼ fXY (x, y)e j(v1 xþv2 y) dx dy

(11:1:9)

1 1

B 11.2

EXAMPLES OF CHARACTERISTIC FUNCTIONS

We will now give several examples of characteristic functions for both discrete and continuous random variables and determining their means and variances. Discrete Random Variables

Example 11.2.1 (Binomial Distribution) n k nk b(k: n,p) ¼ pq k

(11:2:1)

The CF is given by n X n k nk jvk pq e k¼0 k n X n pe jvk qnk ¼ ( pe jv þ q)n ¼ k k¼0

FX (v) ¼

(11:2:2)

We ﬁnd that FX(0) ¼ ( p þ q) ¼ 1 Mean: F0X (v) ¼ n( pe jv þ q)n1 jpe jv F0X (0) ¼ jnp

or E{X ¼ k} ¼ m1 ¼ np

(11:2:3)

Variance. We ﬁrst ﬁnd EfX 2 ¼ k 2g ¼ m2: F00X (v) ¼ n(n 1)( pe jv þ q)n2 ( jpe jv )2 þ n( pe jv þ q)n1 j2 pe jv F00X (0) ¼ n(n 1)p2 np and

m2 ¼ (np)2 þ npq

Hence var½X ¼ m2 m21 ¼ npq

(11:2:4)

158

Characteristic Functions and Generating Functions

Example 11.2.2 (Poisson Distribution) p(k; l) ¼ el FX (v) ¼

1 X

lk k! el

lk jvk e k!

el

le jv jv jv ¼ el ele ¼ el(1e Þ k!

k¼0

¼

1 X

(11:2:5)

k

k¼0

(11:2:6)

and FX(0) ¼ 1. Mean: jv

F0X (v) ¼ el(1e ) l(e jv )j F0X (0) ¼ lj

(11:2:7)

or E{X ¼ k} ¼ m1 ¼ l

Variance. We ﬁrst ﬁnd EfX 2 ¼ k2g ¼ m2: jv

jv

F00X (v) ¼ el(1e ) l(e jv )j l(e jv )j el(1e ) lj(e jv )j F00X (0) ¼ l2 l or

m2 ¼ l2 þ l

Hence var{X} ¼ m2 m21 ¼ l

(11:2:8)

Example 11.2.3 (Geometric Distribution) p{k : p} ¼ P{k failures followed by a success} ¼ qk p FX (v) ¼

1 X

pqk e jvk ¼ p

k¼0

1 X

(11:2:9)

(qe jv )k

k¼0

¼

p 1 qe jv

(11:2:10)

and FX(0) ¼ 1. Mean: F0X (v) ¼

p jpqe jv jv (jqe ) ¼ (1 qe jv )2 (1 qe jv )2

F0X (0) ¼

jpq p2

or E{x ¼ k} ¼ m1 ¼

q p

(11:2:11)

Variance. We ﬁrst ﬁnd EfX 2 ¼ k 2g ¼ m2. F00X (v) ¼ F00X (0)

2jpqe jv j2 pqe jv jv (jqe ) þ (1 qe jv )3 (1 qe jv )2

2pq2 pq ¼ 2 p3 p

2q2 q or m2 ¼ 2 þ p p

(11:2:12)

11.2

Examples of Characteristic Functions

159

Hence variance is m2 m2X ¼

2q2 q q2 q(p þ q) q þ 2¼ ¼ 2 p2 p p p2 p

(11:2:13)

Continuous Random Variables

Example 11.2.4 (Uniform Distribution) 1 ½u(x b) u(x a) ba ð b jvx e e jvb e jva dx ¼ FX (v) ¼ jv(b a) aba fX (x) ¼

(11:2:14) (11:2:15)

This is one of the more difﬁcult CFs to obtain the mean and the variance by differentiating Eq. (11.2.15) because of the limiting processes involved. Instead, we expand Eq. (11.2.15) using Taylor series and differentiate the resulting expression. Expansion of Eq. (11.2.15) results in 1 ( jv)2 2 ( jv)3 3 FX (v) ¼ (b a2 ) þ (b a3 ) þ jv(b a) þ 2! 3! jv(b a) jv ( jv)2 2 (b þ ab þ a2 ) þ (11:2:16) (b þ a) þ ¼ 1þ 3! 2! Mean: F0X (v)

j 2j2 v 2 2 ¼ (b þ a) þ (b þ ab þ a ) þ 2! 3!

F0X (0)

j(b þ a) ¼ 2

(11:2:17)

(b þ a) or m1 ¼ 2

Variance: F00X (v) ¼ F00X (0) ¼

2j2 2 (b þ ab þ a2 ) þ 3!

(b2 þ ab þ a2 ) 3

or

m2 ¼

b2 þ ab þ a2 3

Hence var½X ¼ m2 m21 ¼

(b a)2 12

(11:2:18)

In most cases it is easier to ﬁnd the mean and variance using CF. However, in this case it is easier to obtain mean and variance by integration techniques using their deﬁnitions.

Example 11.2.5 (Exponential Distribution) fX (x) ¼ lelx u(x)

(11:2:19)

160

Characteristic Functions and Generating Functions

From Example 11.1.1 the CF is ð1 FX (v) ¼

lelx e jvx dx ¼

0

l l jv

(11:2:20)

Mean: F0X (v) ¼ F0X (0)

jl (l jv)2

j ¼ l

(11:2:21)

1 m1 ¼ l

or

Variance: F00X (v) ¼

2l (l jv)3

F00X (0) ¼

2 l2

or

m2 ¼

(11:2:22)

2 l2

Hence var½X ¼ m2 m21 ¼

1 l2

(11:2:23)

Example 11.2.6 (Gaussian Distribution) 2 1 2 fX (x) ¼ pﬃﬃﬃﬃﬃﬃ e(xm) =2s 2p s ð1 2 1 2 p ﬃﬃﬃﬃﬃﬃ e(xm) =2s þjvx dx FX (v) ¼ 2p s 1

(11:2:24) (11:2:25)

Completing the squares in the exponent of Eq. (11.2.25), we obtain the resulting expression as follows: ð1 1 (x m jvs2 )2 jvm½(s2 v2 )=2 pﬃﬃﬃﬃﬃﬃ FX (v) ¼ e exp dx 2s2 2p s 1 The value of the integral within braces is 1 and hence the CF of a Gaussian is 2

FX (v) ¼ e jvm½(s v

2

)=2

(11:2:26)

Mean: 2

2

F0X (v) ¼ e jvm½(s v )=2 ( jm s2 v) F0X (0) ¼ jm or m1 ¼ m

(11:2:27)

Variance: 2

F00X (v) ¼ e jvm½ðs v F00X (0)

2

¼ m s

2

2

Þ=2

( jm s2 v)2 e jvm½(s

or

2

m2 ¼ m þ s

2

2

v2 )=2 2

s

11.3 Generating Functions

161

Hence var½X ¼ m2 m21 ¼ s2

B 11.3

(11:2:28)

GENERATING FUNCTIONS

Certain simpliﬁcations are possible in the deﬁnition of the characteristic functions for discrete random variables taking positive integral values. This is called the generating function (GF) and is deﬁned for a discrete random variable X taking positive discrete values as PX (z) ¼ E{zk } ¼

1 X

zk P{X ¼ k}

(11:3:1)

k¼0

with PX(1) ¼ 1. The inﬁnite series of Eq. (11.3.1) will converge for values of jzj , 1. Equation (11.3.1) can be recognized as the z transform of discrete data with the sign reversal for z. Example 11.3.1 Six dice are tossed. Using the generating function, we want to ﬁnd the probability that the sum of the faces of the dice add to 24. The GF for the toss of a single die is 1 PX (z) ¼ ½z þ z2 þ z3 þ z4 þ z5 þ z6 6 and in closed form this can be written as z 1 z6 PX (z) ¼ 6 1z Since the tosses are independent, the GF for the toss of 6 dice is [PX(z)]6 given by QX (z) ¼ ½PX (z)6 ¼

z 6 1 z6 6 6

1z

To ﬁnd Pffaces add up to 24g, we have to ﬁrst ﬁnd the coefﬁcient of z 24 in the power series expansion of Q(z). Alternately, we have to ﬁnd the coefﬁcient of z 18 in the expansion of RX (z) ¼

(1 z6 )6 (1 z)6

The expansion of the numerator of RX(z) is (1 z)6 ¼ 1 6z6 þ 15z12 20z18 þ 15z24 6z30 þ z36 and the expansion of the denominator of RX(z) is given by the negative exponential (1 z)6 ¼

1 X (5 þ k 1)! k¼1

5!(k 1)!

zk1

162

Characteristic Functions and Generating Functions

We now multiply the two expansions above and ﬁnd the coefﬁcient z 18 in the resulting product. The terms containing z 18 are (5 þ 7 1) (5 þ 13 1)! (5 þ 19 1)! z18 20 þ 15 6 þ1 5!6! 5!12! 5!18! ¼ 3451z18 Hence the probability that the sum of the faces add to 24 is P{sum ¼ 24} ¼

3451 ¼ 0:073967 66

a result that would have been very difﬁcult to obtain without generating functions.

Moment Generating Properties of GF

It is easier to use the GF in ﬁnding the moments of a discrete random variable taking positive integral values than ﬁnding it through CF. We can differentiate Eq. (11.3.1) and obtain PX (z) ¼

1 X

zk P{X ¼ k}

k¼0

P0X (z) ¼

1 X

kzk1 P{X ¼ k}

(11:3:2)

k¼0

P0X (1) ¼

1 X

kP{X ¼ k} ¼ m1

k¼0

Differentiating PX0 (z) again, we obtain P00X (z) ¼

1 X

k(k 1)P{X ¼ k}

k¼0

¼

1 X

k2 P{X ¼ k}

k¼0

1 X

kP{X ¼ k}

(11:3:3)

k¼0

P00X ð1Þ ¼ m2 m1 Hence the variance is given by

2 var½X ¼ P00X (1) þ P0X (1) P0X (1)

(11:3:4)

Unlike the CF, there is no general formula for the kth moment in the case of GF. In particular P000 X (1) ¼ m3 3m2 þ 2m1

B 11.4

(11:3:5)

EXAMPLES OF GENERATING FUNCTIONS

We will now derive generating functions for all discrete random variables discussed in Section 11.2 and determine their means and variances.

11.4 Examples of Generating Functions

163

Example 11.4.1 (Binomial Distribution) b(k: n,p) ¼

n k nk pq k

(11:4:1)

The GF is given by PX (z) ¼

N X n pzk qnk ¼ (pz þ q)N k k¼0

(11:4:2)

Clearly PX(1) ¼ 1. Mean: P0X (z) ¼ N(pz þ q)N1 p P0X (1) ¼ Np

or m1 ¼ Np

(11:4:3)

Variance: P00X (z) ¼ N(N 1)(pz þ q)N2 p2 P00X (1) ¼ N(N 1)p

or m2 ¼ N(N 1)p2 Np

Hence var½X ¼ m2 m21 ¼ Npq

(11:4:4)

This result has been obtained with comparative ease. Example 11.4.2 (Poisson Distribution) p(k; l) ¼ el

lk k!

(11:4:5)

The GF is given by PX (z) ¼

1 X

el

k¼0

lzk ¼ el(1z) k!

(11:4:6)

Note that PX(1) ¼ 1. Mean: P0X (z) ¼ el(z1) ¼ el(z1) l P0X (1) ¼ l or m1 ¼ l

(11:4:7)

a result that has been trivially obtained compared to Eq. (11.2.7). Variance: P00X (z) ¼ el(z1) l2 P00X (1) ¼ l2

or

m2 ¼ l2 l

(11:4:8)

Hence var½X ¼ m2 m21 ¼ l A result that has been obtained much more easily than using Eq. (11.2.8).

(11:4:9)

164

Characteristic Functions and Generating Functions

Example 11.4.3 (Geometric Distribution) p{k: p} ¼ qk p

(11:4:10)

The GF is PX (z) ¼

1 X

qzk p ¼

k¼0

p 1 qz

(11:4:11)

Here also PX(1) ¼ 1. Mean: pq (1 qz)2 q P0X (1) ¼ p P0X (z) ¼

(11:4:12)

Variance: P00X (z) ¼ 2 P00X (1)

pq2 (1 qz)3

pq2 2q2 ¼2 ¼ 2 3 p (1 qz)

(11:4:13)

or m2 ¼

2q2 q þ p2 p

Hence var½X ¼ m2 m21 ¼

B 11.5

q p2

(11:4:14)

MOMENT GENERATING FUNCTIONS

The moment generating function (MGF) MX(t) if it exists is deﬁned by ð1 fX (x)etx dx MX (t) ¼ E(etX ) ¼

(11:5:1)

1

where t is a real variable. This is similar to the characteristic function (CF) [Eq. (11.1.1)], except that jv has been replaced by the real variable t. Unlike the CF, which always exists, the MGF may or may not exist. Expanding E(e tX) in a power series about the origin and taking expectations, if they exist, we can write t2 X 2 tn X n þ þ þ MX (t) ¼ E(etX ) ¼ E 1 þ tX þ 2! n! ¼ 1 þ tm þ

t2 m2 t n mn þ þ þ 2! n!

(11:5:2)

11.5

Moment Generating Functions

165

If MX(t) exists, Eq. (11.5.2) can be differentiated term by term to obtain d tm2 tn1 mn þ þ þ MX (t) ¼ m þ 2! n! dx Substituting t ¼ 0 in Eq. (11.5.3), we obtain MX0 (t)t¼0 ¼ m

(11:5:3)

(11:5:4)

Differentiating Eq. (11.5.2) successively with respect to t and setting t ¼ 0 in the resulting equation, we obtain the moments as follows: (11:5:5) MX(2) (t)t¼0 ¼ m2 , . . . , m(n) X (t)jt¼0 ¼ mn , . . . These results are very similar to those in Eq. (11.1.5) except for the absence of pﬃﬃﬃﬃﬃﬃ ﬃ j ¼ 1. Since the methodology of ﬁnding the moments using MGF is very similar to that of using CF, we will give only three examples. Example 11.5.1 (Negative Binomal—Pascal) pmf is

If N is the random variable, then the

nb{n; k,p} ¼ P{n trials produce k successes} ¼

n 1 k nk p q : n ¼ k, k þ 1, . . . k1

Or substituting m ¼ n 2 k, we can write the pmf with M as the random variable: nb{m; k, p} ¼ P{(m þ k) trials produce k successes} kþm1 k m ¼ p q , m ¼ 0, 1, . . . k1 Since

kþm1 k1

¼

kþm1 m

we can write the MGF for the random variable M: 1 X kþm1 MM (t) ¼ (qet )m pk m m¼0 The negative binomial expansion of (1 2 qe t)2k can be written as follows: (1 qet )k ¼

1 X kþm1 (qet )m m m¼0

Substituting this equation in the previous one, we obtain the MGF: MM (t) ¼ pk (1 qet )k Clearly MM(0) ¼ 1.

166

Characteristic Functions and Generating Functions

Mean: 0 MM ðtÞ ¼ pk qk(1 qet )k1 et 0 MM (0) ¼ E(M) ¼

kq p

and E(N) ¼

kq k þk ¼ p p

Variance. We ﬁrst ﬁnd EfM 2g ¼ m2: 00 MM (t) ¼ pk qk(1 qet )k2 (kqet þ 1)et 00 MM (0) ¼ E(M 2 ) ¼

qk(1 þ qk) p2

Hence the variance of M is var½M ¼

qk(1 þ qk) (qk)2 qk 2 ¼ 2 p p2 p

The variance of N can be calculated by using N ¼ M þ k in the equations above. The MGF of the Gaussian distribution is

Example 11.5.2 (Gaussian Distribution) given by 1 MX (t) ¼ pﬃﬃﬃﬃﬃﬃ 2p s

ð1

e½(xm)

2

=2s2 þtx

dx

1

Completing the squares, we obtain

MX (t) ¼ e

tmþ½ðs2 t2 Þ=2

ð1 1 (x m ts2 )2 pﬃﬃﬃﬃﬃﬃ exp dx 2s2 2p s 1

The value of the integral within braces is 1, and hence the MGF of a Gaussian is 2 2

MX (t) ¼ etmþ½(s t

)=2

Mean: MX0 (t) ¼ etm½(s

2 2

t )=2

(m þ s2 t)

MX0 (0) ¼ m Variance: MX00 (t) ¼ etmþ½(s

2 2

t )=2

MX00 (0) ¼ m þ s2

2 2

(m þ s2 t)2 þ etmþ½(s t

and

var½X ¼ s2

)=2 2

s

11.6

Example 11.5.3 (Gamma Distribution) given by

167

We will ﬁnd the MGF of a gamma density

l(lx)a1 elx G(a)

fX (x) ¼

Cumulant Generating Functions

a, l; x . 0

From Eq. (11.5.1), we have ð1

l(lx)a1 elx tx e dx, a, l . 0 G(a) 0 ð la 1 a1 (lt)x ¼ x e dx G(a) 0

MX (t) ¼

Substituting (l 2 t)x ¼ y and using the deﬁnition of a gamma function [Eq. (7.5.2)], we can write this equation as ð1 la ya1 ey dy MX (t) ¼ G(a)(l t)a 0 MX (t) ¼

la la a G(a) ¼ G(a)(l t) (l t)a

Mean: la (a)(1) ala ¼ (l t)aþ1 (l t)aþ1 a MX0 (0) ¼ l MX0 (t) ¼

Variance. We ﬁrst ﬁnd EfX 2g ¼ m2: MX00 (t) ¼

a(a þ 1)la (l t)aþ2

MX00 ð0Þ ¼ m2 ¼

a(a þ 1)la a2 þ a ¼ l2 laþ2

Hence the variance is m2 m2X ¼

a2 þ a a2 a 2¼ 2 2 l l l

The means and variances of most of the commonly occurring discrete (Chapter 4) and continuous (Chapters 6 and 7) distributions are shown in Section 11.7.

B 11.6

CUMULANT GENERATING FUNCTIONS

The cumulant generating function (CGF) KX(t) is deﬁned as the logarithm of the moment generating function MX(t), or KX (t) ¼ ln½MX (t). The cumulants also called semiinvariants {ki } are the coefﬁcients of CGF: KX (t) ¼ ln½MX (t) ¼ tk1 þ

1 i X t2 tn t k2 þ þ kn þ ¼ ki 2! n! i! i¼0

(11:6:1)

168

Characteristic Functions and Generating Functions

We will now ﬁnd the cumulants in terms of the moments mk. From Eq. (11.5.2), we can expand ln[MX(t)] in terms of a Taylor series as follows: t 2 m2 t n mn þ þ þ KX (t) ¼ ln½MX (t) ¼ ln 1 þ tm1 þ 2! n! t2 m2 t n mn ¼ tm1 þ þ þ þ 2! n! 2 1 t 2 m2 tn mn tm1 þ þ þ þ 2 2! n! 3 1 t 2 m2 tn mn þ þ þ + tm1 þ þ 2! n! 3

(11:6:2)

Collecting terms in t, we have KX (t) ¼ tm1 þ

t2 (m2 m21 ) 2!

þ

t3 (m3 3m1 m2 þ 2m31 ) 3!

þ

t4 (m4 3m22 4m1 m3 þ 12m21 m2 6m41 ) 4!

þ

t5 (m5 5m4 þ 30m1 m22 þ 40m21 m3 60m31 m2 24m51 ) 5!

þ

(11:6:3)

Equating coefﬁcients of ti in Eqs. (11.6.1) and (11.6.3), the cumulants have been expressed in terms of the moments fmig in Eq. (11.6.4), where they are also expressed in terms of the central moments vi ¼ E½X mX i : k1 ¼ m1 ¼ mX k2 ¼ m2 m21 ¼ s2X ¼ v2 k3 ¼ m3 3m1 m2 þ 2m31 ¼ v3

(11:6:4)

k4 ¼ m4 3m22 4m1 m3 þ 12m21 m2 6m41 ¼ v4 3v2 k5 ¼ m5 5m4 þ 30m1 m22 þ 40m21 m3 60m31 m2 24m51 ¼ v5 10v2 v3 þ Skewness measures the lack of symmetry of the tails of a density. A density is symmetric if it looks the same on both the left and the right of the centerpoint. Skewness is deﬁned as v3 (11:6:5) g1 ¼ 3=2 v2 Kurtosis is deﬁned as a normalized form of the fourth central moment of a distribution. It is a measure of whether the density is peaked or ﬂat relative to a normal distribution. High-kurtosis density functions will have a high peak near the mean, and low-kurtosis

11.6

Cumulant Generating Functions

functions will have a ﬂat top. Kurtosis is deﬁned as v4 g2 ¼ 2 v2

169

(11:6:6)

For Gaussian distributions the kurtosis is 3. Example 11.6.1 We will ﬁnd the cumulants of the Poisson distribution given by p(k; l) ¼ el (lk =k!). t The MGF of the Poisson distribution is MX (t) ¼ el(1e ) , and the CGF is given by t KX (t) ¼ ln el(1e ) ¼ l þ let Expanding KX (t) in Taylor series, we have t2 t3 t4 t5 l þ let ¼ l t þ þ þ þ þ 2! 3! 4! 5! Thus, all the cumulants are equal to l, or ki ¼ l, i ¼ 1,2, . . . : Example 11.6.2

We will ﬁnd the cumulants of a Gaussian distribution " # 1 1 x mX 2 fX (x) ¼ pﬃﬃﬃﬃﬃﬃ exp sX 2 2psX 2 2

The MGF is given by MX (t) ¼ etmX þ½(t sX )=2 and the CGF, by h i t2 s2X 2 2 KX (t) ¼ ln etmX þ½(t sX )=2 ¼ tmX þ 2 and the cumulants are k1 ¼ mX , k2 ¼ s2X and ki ¼ 0 for i . 2. Thus, for the Gaussian distribution, there are only two cumulants. Joint Cumulants

By analogy to Eq. (11.6.4), we can deﬁne joint cumulants for one to four random variables, X1, X2, X3, X4, as follows: k1 ¼ E½X1 ¼ mX k12 ¼ E½X1 X2 E½X1 E½X2 ¼ sX1 X2 k123 ¼ E½X1 X2 X3 E½X1 E½X2 X3 E½X2 E½X1 X3 E½X3 E½X1 X2 þ 2E½X1 E½X2 E½X3 k1234 ¼ E½X1 X2 X3 X4 E½X1 X2 E½X3 X4 E½X1 X3 E½X2 X4 E½X1 X4 E½X2 X3 E½X1 E½X2 X3 X3 E½X2 E½X1 X3 X4 E½X3 E½X1 X2 X4 E½X4 E½X1 X2 X3 þ 3E½X1 E½X2 E½X3 X4 þ 3E½X2 E½X3 E½X1 X4 þ 3E½X3 E½X4 E½X1 X2 þ 3E½X1 E½X4 E½X2 X3 6E½X1 E½X2 E½X3 E½X4

(11:6:7)

170

Characteristic Functions and Generating Functions

If the mean values of random variables Z1, Z2, Z3, Z4 are zero, then Eq. (11.6.7) can be considerably simpliﬁed as follows: k1 ¼ E½Z1 ¼ 0 k12 ¼ E½Z1 Z2 k123 ¼ E½Z1 Z2 Z3 k1234 ¼ E½Z1 Z2 Z3 Z4 E½Z1 Z2 E½Z3 Z4

(11:6:8)

E½Z1 Z3 E½Z2 Z4 E½Z1 Z4 E½Z2 Z3 If X1, X2, Y1, Y2 are four random variables, not necessarily independent, we will express the cov[X1 X2, Y1 Y2] in terms of the joint cumulant k1234 : cov½X1 X2 , Y1 Y2 ¼ E{½X1 X2 E(X1 X2 )½Y1 Y2 E(Y1 Y2 )} ¼ E½X1 X2 Y1 Y2 E½X1 X2 E½Y1 Y2

(11:6:9)

Expressing E½X1 X2 Y1 Y2 in terms of the fourth cumulant k1234 given in Eq. (11.6.8), we can rewrite Eq. (11.6.9) as follows: cov½X1 X2 ; Y1 Y2 ¼ k1234 þ E½X1 Y1 E½X2 Y2 þ E½X1 Y2 E½X2 Y1

B 11.7

(11:6:10)

TABLE OF MEANS AND VARIANCES pþq¼1

Random Variable

Mean

Variance Discrete Distributions

Bernoulli

p

pq

Binomial

Np

Npq or Np(1 2 p)

Multinomial

npi , q p

Geometric

i ¼ 1, . . . , m

Negative binomial E(M) ¼ Hypergeometric

Kn N

Poisson

l

Logarithmic— Benford

kq p

npi (1 pi ), i ¼ 1, . . . , m q p2 kq for M ¼ N k var(M) ¼ 2 for M ¼ N k p nK(N K)(N n) N 2 (n 1) l

2 8 2 :5 ﬃ 3:44 log 7 : 34 (d ¼ 1, . . . , 9)

log

2 8 2 28 : 572 2 :5 ﬃ 6:06 log 13 50 7 :3 7 : 34

(d ¼ 1, . . . , 9)

Continuous Distributions Uniform

bþa 2

(b a)2 12 (Continued )

11.7

Random Variable

Table of Means and Variances

Mean 1 l

Variance 1 l2

Gaussian

m

Triangular

b

s2 (b a)2 6 2 l2 n , n integer l2 a l2 1 aþ2 2 aþ1 G G a a l2

Exponential

Laplace—double- 0 exponential n , n integer Erlang l a Gamma l 1 aþ1 Weibull mþ G l a Chi-square

Chi

Rayleigh Maxwell

ns2

2ns4

pﬃﬃﬃ nþ1 2sG 2 n G 2 rﬃﬃﬃﬃ p s 2 rﬃﬃﬃﬃ 2 2s p

mR ¼ s2

Rice

171

2 6 s2 6 4n

3 nþ1 7 2 n 7 5 G2 2

2G2

p s2 2 2 8 2 s 3 p

rﬃﬃﬃﬃ 2 2 p (m2 =4s2 ) m2 m m2 m 1 þ 2 I0 I þ e 1 2s 4s2 2s2 4s2 2

s2R ¼ m2 þ 2s2 m2R ; m: noncentrality parameter

Nakagami

1 1=2 G m þ V 2 m G(m)

3 2 1 2 G mþ V6 2 7 7 V 6 4 m G(m) 5 n , n.2 v2

Student-t

0

Snedecor F

n2 , n2 2

Lognormal

e½mþ(s =2)

2v22 (n1 þ n2 2) , n1 (n2 2)2 (n2 4) 2 2 e(2mþs ) es 1

Beta

a aþb

ab (a þ b þ 1)(a þ b)2

2

n2 . 2

v2 . 4

(Continued )

172

Characteristic Functions and Generating Functions

Random Variable

Mean

Variance

Cauchy

Not deﬁned

Not deﬁned

Pareto

b , a1

ab2 , a.2 (a 1)2 (a 2)

a.1

CHAPTER

12

Functions of a Single Random Variable

B 12.1

RANDOM VARIABLE g(X )

In many probability problems, we have to ﬁnd the pdf of a function of random variables. We have already deﬁned (in Chapter 5) a random variable X as the mapping from the sample space S to space of real numbers Rx; that is, it is the set of all outcomes j belonging to the sample space S such that the unique number X(j) belongs to the real line Rx. Thus, the random variable X is a mapping of the probability space {S, F, P} to the probability space on the real line {Rx , FX , PX }: X: {S, F, P} ! {Rx , FX , PX }

(12:1:1)

We can now deﬁne a random variable Y as a function of another random variable X, or Y ¼ g(X ), as the mapping from the real line Rx to another real line Ry, such that it induces a probability space {Ry , FY , PY }. In other words, the random variable Y is a composite function of the random variable X such that Y(j) ¼ g(X ) ¼ g[X (j)]. The mapping is given by Y: {S, F, P} ! {Rx , FX , PX } ! {Ry , FY , PY }

(12:1:2)

and shown in Fig. 12.1.1. Conditions on Y 5 g(X ) to be a Random Variable 1. The domain of g(x) must include the range of the random variable (RV) X. 2. For each y [ Iy, fg(x) yg must form a ﬁeld FY ; that is, it must consist of the unions and intersections of a countable number of intervals of the real line Ry. 3. The probability of the event fg(x) ¼ +1g is 0. In essence, the transformation Y ¼ g(X ) induces a probability measure PY on FY , and we have to ﬁnd this probability measure. From Fig. 12.1.1 we can write PY {Y} ¼ PX {g1 ( Y)} ¼ PX {x : g1 ( Y) , Ix }

(12:1:3)

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

173

174

Functions of a Single Random Variable

FIGURE 12.1.1

The set Ix consists of all the countable unions and intersections of the type a , x b. Without loss in generality, we will assume that Ix ¼ {X y}. Thus, Eq. (12.1.3) becomes PY {Y} ¼ P{Y y} ¼ FY ( y) ¼ PX {x : g1 ( Y) y} ¼ PX {x : g(x) y}

(12:1:4)

In conclusion, we can say that the probability of Y y is given by the probability of the set of all values of x such that g(x) y.

B 12.2

DISTRIBUTION OF Y 5 g(X )

We will ﬁnd the distribution FY ( y) of the RV Y ¼ g(X) in terms of the distribution FX (x) of the RV X and the function g(x). We shall do this with an arbitrary function g(x) bounded between a and b as shown in Fig. 12.2.1. As y traverses from 21 to þ1 along the g(x) axis, we have to ﬁnd the values of x along the x axis that satisfy g(x) y. In Fig. 12.2.1 there are ﬁve distinct regions for y corresponding to the intersections of y with g(x): 1. y a, no points of intersection 2. a , y y1, one point of intersection x1 3. y1 , y y2, three points of intersection x2, x20 , x002 4. y2 , y b, one point of intersection x3 5. y . b, no points of intersection

FIGURE 12.2.1

12.2 Distribution of Y 5 g(X )

175

We have to ﬁnd FY ( y) ¼ probability of g(x) y in each of these ﬁve regions. 1. y a: There are no points along the x axis for which g(x) y, or IX ¼ {1}; hence FY ( y) ¼ 0. 2. a , y y1: The set of all points along the x axis for which g(x) y, or Ix ¼ (x1, 1); hence FY ( y) ¼ 1 2 FX(x1). 3. y1 , y y2: The set of all points along the x axis for which g(x) y, or Ix ¼ (x2, x20 ] < (x002, 1); hence FY ( y) ¼ FX (x20 ) 2 FX (x2) þ 1 2 FX (x002). 4. y2 , y b: The set of all points along the x axis for which g(x) y or Ix ¼ (x3, 1); hence FY ( y) ¼ 1 2 FX(x3). 5. y . b: Here the entire g(x) is below y; and hence Ix ¼ (21, 1); hence FY ( y) ¼ 1. Thus, we have used Eq. (12.1.4) to ﬁnd PfY y) along the entire g(x) axis. Special Case

We now consider a situation where g(x) ¼ y1 for x1 , x x2. In this case we have P{Y ¼ y1 } ¼ P{x1 , X , x2 } ¼ FX (x2 ) FX (x1 )

(12:2:1)

Hence FY ( y) is discontinuous at y ¼ y1, and the discontinuity is equal to FX (x2 ) FX (x1 ). Summary of Steps for Finding FY( y) 5 PfY y) 1. Graph the function g(x) along the x and g(x) axes. Do not label the g(x) axis as y. 2. Draw y parallel to the x axis. 3. Find the distinct regions along the g(x) axis. The regions are determined by the number of intersections of y with g(x) as y goes from 21 to 1 along the g(x) axis. Let these be y1, y2, . . . . 4. In each region solve for all values of x for which y ¼ g(x). Let these be x, x0 , x00 , . . . . 5. For each of these regions, ﬁnd Ix such that g(x) y. 6. Find FY( y) ¼ PfY yg ¼ Pfx [ Ix) ¼ Pfx : g(x) yg. We will take a number of examples of increasingly difﬁcult transformations g(x) and solve for PfY yg using the steps listed above. We shall assume that in all the following examples FX(x) or fX(x) is given and we have to ﬁnd FY( y). Example 12.2.1 Given that Y ¼ aX þ b with a . 0, we have to ﬁnd FY( y). We follow the steps listed above: 1. The function g(x) ¼ ax þ b has been graphed. 2. y parallel to the x axis has been drawn in Fig. 12.2.2. 3. In this simple example there is only one region 21 , y , 1. 4. In this region, solving for y ¼ ax þ b, we have x ¼ ( y 2 b)/a. 5. The region Ix such that g(x) y is given by yb Ix ¼ 1 , x a y b yb 6. ¼ FX FY ( y) ¼ P{x : g(x) y} ¼ P X a a

176

Functions of a Single Random Variable

FIGURE 12.2.2

We can now ﬁnd the density fY ( y) by differentiating FY ( y) with respect to y. The result is dFX (ð y bÞ=a) 1 yb ¼ fX fY ( y) ¼ dy a a Example 12.2.2 before.

We repeat Example 12.2.1 with a 0. We follow the same steps as

1. The function g(x) is graphed in Fig. 12.2.3. 2. y has been drawn parallel to x. 3. In this example also, there is only one region 21 , y ,1. 4. In this region, solving for y ¼ ax þ b, we have x ¼ ( y 2 b)/a or x ¼ [( y 2 b)/jaj]. 5. The region IX such that g(x) y is given by

6.

yb IX ¼ ,x1 jaj yb FY ( y) ¼ P{x : g(x) y} ¼ P X . jaj yb yb ¼1P X . ¼ 1 FX jaj jaj

The corresponding density function fY ( y) is given by differentiating FY ( y) with respect to y, and the result is dFX fY ( y) ¼

yb 1 yb a fX ¼ jaj a dy

The result is very similar to that obtained in Example 12.2.1 except for the absolute value of a.

12.2 Distribution of Y 5 g(X )

177

FIGURE 12.2.3

Example 12.2.3 Let g(x) ¼ (x 2 a)2. This example is different from the previous two examples because there are multiple regions for y. Following similar steps, we obtain 1. The function g(x) is graphed in Fig. 12.2.4 and 2. y has been drawn parallel to x. 3. Here the two regions of y are fy 0g, no points of intersection and fy . 0g, two points of intersection. 4. In the region fy 0g, Ix ¼ 1. In the region fy.0g, we solve for y ¼ g(x) and pﬃﬃﬃ pﬃﬃﬃ obtain x1 ¼ a þ y and x2 ¼ a y. pﬃﬃﬃ pﬃﬃﬃ 5. Thus, Ix ¼ (a y, a þ y , y . 0. 6. In the ﬁrst region FY ( y) ¼ 0 : y 0. In the second region pﬃﬃﬃ pﬃﬃﬃ y , X a þ y) pﬃﬃﬃ pﬃﬃﬃ ¼ FX (a þ y ) FX (a y ) : y . 0

FY ( y) ¼ P(a

FIGURE 12.2.4

178

Functions of a Single Random Variable

The corresponding density function is fY ( y) ¼ 0 : y 0 fY ( y) ¼

d pﬃﬃﬃ pﬃﬃﬃ (FX (a þ y ) FX (a y )) dy

1 pﬃﬃﬃ pﬃﬃﬃ ¼ pﬃﬃﬃ (fX (a þ y ) þ fX (a y )) : y . 0 2 y If X is a zero mean Gaussian distributed random variable given by 1 2 2 fX (x) ¼ pﬃﬃﬃﬃﬃﬃ e(x =2s ) s 2p and if Y ¼ X 2, then fY ( y) is given by 1 1 1 2 2 2 fY ( y) ¼ pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ e( y=2s ) þ e( y=2s ) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ e( y=2s ) 2 y s 2p s 2py which is a chi-square distribution with one degree of freedom [Eq. (7.7.1)].

Example 12.2.4 This example is a complement of Example 12.2.3. Here g(x) ¼ 1=(x a)2 and we have to ﬁnd FY ( y). We follow the same procedure laid out earlier. 1. The function g(x) is graphed in Fig. 12.2.5. 2. y has been drawn parallel to the x axis. 3. Here the two regions of y are fy 0g, no points of intersection and fy . 0g, two points of intersection. 4. In the region f y 0g, Ix ¼ {1} and FY( y) ¼ 0. In the region f y . 0g, we solve for pﬃﬃﬃ pﬃﬃﬃ y ¼ g(x) and obtain x1 ¼ a þ ð1= y Þ and x2 ¼ a (1= y ). 5. Thus 1 1 Ix ¼ 1, a pﬃﬃﬃ < a þ pﬃﬃﬃ , 1 y y

FIGURE 12.2.5

12.2 Distribution of Y 5 g(X )

179

6. In the region indicated in item 5 (above), we obtain 1 1 FY ( y) ¼ P X a pﬃﬃﬃ þ 1 P X a þ pﬃﬃﬃ y y 1 1 ¼ FX a pﬃﬃﬃ þ 1 FX a þ pﬃﬃﬃﬃ : y . 0 y Y The corresponding density function is fY ( y) ¼ 0 : y 0 d 1 1 FX a pﬃﬃﬃ þ 1 FX a þ pﬃﬃﬃ fY ( y) ¼ dy y y 1 1 1 ¼ 3=2 fX a pﬃﬃﬃ þ fX a þ pﬃﬃﬃ : y . 0 2y y y Example 12.2.5 The previous examples illustrated continuous functions of g(x). In this example g(x) is a discrete function as shown in Fig. 12.2.6. Note that g(x) is not a true function because there are many points of y at x ¼ +a. We can still ﬁnd FY ( y) given FX (x). The procedure is no different from that in the previous examples. 1. The function g(x) has already been graphed in Fig. 12.2.6. 2. y has been drawn parallel to the x axis. 3. There are four region of y, given by (1) y 2b, (2) 2b , y 0, (3) 0 , y b, (4) y . b. 4. The intersection points are (1) none, (2) x1 ¼ 2a, (3) x2 ¼ a, (4) none. 5. The corresponding four regions along the x axis are (1) Ix ¼ {1}, (2) Ix ¼ {1, a}, (3) Ix ¼ f1, a}, (4) Ix ¼ {1, 1}. 6. The distribution function FY( y) in the various regions are (1) y 2 b : FY ( y) ¼ 0, (2) 2b , y 0 : FY ( y) ¼ FX (2a), (3) 0 , y b : FY ( y) ¼ FX (a), (4) y . b : FY ( y) ¼ 1. The distribution function FY( y) is a “staircase” function, indicating that Y is a discrete random variable and the density function fY ( y)

FIGURE 12.2.6

180

Functions of a Single Random Variable

FIGURE 12.2.7

is (a) fY( y) ¼ 0, (b) fY( y) ¼ FX (2a)d(x þ b), (c) fY( y) ¼ [FX(a) 2 FX(2a)]d(x), (d) [1 2 FX(a)]d(x 2 b). The distribution and density functions are shown in Fig. 12.2.7.

Example 12.2.6 We shall now take an example of a piecewise-continuous g(x) with a speciﬁed fX(x) and FX(x). The functions g(x), fX(x) and FX(x) are shown in Fig. 12.2.8.

FIGURE 12.2.8

12.2 Distribution of Y 5 g(X )

181

We have to ﬁnd FY( y) given that fX(x) is deﬁned by Fig. 12.2.8 as 2

3 1 : 3 , x 2 5 fX (x) ¼ 4 5 0: otherwise Since fX(x) is speciﬁed, FX(x) is determined by integrating fX(x). The result is 2

0: 6x 3 FX (x) ¼ 4 þ : 5 5 1:

x 3

3

7 3 , x 2 5 x.2

and is shown in Fig. 12.2.8. In this example, the range of fX(x), x [ f23, 2g has to be factored in determining FY ( y). Other than that, we will still follow the procedure laid out earlier. 1. The function g(x) has already been graphed in Fig. 12.2.8. 2. y will scan from 21 to 1 along the g(x) axis. 3. The regions of y are given by (a) y 24, (b) 24 , y 22, (c) 22 , y21, (d) 21 , y 0, (e) 0 , y 1, (f) 1 , y 2, (g) 2 , y 4, (h) y . 4. 4. The points of intersection of y with g(x) are (a) none, (b) x ¼ 22, (c) x ¼ y, (d) x ¼ 0, (5) x ¼ 1, (6) x ¼ y, (7) x ¼ 2, (8) none. 5. The corresponding regions Ix are (a) Ix ¼ f1g, (b) Ix ¼ f21, 22g, (c) Ix ¼ f21, yg, (d) Ix ¼ f21, 0g, (e) Ix ¼ f21, 1g, (f) Ix ¼ f21, yg, (g) Ix ¼ f21, 2g, (h) Ix ¼ f21, 1g. 6. The distributions functions in the six regions are (a) y 4 : FY ( y) ¼ 0 1 (b) 4 , y 2 : FY ( y) ¼ FX (2) FX (4) ¼ 5 y 3 (c) 2 , y 1 : FY ( y) ¼ FX ( y) ¼ þ 5 5 3 (d) 1 , y 0 : FY ( y) ¼ FX (0) ¼ 5 4 (e) 0 , y 1 : FY ( y) ¼ FX (1) ¼ 5 y 3 (f) 1 , y 2 : FY ( y) ¼ FX ( y) ¼ þ 5 5 (g) 2 , y . 4 : FY ( y) ¼ FX (2) ¼ 1 since FX (x) ¼ 1 for all values of x . 2 (h) y . 4 : FY ( y) ¼ 1 By differentiating FY( y) in every region, we can express the density function fY ( y) as 1 1 1 1 fY ( y) ¼ d( y þ 3) þ ½u( y þ 1) u( y þ 2) þ d( y þ 1) þ d( y) 5 5 5 5 1 þ ½u( y 2) u( y 1) 5 where u( y) is the usual step function. FY ( y) and fY ( y) are graphed in Fig. 12.2.9.

182

Functions of a Single Random Variable

FIGURE 12.2.9

Example 12.2.7 This example is similar to the previous one with a piecewisecontinuous function g(x). The functions g(x), fX(x) and FX(x) are given by 2

3 0: x 2 "1 # 6 1: 2 , x 0 7 6 7 : 2 , x 2 7 g(x) ¼ 6 ; 6 2 x: 0 , x 3 7; fX (x) ¼ 4 4 5 0: otherwise 3 2: x.3 2 3 0: x 2 6x 1 7 FX (x) ¼ 4 þ : 2 , x 2 5 4 2 1: x,2

The functions g(x), fX(x), and FX(x) are shown in Fig. 12.2.10. We have to ﬁnd FY( y). Following the same procedure as before 1. The function g(x) has been graphed in Fig. 12.2.10. 2. y has been drawn, and the intersection of y with g(x) is given by x ¼ 32y. 3. The regions of y are given by (a) y 21, (b) 21 , y 0, (c) 0 , y 43, (d) 43 , y 2, (e) y . 2. Even though g(x) extends upto x ¼ 3, the function fX(x) terminates at x ¼ 2 and FX(x) ¼ 1 beyond x ¼ 2. 4. The points of intersection of y with g(x) are (a) none, (b) x ¼ 22,0, (c) x ¼ 32y, (d) x ¼ 32y, (e) none. 5. The corresponding regions Ix are (a) Ix ¼ f1g, (b) Ix ¼ (– 2, 0], (c) Ix ¼ (21, 23 x], (d) Ix ¼ (21, 23x], (e) Ix ¼ (21, 1]. 6. The distribution functions in the four regions are (a) y 21: FY ( y) ¼ 0, (b) 21 , y 0: FY( y) ¼ FX(0) 2 FX(22) ¼ 12, (c) 0 , y 43: FY( y) ¼ FXð32 yÞ ¼ 3 1 4 8 y þ 2, (d) 3 , y 2: FY ( y) ¼ 1, (e) y 2: FY( y) ¼ 1.

12.2 Distribution of Y 5 g(X )

FIGURE 12.2.10

The corresponding density function fY ( y) is given by 1 3 4 fY ( y) ¼ d( y þ 1) þ u y u( y) 2 8 3 The functions FY ( y) and fY ( y) are shown in Fig. 12.2.11.

Example 12.2.8 We shall now take a more complicated example and ﬁnd FY ( y). In this example g(x) is deﬁned by g(x) ¼

x2

1 1

for all x

FIGURE 12.2.11

183

184

Functions of a Single Random Variable

FIGURE 12.2.12

The probability density function fX(x) and the distribution function FX(x) are given by 2

2 fX (x) ¼ 4 9 0

3 if 1:5 , x 3 5 ; otherwise

2

0 1 62 FX (x) ¼ 4 x þ 9 3 1

3

if

x 1:5

if

7 1:5 x 3 5

if

x.3

The function g(x) is shown in Fig. 12.2.12 and fX(x) and FX(x) are shown in Fig. 12.2.13. 1. g(x) has been graphed in Fig. 12.2.12. 2. y scans from 21 to þ1 along the g(x) axis. 3. The regions of y are given by (a) y 21, two points of intersection in the range of fX(x), (21.5, 3]; (b) 21 , y 0, no points of intersection; (c) 0 , y 18, two points of intersection beyond the range of fX(x), (21.5, 3]; (d) 18 , y 45, one point of intersection in the range of fX(x), (21.5, 3], (e) y . 45, two points of intersection in the same range. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ (1=y) and 4. The p points of ﬃintersection of y with g(x) areﬃ (a) x1 ¼pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ (1=y) beyond x2 ¼ 1 þ (1=y), (b) none, (c) x1 ¼ 1 þ (1=y) and x2 p ﬃ the range of (21.5, 3], (d) one point of intersection x2 ¼ 1 þ (1=y) within the

FIGURE 12.2.13

12.2 Distribution of Y 5 g(X )

185

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rangep and anotherﬃ x1 ¼ 1 þ (1=y) beyond the range, (e) x1 ¼ 1 þ (1=y) and ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x2 ¼ 1 þ (1=y). 5. The corresponding regions Ix are sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ) (sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ) ( 1 1 < 1þ , 1 ðaÞ Ix ¼ 1, 1 þ y y ðbÞ Ix ¼ (1, 1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ) (sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ) 1 1 Ix ¼ 1, 1 þ < {1, 1} 1þ , 1 y y (

ðcÞ

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ) (sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ) 1 1 ðdÞ Ix ¼ 1, 1 þ < {1, 1} 1þ , 1 y y (

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ) (sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ) 1 1 Ix ¼ 1, 1 þ < {1, 1} 1þ , 1 y y (

ðeÞ

6. The distribution functions in the various regions are sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 1 FX (1) þ FX (1) FX 1þ ðaÞ y 1 : FY ( y) ¼ FX 1 þ y y sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 1 2 1 2 1 ¼ 1þ þ þ þ þ 9 y 3 9 3 9 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! 2 1 1 1þ þ 9 y 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 4 1 1þ ¼ 9 9 y

ðbÞ

ðcÞ

1 , y 0 : FY ( y) ¼ FX (1) þ FX (1) ¼

2 1 2 1 4 þ þ ¼ 9 3 9 3 9

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 1 FX (1) þ FX (1) FX (1) 0 , y : FY ( y) ¼ FX 1 þ 8 y sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 1þ þ FX (1) FX y 4 4 ¼00þ þ11¼ 9 9

186

Functions of a Single Random Variable

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 4 1 , y : FY ( y) ¼ FX 1 þ FX (1) þ FX (1) FX (1) ðdÞ 8 5 y sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 1þ þ FX (1) FX y sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! 4 2 1 1 10 2 1 1þ þ 1þ ¼ ¼00þ þ1 9 9 y 3 9 9 y sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 4 1 FX (1) þ FX (1) FX (1) ðeÞ y . : FY ( y) ¼ FX 1 þ 5 y sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 1þ þ FX (1) FX y sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! ! 2 1 1 4 2 1 1 ¼ 1þ þ 0þ þ1 1þ þ 9 y 3 9 9 y 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 13 4 1 1þ ¼ 9 9 y The density function fY ( y) can be obtained by differentiating FY ( y). Functions FY ( y) and fY ( y) are shown in the following equation. FY ( y) is graphed in Fig. 12.2.14a, and fY ( y) is graphed in Fig. 12.2.14b. sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 3 4 4 1 1þ if y 1 7 6 6 9 9 7 y 6 7 64 17 6 if 1 , y 7 69 87 6 7 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7; FY ( y) ¼ 6 6 10 2 1 1 4 7 6 7 6 9 9 1 þ y if 8 , y 5 7 6 7 6 7 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 6 7 4 13 4 5 1 4 1þ if y. 9 9 y 5 22 3 1 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ if y 1 69 2 7 6 y 1þ1 7 6 7 y 6 7 6 17 60 if 1 , y 7 6 87 6 7 61 1 1 4 7 fY ( y) ¼ 6 7 6 9 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 if 8 , y 5 7 6 2 7 6 y 1þ 7 6 7 y 6 7 62 1 4 7 6 7 y. 4 9 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 if 5 5 2 y 1þ y

12.2 Distribution of Y 5 g(X )

187

FIGURE 12.2.14

Example 12.2.9 In Example 12.2.8 we will ﬁnd the conditional probability of Y conditioned on the event A ¼ jXj . 1. Since jXj . 1, the part of the function g(x) that lies between 21 and 1 has no effect on the conditional distribution. Thus, g(x) is given by

g(x) ¼

1 x2 1

for

jxj , 1

The conditional density function fX(x j A) with A ¼ jXj . 1 is given by

fX (x j A) ¼

fX (x) fX (x) ¼ p(jXj . 1) FX (1) þ 1 FX (1) ¼

9 ¼ fX (x) 2 1 2 1 5 þ þ1 þ 9 3 9 3

fX (x)

for

jxj . 1

188

Functions of a Single Random Variable

Hence, the conditional density fX(x j A) and the conditional distribution FX(x j A) can be written as 22 3 if 1:5 , x 1 65 7 6 7 fX (x j A) ¼ 6 2 7; if 1,x3 5 4 5 0 otherwise 2 3 0 if x , 1:5 6 2x þ 3 7 if 1:5 , x 1 7 6 6 5 7 6 1 7 6 7 if 1 , x 1 7 FX (x j A) ¼ 6 6 5 7 6 2x 1 7 6 if 1,x3 7 4 5 5 1 if x.3 The function g(x j A) is shown in Fig. 12.2.15. f (x j A) and FX(x j A) are shown in Fig. 12.2.16. Using exactly the same procedure as in Example 12.3.9, we can ﬁnd the conditional distribution FY ( y j A) and the conditional density fY ( y j A) for the regions (a) y 18, (b) 18 , y 45, and (c) y . 45. The results are shown below: 2 3 1 if y 60 7 8 6 7 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 6 7 1 1 47 6 2 6 7 3 1 þ if , x FY ( y j A) ¼ 6 5 y 8 57 6 7 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 6 7 61 7 1 4 4 5 94 1þ if y . 5 y 5 2

0

6 1 6 0 6 6 C 61B 1 6 B rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃC @ 65 1A 6 y2 1 þ fY ( y j A) ¼ 6 y 6 0 1 6 6 6 B C 1 62B 6 @ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃC 45 1A y2 1 þ y

if

if

if

y

1 8

3

7 7 7 7 1 47 ,x 7 8 57 7 7 7 7 7 7 4 7 7 y. 5 5

FY ( y j A) and fY (Y j A) are also graphed in Fig. 12.2.17.

Example 12.2.10 In the previous examples g(x) was explicitly given, and we had to ﬁnd FY ( y) given fX(x). In this example g(x) is not explicitly given, but we will have to determine it from the problem. Points are uniformly distributed along the circumference of a unit circle. From each point a tangent is drawn that intersects the x axis (Fig. 12.2.18). We have to ﬁnd the distribution of the points of intersection along the x axis. Since the

12.2 Distribution of Y 5 g(X )

189

FIGURE 12.2.15

points are uniformly distributed, the angle Q that the point subtends at the center of the circle is a random variable that is uniformly distributed as U(0,2p). The transformation mapping X ¼ g(Q) between Q and X is found as follows. Since the radius of the circle is 1, the cosine of the angle u subtended by any point on the unit circle at the center of the circle is given by cos(u) ¼ (1/x). Hence, the transformation between the random variable Q and the random variable X is given by X ¼ g(Q) ¼ sec(Q) The function g(u) is graphed in Fig. 12.2.19. The density function fQ(u) and the distribution function FQ(u) of the random variable Q are given by 2 3 0 if x 0 " 1 # for 0 , u 2p 6 u 7 ; FQ (u) ¼ 4 fQ (u) ¼ 2p if 0 , u 2p 5 2p 0 otherwise 1 if x . 2p They are shown in Fig. 12.2.20.

FIGURE 12.2.16

190

Functions of a Single Random Variable

FIGURE 12.2.17

The usual procedure can now be followed to determine the distribution function FX(x). 1. The function g(u) has been graphed. 2. The line x has been drawn in Fig. 12.2.19.

FIGURE 12.2.18

12.2 Distribution of Y 5 g(X )

191

FIGURE 12.2.19

3. The regions along the x axis are given by (a) x21, two points of intersection; (b) 21 , x 1, no points of intersection; (c) x . 1, two points of intersection. 4. Intersection points of x with g(u) are (a) u1 ¼ cos21 (1/x) and u10 ¼ 2p 2 cos21(1/x), (b) no points of intersection, (c) u2 ¼ cos21(1/x) and u20 ¼ 2p 2 cos21(1/x). 5. The corresponding regions Iu are p 1 S 1 3p 2p cos1 (a) Iu ¼ , , cos1 2 x x 2 p 3p (b) Iu ¼ , 2 2 S p 3p S 1 1 1 1 2p cos , , 2p (c) Iu ¼ 0, cos x 2 2 x 6. The distribution functions in the various regions are 1 1 1 3 1 1 1 1 1 cos1 þ 1 cos1 (a) FX (x) ¼ ¼ cos1 p 2p x 4 4 2p x x 2 3 1 1 (b) FX (x) ¼ ¼ 4 4 2

FIGURE 12.2.20

192

Functions of a Single Random Variable

FIGURE 12.2.21

(c) FX (x) ¼

1 1 1 1 1 1 1 1 cos1 þ þ 1 1 cos1 ¼ cos1 þ p 2p x 2 2p x x 2

This distribution is called the arcsine law. The density function fX(x) is obtained by differentiating the distribution function FX(x). They are both shown below, and the graphs are shown in Fig. 12.2.21: 2

3 1 1 1 cos1 if x 1 6p 7 x 2 6 7 61 7 FX (x) ¼ 6 if 1 , x 1 7; 62 7 41 5 1 1 cos1 þ if x.1 p x 2 3 2 1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ p if x 1 7 6 px x2 1 7 6 7 fX (x) ¼ 6 0 if 1 , x 1 7 6 5 4 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ if x.1 px x2 1 Example 12.2.11 As a ﬁnal example, we will take the case when g(u) ¼ cos(u) in the region 2p , u p. In the previous example g(x) ¼ sec(u), and here g(x) ¼ cos(u). We will also assume as in the previous example that fQ(u) is uniformly distributed in

193

12.2 Distribution of Y 5 g(X )

FIGURE 12.2.22

(2p, p]. Thus, fQ(u) and FQ(u) are given by " 1 fQ (u) ¼ 2p 0

for

p , u p otherwise

2

# ;

0 1 6 u FQ (u) ¼ 4 þ 2p 2 1

if if if

x0

3

7 p , u p 5 x.p

The function g(u) ¼ cos(u) is shown in Fig. 12.2.22. The density and distribution functions are graphed in Fig. 12.2.23. Following the usual procedure 1. g(u) has been graphed. 2. The line x has been drawn (Fig. 12.2.22). 3. The regions along the x axis are given by (a) x 21, no points of intersection; (b) 21 , x 1, two points of intersection; (c) x . 1, no points of intersection. 4. Intersection points of x with g(u) are (a) no points of intersection, (b) u1 ¼ 2cos21(x) and u10 ¼ cos21(x), (c) no points of intersection. 5. The corresponding regions Iu are (a) Iu ¼ {1}, (b) Iu ¼ { p, cos1 (x)} {cos1 (x), p}, (c) Iu ¼ f.

FIGURE 12.2.23

194

Functions of a Single Random Variable

FIGURE 12.2.24

6. The distribution functions in the various regions are (a) FX (x) ¼ 0 1 1 p 1 (b) 1 cos (x) þ þ FX (x) ¼ 2p 2 2p 2 p 1 1 1 1 1 þ cos (x) þ ¼ 1 cos1 (x) þ p 2p 2 2p 2 (c) FX (x) ¼ 1 Here also the distribution is the arcsine law. The density function fX(x) is obtained by differentiating the distribution function FX(x). The distribution and density functions are given below and graphed in Fig. 12.2.24. 2

0

if

1 6 FX (x) ¼ 4 1 cos (x) if p 1 if 2 0 if 6 1 ﬃ if fX (x) ¼ 6 4 ppﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 x2 0 if

x 1

3

7 1 , x 1 5; x.1 x 1

3

7 1 , x 1 7 5 x.1

B 12.3 DIRECT DETERMINATION OF DENSITY fY ( y) FROM fX (x) In the previous section, for any given Y ¼ g(X ), we determined fY ( y) from fX (x) through a circuitous route by converting fX (x) to FX(x), using the transformation g(X ) to ﬁnd FY ( y), and then differentiating FY ( y) to obtain fY ( y) as shown in the schematic representation in Fig. 12.3.1. Consider the function g(x) given in Fig. 12.3.2. The probability of the random variable Y lying between y and y þ dy is given by fY ( y)dy ¼ P{y , Y y þ dy}

(12:3:1)

12.3 Direct Determination of Density fY ( y) from fX (x)

195

FIGURE 12.3.1

In Fig. 12.3.2 y ¼ g(x) has three real solutions: x1, x2, x3. Hence the set of values for which {y , g(x) y þ dy} is given by the union of the set of values corresponding to the three points of intersection, {x1 , X x1 þ dx1 } {x2 þ dx2 , X x2 } {x3 , X x3 þ dx3 }, where dx1 . 0, dx3 . 0, and dx2 , 0. By the conservation of probability measure, we have fY ( y)dy ¼ P{y , Y y þ dy} ¼ P{x1 , X x1 þ dx1 } þ P{x2 þ dx2 , X x2 } þ P{x3 , X x3 þ dx3 } ¼ fX (x1 )dx1 þ fX (x2 ) j dx2 j þ fX (x3 )dx3

(12:3:2)

We can now write fY ( y) ¼ fX (x1 )

dx1 jdx2 j dx3 þ fX (x2 ) þ fX (x3 ) dy dy dy

(12:3:3)

jdx2 j 1 , ¼ 0 dy jg (x2 )j

(12:3:4)

But dx1 1 ¼ 0 , dy jg (x1 )j

FIGURE 12.3.2

dx3 1 ¼ 0 dy jg (x3 )j

196

Functions of a Single Random Variable

Substituting Eq. (12.3.4) into Eq. (12.3.3), we obtain the result for fY ( y) directly from fX(x): fY ( y) ¼

fX (x1 ) fX (x2 ) fX (x3 ) þ 0 þ 0 0 jg (x1 )j jg (x2 )j jg (x3 )j

(12:3:5)

Equation (12.3.5) can be generalized to the case when there are n real solutions to the equation y ¼ g(x), given by {x1 , x2 , . . . , xn }: fY ( y) ¼

n X fX (xi ) 0 (x )j jg i i¼1

(12:3:6)

From the derivation it is clear that certain restrictions have to be satisﬁed if we have to determine fY ( y) directly from fX(X ). Conditions for Direct Determination of fX(X) 1. The derivative of g(x) must exist. 2. There can be no straight-line solutions to g(x) ¼ y. We now formulate the following steps for the direct determination of fY ( y) from fX(x): Steps for Direct Determination of fY ( y) 1. Check whether g(x) satisﬁes restrictions 1 and 2. 2. Solve for y ¼ g(x). Let the n real roots be fx1, x2, . . . , xng. 3. Determine

dg(x)

i ¼ 1, 2, . . . , n

dx , x¼xi 4. Calculate fY ( y) ¼

n X fX (xi ) jg0 (xi )j i¼1

Thus, if the restrictions on g(x) are satisﬁed, it is easier to calculate fY ( y) directly from fX(x). We will take some of the examples in the previous section and solve for fX(x) directly. Example 12.3.1

We will ﬁnd fY ( y) for Y ¼ aX þ b with a . 0 or, a 0.

1. g(x) satisﬁes the restrictions. 2. We solve for y ¼ ax þ b. The solution yields x ¼ ( y 2 b)/a.

dg(x) d(ax þ b)

¼ jaj

¼ 3.

dx dx

4. fY ( y) ¼ 1 fX y b jaj a Example 12.3.2

(12:3:7)

Y ¼ g(X ) ¼ (X 2 a)2:

1. g(x) satisﬁes the restrictions. pﬃﬃﬃ pﬃﬃﬃ 2. Solving for x in y ¼ (x 2 a)2, we obtain two roots: x1 ¼ a y, x2 ¼ a þ y

12.3 Direct Determination of Density fY ( y) from fX (x)

3.

dg(x) d(x a)2

pﬃﬃﬃ

¼

dx dx ¼ j2(x a)j ¼ 2 y

4.

1 pﬃﬃﬃ pﬃﬃﬃ fY ( y) ¼ pﬃﬃﬃ ½ fX (a y ) þ fX (a þ y ); 2 y

Example 12.3.3

y.0

197

(12:3:8)

Y ¼ g(X) ¼ 1=(X a)2 .

1. g(x) satisﬁes the restrictions.

pﬃﬃﬃ 2. Solving for x in y ¼ 1=(x a)2 , we obtain two roots x1 ¼ a (1= y), pﬃﬃﬃ x2 ¼ a þ (1= y).

1

1

3.

dg(x)

d 3=2 ¼ ¼ 2

dx dx (x a)2 (x a)3 ¼ j2y j 1 1 1 4. (12:3:9) fY ( y) ¼ 3=2 fX a pﬃﬃﬃ þ fX a þ pﬃﬃﬃ ; y . 0 2y y y

Example 12.3.4 (See Example 12.2.8) The probability density functions fX(x) and g(x) and are given by 2 3 2 1 if 1:5 , x 3 5 g(x) ¼ 2 for all x fX (x) ¼ 4 9 1 x 0 otherwise Even though g(x) is deﬁned from 21 to 1, the range of fX(x) plays a critical role in determining fY ( y). Figure 12.2.12 is redrawn in Fig. 12.3.3 to indicate the ranges. 1. g(x) satisﬁes the restrictions.

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2. Solving for x in pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ y ¼ 1/(x 2 1), we obtain two roots: x1 ¼ 1 þ (1=y), x2 ¼ 1 þ (1=y).

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2

3. dg(x) d 1 x 1

dx ¼ dx (x2 1) ¼ 2 (x2 1)2 ¼

2y 1 þ y

4. fY ( y) have to be calculated in the four ranges of y as shown in Fig. 12.3.3. The density function fY ( y) is calculated as follows: Region I: y 21, two points of intersection: 1 2 2 2 1 þ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

fY ( y) ¼ ¼

9 9

9 1 1

2y2

1 þ

y2

1 þ

y y 1 Region II: 1 , y , no points of intersection: 8 fY ( y) ¼ 0 1 4 Region III: , y , one point of intersection: 8 5 1 2 1 1 fY ( y) ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

¼

9

9 1 1

2y2

1 þ

y2

1 þ

y y

198

Functions of a Single Random Variable

FIGURE 12.3.3

4 Region IV: y . , two points of intersection: 5 1 2 2 2 1 þ fY ( y) ¼ ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

9 9

9 1 1

2y2

1 þ

y2

1 þ

y y Here we have found the result far more easily than in Example 12.2.8. Example 12.3.5 (See Example 12.2.10) by g(u) ¼ sec(u) and fQ(u) is given by " 1 fQ (u) ¼

2p 0

In Example 12.2.10, the function g(u) is given

for 0 , u 2p

#

otherwise

We will now ﬁnd fX(x) directly from fQ(u). 1. g(u) satisﬁes the restrictions. 2. Solving for u in x ¼ sec(u), we have u1 ¼ sec21(x) and u10 ¼ 2p 2 sec21(x).

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

3.

dg(u)

d

¼ j sec (u) tan (u)j ¼

x x2 1

¼ sec (u)

du du 4. From Fig. 12.2.19, the regions of x are (a) x 21, (b) 2 1 , x 1, (c) x . 1. (a) x 21, two points of intersection: 1 1 1 1 fX (x) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x x2 1 2p 2p px x2 1 (b) 21 , x 1, no points of intersection, fX(x) ¼ 0.

199

12.3 Direct Determination of Density fY ( y) from fX (x)

(c) x . 1, two points of intersection: 1 1 1 1 p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ þ fX (x) ¼ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2p 2p x x 1 px x2 1 These are the same results obtained in Example 12.2.10.

Example 12.3.6 (See Example 12.2.11) " 1 fQ (u) ¼ 2p 0

Here g(u) ¼ cos(u) and fQ(u) is given by for p , u p

#

otherwise

The three regions of x are (a) x 2 1, (b) 21 , x 1, (c) x . 1. 1. g(u) satisﬁes the restrictions. 2. The solutions for x ¼ cos(u) are u1 ¼ 2cos21(x) and u10 ¼ cos21(x).

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

3.

dg(u)

d

¼ cos (u) ¼ j sin (u)j ¼

1 x2

du du 4. fX(x) can now be found in the three regions. (a) x 21, no points of intersection: fX (x) ¼ 0: (b) 21 , x 1, two points of intersection: 1 1 1 1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ p þ fX (x) ¼ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2p 2p 1x p 1 x2

(12:3:10)

(c) x . 1, no points of intersection: fX (x) ¼ 0: These are the same results obtained in Example 12.2.11.

Example 12.3.7 This example will have relevance in the next section. Here g(x) and fX(x) are given by 2

0

6 4 6 2 þ ln g(x) ¼ 4 5x 0

3

if

x1

if

7 1 , x , 57 5

if

x5

"1 fX (x) ¼

if

1,x5

#

4 0 otherwise

The functions g(x) and fX(x) are shown in Fig. 12.3.4. 1. g(x) satisﬁes the restrictions. 2. There are no points of intersection in the region y 2. Solving for y ¼ g(x) in the region y . 2, we obtain y ¼ 2 þ ln

4 4 ; y 2 ¼ ln 5x 5x

or 5 x ¼ 4e( y2)

200

Functions of a Single Random Variable

FIGURE 12.3.4

or x ¼ 5 4e( y2)

for

y.2

4

¼ 1 ¼ 4e( y2) ; 3. dg(x) ¼ d 2 þ ln

dx dx 5x 5x

y.2

4. Since there are no points of intersection of y with g(x) in the region y 2, it follows that fY ( y) ¼ 0. In the region y . 2, 1 fY ( y) ¼ 4e( y2) ¼ e( y2) 4

for y . 2

B 12.4 INVERSE PROBLEM: FINDING g(x) GIVEN fX(x) AND fY ( y) In the previous sections we studied ﬁnding fY ( y) given fX(x) and g(x). In this section we will ﬁnd g(x) given fX(x) and fY ( y). We will use the conservation of probability measure given by an equation similar to Eq. (12.3.3): fY (h)Dh ¼ fX (j1 )Dj1 þ fX (j2 )Dj2 þ þ fX (jn )Djn

(12:4:1)

Taking the limit as Dh, Dj1, Dj2, . . . , Djn ! 0, all the jk coalesce to j, and integrating h between 21 to y and j between 21 to x, we obtain ðx ðy fY (h)dh ¼ fX (j)dj (12:4:2) 1

1

The lefthand side of Eq. (12.4.2) is a function of y, the righthand side is a function of x, and the equation can be solved for y, which yields g(x) as the transforming function. Example 12.4.1 Given fX (x) ¼ 2e2(x2) u(x 2) and fY ( y) ¼ 14 ½u( y 2) u( y 6), we have to ﬁnd the transforming function g(x). The functions fX(x) and fY ( y) are shown in Fig. 12.4.1.

12.4

Inverse Problem: Finding g(x) Given fX(x) and fY ( y)

201

FIGURE 12.4.1

Integrating both fX(x) and fY ( y), we obtain the following: ðy ðy 1 y2 dh ¼ for fY (h)dh ¼ 4 4 ð2x ð x2 fX (j)dj ¼ 2e2(j2) dj ¼ 1 e(x2) for 2

2,y6 x.2

2

Thus y2 ¼ 1 e(x2) 4

for

2 , y 6 and x . 2

or y ¼ g(x) ¼ 6 4e(x2)

for

x.2

The function g(x) is graphed in Fig. 12.4.2. Using the same techniques as in the previous section, we can show that the transformation function g(x) indeed gives a uniform distribution in (2,6], given that fX(x) is an exponentially distributed function starting at y ¼ 2. Example 12.4.2 (See Example 12.3.7) We will revisit Example 12.3.7, where fX(x) is uniformly distributed in the interval (1,5] and fY ( y) is exponentially distributed as

FIGURE 12.4.2

202

Functions of a Single Random Variable

e 2( y22) starting at y ¼ 2. We have to ﬁnd the transforming functions. Proceeding as in the previous example, we have ðx ðy 1 x1 (h2) e dh ¼ dj or 1 e( y2) ¼ , y . 2, x . 1 4 4 2 1 or e( y2) ¼

5x , 4

y . 2, x . 1

Taking logarithm on both sides, we obtain 5x 4 ( y 2) ¼ ln or y ¼ g(x) ¼ 2 þ ln , 4 5x

x.1

g(x) is exactly the same function that we used in Example 12.3.7 and is shown in Fig. 12.3.4.

B 12.5 MOMENTS OF A FUNCTION OF A RANDOM VARIABLE Expectation

We will now ﬁnd the expectation of the random variable Y ¼ g(X ). We have already found the density function fY ( y). We can use the usual deﬁnition of expectations and write ð1 y fY ( y)dy (12:5:1) mY ¼ E½Y ¼ 1

However, if only the expected value of Y, and not the density function fY ( y) is required, we can ﬁnd the result from the knowledge of fX(x) and g(x). Using the law of conservation of probability measure [Eq. (12.3.2)], we have ð1 ð1 y fY ( y)dy ¼ y fX (x)dx (12:5:2) E½Y ¼ 1

1

Substituting y ¼ g(x) in Eq. (12.5.2), we obtain the equation for E[Y ] in terms of the density function fX(x): ð1 E½Y ¼ g(x)fX (x)dx (12:5:3) 1

If the random variable is of the discrete type, then X X E½Y ¼ yi P( y ¼ yi ) ¼ g(xi ) P(x ¼ xi ) i

(12:5:4)

i

Variance

In a similar manner, the variance of Y ¼ g(X ) can be deﬁned as ð1 2 2 sY ¼ E½Y mY ¼ ( y mY )2 fY ( y) dy ð1 ¼ 1

1

( y mY )2 fX (x)dx ¼

ð1 1

½g(x) mY 2 fx (x)dx

(12:5:5)

12.5

Moments of a Function of a Random Variable

203

Higher-Order Moments

The rth moment of Y ¼ g(X ) can be deﬁned as follows: mr ¼ E½Yr ¼ ð1

ð1

( y)r fY ( y)dy

1 r

ð1

( y) fX (x)dx ¼

¼ 1

½g(x)r fX (x)dx

(12:5:6)

1

Example 12.5.1 We will take Example 12.2.7 and calculate the expected values by using the density functions fY ( y) and fX(x). Function fX(x) is uniformly distributed in (22, 2] as shown in Fig. 12.2.10. The density function fY ( y) for that example is given by 1 3 4 u( y) fY ( y) ¼ d( y þ 1) þ u y 2 8 3 and g(x), fX(x), and fY ( y) are shown in Fig. 12.5.1. Using Eq. (12.5.1) to obtain E[Y ], we have

1 3 4 u( y) dy d( y þ 1) þ u y 2 8 3 1 ð 4=3 1 3 1 3 16 1 ¼ þ y dy ¼ þ ¼ 2 8 2 8 9:2 6 0 ð1

E½Y ¼

y

FIGURE 12.5.1

204

Functions of a Single Random Variable

Using Eq. (12.5.3) to obtain E[Y ], we have ð ð2 1 0 2 E½Y ¼ 1 dx þ x dx 4 2 03 2 1 2 4 1 1 1 ¼ þ ¼ þ ¼ 4 4 3 2 2 3 6 This result could just as well have been obtained by inspection of g(x) by adding the areas of the rectangle (22) and the triangle 43 and dividing them by 14, yielding 16. All three of these methods yield exactly the same result. We will now compute the variance of Y from fY ( y) using Eq. (12.5.5): ð1 1 2 1 3 4 d( y þ 1) þ u y yþ var½Y ¼ u( y) dy 6 2 8 3 1 2 ð 4=3 1 5 3 1 2 25 91 83 þ dy ¼ þ þ yþ ¼ ¼ 2 6 6 72 216 108 0 8 Computing var[Y ] from fX(x) and using var[Y ] ¼ E[Y ]2 2 m2Y, we have ð ð 1 0 1 2 2 2 1 2 (1) dx þ x dx var½Y ¼ 4 2 4 0 3 36 ¼

2 1 32 1 83 þ ¼ 4 4 27 36 108

and the two results are the same. Example 12.5.2 We are given fX(x) and fY ( y), and we have to ﬁnd g(x) and the expectation and variance of Y using fY ( y) and fX(x): 2 2 3 "1 # 3y 2,x6 ; fY ( y) ¼ 4 8 0 , y 2 5 fX (x) ¼ 4 0 otherwise 0 otherwise Integrating fX(j) from 2 to x and fY (h) from 0 to y, we have ðx ðy 2 1 3h dh dj ¼ 4 2 0 8 1 y3 (x 2) ¼ ; 2 , x 6, 0 , y 2 8 4 Hence, y ¼ g(x) ¼ ½2(x 2)1=3 . We also note the following: Expectation with fY ( y): ð2 E½y ¼

y 0

3y2 3 dy ¼ 8 2

Expectation with fX(x): ð6 E½ y ¼

1 21=3 3 3 41=3 ¼ ½2(x 2)1=3 dx ¼ 4 2 24

12.5

Moments of a Function of a Random Variable

205

FIGURE 12.5.2

Variance with fY ( y): ð2 var½ y ¼ 0

3 y 2

2

3y2 3 dy ¼ 8 20

Variance with fX(x), using var[Y ] ¼ E[Y ]2 2 m2Y: ð6 1 9 3 9 3 var½ y ¼ ½2(x 2)2=3 dx ¼ 82=3 ¼ 4 4 5 4 20 2 Both methods give the same results. Indicator Functions

An indicator function IA(j) is a real-valued function deﬁned on the sample space S, having a value 0 or 1 depending on whether the j point is in the event A , S. Or 1 j[A IA (j) ¼ (12:5:7) 0 jA The indicator function is a unit step for sets. It is shown in Fig. 12.5.2. The indicator function is a zero – one random variable, and if P{j [ A} ¼ p P{j A} ¼ 1 p ¼ q

(12:5:8)

then the expected value and the variance of the indicator function IA are E½IA ¼ 1 p þ 0 q ¼ p var½IA ¼ 12 p p2 ¼ pq

(12:5:9)

Properties of Indicator Functions 1. A , B () IA IB 2. IA ¼ IA2 ¼ ¼ IAn 3. Is ¼ 1: sample space 4. IAB ¼ min (IA , IB ) 6. IA ¼ 1 IA : A ¼ complement of A

(12.5.10)

CHAPTER

13

Functions of Multiple Random Variables

B 13.1 FUNCTION OF TWO RANDOM VARIABLES, Z 5 g(X,Y ) In the previous chapter we discussed a single function of a single random variable. Here we will ﬁnd the probability fZ(z) of a single function Z of two random variables X and Y deﬁned by the joint probability density function fXY(x, y). The analysis is very similar to that in the previous chapter. Following similar analysis given in Chapter 12, we have Z ¼ g(X, Y ) as a function that maps the domain space of Z given by fCxy,FXY,PXYg into the range space fRz,FZ ,PZg, which is a real line, as shown in Fig. 13.1.1 {Z ¼ g(X,Y)} : {Cxy ,FXY ,PXY } ! {Rz ,FZ ,PZ }

(13:1:1)

and the mapping of fX,Y g is from the probability space fS, F, Pg (where, S is the product space S ¼ SX SY) into the domain space of Z given by fCxy,FXY,PXYg X,Y: {S,F,P} ! {Cxy ,FXY ,PXY }

(13:1:2)

Alternately, if Dxy , Cxy and Iz , Rz, then Iz is the image of Dxy under the transformation g and Dxy is the inverse image of Iz under the transformation g – 1. Hence Iz is the set of all points (x,y) belonging to Dxy such that g(x,y) belongs to Iz. This is represented by Dxy ¼ {x,y : g(x,y) [ Iz }

(13:1:3)

We also have conditions similar to those in the last chapter for Z to be a random variable. Conditions on Z 5 g(X,Y) to Be a Random Variable 1. The domain of g(x,y) must include the range of the RV Z. 2. For each z [ Iz, fg(x,y) zg must form a ﬁeld FZ; that is, it must consist of the unions and intersections of a countable number of intervals. 3. The probability of the event fg(x,y) ¼ +1g is 0.

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

206

Function of Two Random Variables, Z 5 g(X,Y )

13.1

207

FIGURE 13.1.1

In essence, the transformation Z ¼ g(X,Y ) induces a probability measure PZ on FZ , and we have to ﬁnd this probability measure. From Fig. 13.1.1, we can write PZ (Z) ¼ P(Z [ Iz ) ¼ PXY (g1 (Z) [ Dxy ) ¼ PXY (X,Y [ Dxy )

(13:1:4)

The set Iz consists of all the countable unions and intersections of the type a , Z b. Without loss in generality, we will assume that Iz ¼ fZ zg, and substituting for Dxy from Eq. (13.1.3) in Eq. (13.1.4), we have PZ (Z) ¼ PXY ½(X,Y) [ Dxy ¼ PXY ½x,y:g(x,y) [ Iz ¼ PXY ½x, y:g(x,y) z ¼ FXY (z) ¼ P(Z z) ¼ FZ (z)

(13:1:5)

Equation (13.1.5) can also be written as ð

ð

PZ (Z) ¼ PXY ½(X,Y) [ Dxy ¼ FZ (z) ¼

fXY (j,h)dj dh

(13:1:6)

Dxy

where fXY (x,y) is the joint density of the random variables X and Y. In conclusion, we can say that the probability of Z z is given by the probability of the set of all values of x and y such that g(x,y) z, or by integrating the joint density fXY (x,y) over the domain Dxy. We will use both these interpretations in solving problems. Summary of Steps for Finding FZ(z) 5 P(Z z) 1. Graph the function g(x,y) in the x– y plane. 2. Equate g(x,y) ¼ z 3. Find Dxy, the distinct regions in the x– y plane such that g(x,y) z. 4. Find FZ(z) from Eq. (13.1.5) or (13.1.6). Example 13.1.1 Given Z ¼ g(X,Y ) ¼ min(X,Y ), we will ﬁnd FZ(z) given the joint distribution FXY(x,y). We follow the scheme outlined above. 1. g(x,y) is shown in Fig. 13.1.2. 2. g(x,y) ¼ z is marked on the diagram. 3. The region Dxy is the set of all points in the x –y plane such that either x z or y z. It is shown by the region ABHCDF in Fig. 13.1.2.

208

Functions of Multiple Random Variables

FIGURE 13.1.2

4. The probability of Dxy is given by Pf(X z) or (Y z)g, which can be written as Pfrectangle ABEFg þ Pfrectangle GCDFg 2 Pfrectangle GHEFg. We subtract Pfrectangle GHEFg since it occurs twice in the addition of the ﬁrst two terms. Hence FZ (z) ¼ FY (z) þ FX (z) FXY (z,z)

(13:1:7)

If X and Y are independent, then Eq. (13.1.7) becomes FZ (z) ¼ FY (z) þ FX (z) FX (z)FY (z)

(13:1:8)

and the density function fZ(z) is given by fZ (z) ¼ fY (z) þ fX (z) fX (z)FY (z) FX (z)fY (z)

(13:1:9)

We will deﬁne the joint density function fXY (x,y) in the region f0 , x 4g, f0 , y 2g as shown in Fig. 13.1.3 and ﬁnd the distribution FZ(z).

FIGURE 13.1.3

13.1

Function of Two Random Variables, Z 5 g(X,Y )

209

The joint density fXY(x,y) and the distribution FXY(x,y) are shown below: 2 0 for x 0, y 0 61 fXY (x,y) ¼ 4 for 0 , x 4, 0 , y 2 ; 8 0 for x . 0, y . 0 2 0 for x 0, y 0 6 xy for 0 , x 4, 0 , y 2 6 68 6x FXY (x,y) ¼ 6 6 4 for 0 , x 4, y . 2 6y 6 for x . 4, 0 , y 2 4 2 1 for x . 0, y . 0 1. In region I, for fz 0g, FZ(z) ¼ 0. 2. In region II, f0 , z 2g or f0 , x 2g, f0 , y 2g we have the following formula from the equation derived earlier: FZ (z) ¼

z z z2 z þ ¼ (6 z) 4 2 8 8

for

0,z2

3. In region III, f2 , z 4g or f2 , x 4g, f2 , y 4g FZ (z) ¼ FX (z) þ FY (2) FXY (z,2) ¼

z z þ1 1¼1 4 4

for

2,z4

The same result can also be obtained from the fact that the random variable Y is always less than 2. Since the region Dxy is either X z or Y z, and z . 2, we have FZ(z) ¼ 1. 4. In region IV, fz . 4g or fx . 4g, fy . 4g, FZ(z) ¼ 1. The density function fZ(z) obtained by differentiating FZ(z) is shown below. 2 0 for z 0 63 z 6 for 0 , z 2 fZ (z) ¼ 6 64 4 for 2 , z , 4 40 0 for z . 4 Example 13.1.2 Given Z ¼ g(X,Y ) ¼ max(X,Y ), we will ﬁnd FZ(z) given the joint distribution FXY(x,y). We follow the scheme outlined above. 1. g(x,y) is shown in Fig. 13.1.4. 2. g(x,y) ¼ max(x,y) ¼ z is marked on the diagram. 3. The region Dxy is the set of all points in the x –y plane such that x z and y z. This region is shown in Fig. 13.1.4. 4. The probability of Dxy is given by Pf(X,Y z)g, which can be written as FZ (z) ¼ FXY (z,z)

(13:1:10)

210

Functions of Multiple Random Variables

FIGURE 13.1.4

If X and Y are independent, then Eq. (13.1.10) becomes FZ (z) ¼ FX (z)FY (z)

(13:1:11)

and the density function fZ(z) is given by fZ (z) ¼ fX (z)FY (z) þ FX (z)fY (z)

(13:1:12)

We will use the same joint density function fXY(x,y) in the region f0 , x 4g, f0 , y 2g as shown in Fig. 13.1.3 and ﬁnd the distribution FZ(z). 1. In region I, for fz 0g, FZ(z) ¼ 0. 2. In region II, f0 , z 2g or f0 , x 2g, f0 , y 2g, FZ(z) is given by FZ (z) ¼

z2 8

for

0 , x 2, 0 , y 2

3. In the region III, f2 , z 4g or f2 , x 4g, f2 , y 4g we have z 2 z FZ (z) ¼ FXY (z, 2) ¼ ¼ 4 2 4

for

2 , x 4, 2 , y 4

We can also integrate the joint density function fXY(x,y) in the ranges f0 , x zg and f0 , y 2g: ðz ð2 FZ (z) ¼ 0

1 dy dx ¼ 08

ðz

2 z dx ¼ 8 4 0

for

2 , x 4, 2 , y 4

4. In region IV, fz . 4g or fx . 4g, fy . 4g, FZ(z) ¼ 1.

13.1

Function of Two Random Variables, Z 5 g(X,Y )

211

The density function fZ(z) obtained by differentiating FZ(z) is shown below. 2 0 for z 0 6 z for 0 , z 2 6 64 fZ (z) ¼ 6 1 6 for 2 , z , 4 4 4 0 for z . 4 Example 13.1.3 edge of fXY(x,y).

Given that Z ¼ g(x,y) ¼ X þ Y, we have to ﬁnd FZ(z) from the knowl-

1. The function g(x,y) is shown in Fig. 13.1.5. 2. The equation x þ y ¼ z is shown in Fig. 13.1.5. 3. Dxy is shown as the shaded region in the ﬁgure. 4. The distribution function FZ(z) is fZ (z) ¼ fX (z)FY (z) þ FX (z)fY (z)

(13:1:13)

The density function fZ(z) is obtained by differentiating FZ(z), and the result is ð 1 ð zx ð1 ð zx d d fZ (z) ¼ fXY (x,y)dy dx ¼ f (x,y)dy dx dz 1 1 1 dz 1 (13:1:14) ð1 fXY (x,z x)dx ¼ 1

If X and Y are independent in Eq. (13.1.14), then ð1 fX (x)fY (z x)dx fZ (z) ¼

(13:1:15)

1

Equation (13.1.15) is a convolution integral, and we can use characteristic functions to write FZ (v) ¼ FX (v)FY (v) fZ(z) can obtained from FZ(v).

FIGURE 13.1.5

(13:1:16)

212

Functions of Multiple Random Variables

Example 13.1.4 We are given X and Y as independent random variables with Z ¼ g(X,Y ) ¼ 2X 2 12Y. The density functions fX(x) and fY( y) are given by fX (x) ¼ 2e2x u(x);

1 fY (y) ¼ ey=2 u(y) 2

We have to ﬁnd fZ(z). Substituting U ¼ 2X and V ¼ 12Y, we can obtain the corresponding densities of U and V from Eq. (12.3.7): 1 u ¼ eu u(u); fV (v) ¼ 2fY (2v) ¼ ev u(v) fU (u) ¼ fX 2 2 With this substitution, Z can be given by Z ¼ U þ (2V ). Since U and V are also independent, we can use Eq. (13.1.16) to ﬁnd fZ(z). The characteristic functions of U and (2V ) are FU (v) ¼

1 1 ; FV (v) ¼ 1 jv (1 þ jv)

Using the independence property, we obtain 1 1 1 1 FZ (v) ¼ FU (v)FV (v) ¼ ¼ þ (1 jv)(1 þ jv) 2 1 jv 1 þ jv Taking the inverse transform, we obtain 1 1 fZ (z) ¼ ½ez u(z) þ ez u(z) ¼ ejzj 2 2 The functions fX(x), fY( y), and fZ(z) are shown in Fig. 13.1.6.

FIGURE 13.1.6

13.1

Example 13.1.5

Function of Two Random Variables, Z 5 g(X,Y )

213

Two independent random variables X and Y are distributed normally as

2 1 fX (x) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e(1=2)½(x2)=3 ; 2p 3

2 1 fY (y) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e(1=2)½(y3)=4 2p 4

Another random variable Z is given by Z ¼ X 2 Y. We have to ﬁnd fZ(z). The characteristic functions of X and (2Y ) are given by FX (v) ¼ e jv

2

½ðv2 9Þ=2

; FY (v) ¼ ejv3½ðv

2

16Þ=2

Using the independence of X and Y, we have 2 2 3 2 2 FZ (v) ¼ e jv ½ðv 9Þ=2 e jv ½ðv 16Þ=2 ¼ ejv½ðv 25Þ=2 and fZ(z) is also normally distributed with mean 21 and variance 25, and is given by 2 1 fZ (z) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e(1=2)½(x1)=5 2p 5

Hence, the probability of a sum of independent Gaussian random variables is also a Gaussian random variable with mean equaling the sum of means, and variance equaling the sum of variances.

Example 13.1.6 This example illustrates when the two random variables X and Y are not independent but are given by a joint Gaussian density function: ( ) 1 1 2 2 (13:1:17) fXY (x,y) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½x 2rxy þ y 2p 1 r2 2 1 r2 where the correlation coefﬁcient is r and the random variable Z ¼ X þ Y. Here we can use Eq. (13.1.14) only. Substituting fXY (x,y) into Eq. (13.1.14), we have ð1 fZ (z) ¼

fXY (x,z x)dx 1

1 2 2 ½x 2rx(z x) þ (z x) dx 2(1 r2 ) 1 ð1 1 1 2 2 2 ½2x ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp 2rxz þ 2rx þ z 2zx dx 2(1 r2 ) 2p 1 r2 1 ð1 1 1 2 2 ½2x exp (1 þ r) 2xz(1 þ r) þ z dx ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2(1 r2 ) 2p 1 r2 1 ð1 1 2 2 z2 =2(1r2 ) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e e½(1þr)=(1r )½x xz dx 2 2p 1 r 1 ð1 1 2 2 2 e½1=(1r)½x xz dx ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ez =2(1r ) 2 2p 1 r 1 ¼

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2p 1 r2

ð1

exp

(13:1:18)

214

Functions of Multiple Random Variables

Completing squares under the integral sign in Eq. (13.1.18), we have ð1 1 1 h z i 2 z2 z2 =2(1r2 ) dx exp x fZ (z) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e 4 1r 2 2p 1 r2 1 ð1 1 1 h z i2 2 2 2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e½z =2(1r )þ½z =4(1r) x exp dx 1r 2 2p 1 r2 1 ( ) ð1 1 z2 =4(1þr) 1 1 x (z=2) 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dx ¼ pﬃﬃﬃﬃﬃﬃ e exp 2 (1 r)=2) 2p 2pð1 r2 Þ 1 ( ) ð1 1 z2 =4(1þr) 1 1 1 x (z=2) 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dx exp ¼ pﬃﬃﬃﬃﬃﬃ e 2 (1 r)=2) 2(1 þ r) 2p 1 r 1 2p 2 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ¼1

1 1 2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ ez =4(1þr) 2(1 þ r) 2p

(13:1:19)

Thus, we see that the sum of jointly Gaussian-distributed zero mean, unit variance random variables with correlation coefﬁcient r is also a zero mean Gaussian with variance 2(1þ r). If r ¼0, we have independent Gaussian random variables and variance equals 2, which is the sum of unit variances. If r ¼ 1, we have essentially Z ¼ 2X and the variance is 4. Example 13.1.7 Two independent Gaussian random variables X and Y are with zero mean and variance s2. The probability density functions of X and Y are 1 1 2 2 (13:1:20) fX (x) ¼ pﬃﬃﬃﬃﬃﬃ e(x=2s ) ; fY (x) ¼ pﬃﬃﬃﬃﬃﬃ e(y=2s ) s 2p s 2p pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Another random variable Z is given by Z ¼ g(X,Y) ¼ X 2 þ Y 2 . We have to ﬁnd the pdf pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fZ(z). The region Dxy ¼ x2 þ y2 z is shown in Fig. 13.1.7. Using Eq. (13.1.6), we have ðð

FZ (z) ¼

1 (j2 þh2 Þ=2s2 e dj dh 2 2p s pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 x þy z

We can solve this integral by using polar coordinates using the following substitution: j ¼ r cos u; h ¼ r sin u; dj dh ¼ r dr du;

u ¼ tan1

h j

Thus ð 2p ð z

1 (r2 =2s2 ) e r dr du ¼ 2 0 0 s 2p h i 2 2 ¼ 1 e(z =2s ) u(z)

FZ (z) ¼

ðz

r (r2 =2s2 ) e dr 2 0s (13:1:21)

13.1

Function of Two Random Variables, Z 5 g(X,Y )

215

FIGURE 13.1.7

and fZ (z) ¼

z (z2 =2s2 ) e u(z) s2

(13:1:22)

where u(z) is the usual unit step function. This is a Rayleigh distribution as given in Eq. (7.8.5) or a chi-square distribution with 2 degrees of freedom. Example 13.1.8 This example is similar to the previous example except that Z ¼ g(X,Y) ¼ X 2+Y 2. The region Dxy ¼ x 2+y 2 z. We can now ﬁnd the pdf fZ(z) using polar coordinates as in the previous example. Thus ð 2p ð pzﬃ ð pzﬃ 1 (r2 =2s2 ) r (r2 =2s2 ) e r dr du ¼ e dr FZ (z) ¼ 2 2 0 s 2p 0 s 0 h i 2 (13:1:23) ¼ 1 e(z=2s ) u(z) and the density function fZ(z) is given by 1 z=2s2 fZ (z) ¼ e u(z) 2s2

(13:1:24)

This is an exponential distribution or a chi-square distribution given by Eq. (7.7.3) with 2 degrees of freedom. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The density functions for Z ¼ X 2 þ Y 2 and Z ¼ X 2 þ Y 2 given by Eqs. (13.1.22) and (13.1.24) are shown in Fig. 13.1.8 for comparison. Example 13.1.9 In Example 13.1.7 we assumed that the independent random variables X and Y are of zero mean. In this example, we will assume that X and Y have the same variance s2 and that their mean values are mX and mY respectively. The density functions are 1 2 fX (x) ¼ pﬃﬃﬃﬃﬃﬃ e½(xmx )=2s ; s 2p

1 2 fY (y) ¼ pﬃﬃﬃﬃﬃﬃ e½(ymy )=2s s 2p

(13:1:25)

216

Functions of Multiple Random Variables

FIGURE 13.1.8

We will now ﬁnd fZ(z) for Z ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X 2 þ Y 2 . Function FZ(z) is now given by

1 (j mX )2 þ (h mY )2 exp dj dh; 2s2 s2 2p pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 ðð

FZ (z) ¼

z.0

(13:1:26)

x þy z

Expanding Eq. (13.1.26) we have, 1 j2 þ h2 2(mX j þ mY h) þ m2X þ m2Y exp dj dh; FZ (z) ¼ 2s2 s2 2p pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 ðð

z.0

x þy z

(13:1:27) Substituting polar coordinates in Eq. (13.1.27), we obtain the following: 2 1 r 2r(mX cos u þ mY sin u) þ m2X þ m2Y exp r dr du 2 2s2 0 0 s 2p 2 2 2 ð e(mX þmY )=s z ð1=2Þðr=sÞ2 e ¼ s2 0 ð 2p 1 r(mX cos u þ mY sin u) exp du r dr; z . 0 2p 0 s2 ð 2p ð z

FZ (z) ¼

(13:1:28)

We now deﬁne qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m2X þ m2Y cos (c); mX ¼ c ¼ tan

1

mY mX

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m2X þ m2Y sin (c) mY ¼ ð13:1:29Þ

Function of Two Random Variables, Z 5 g(X,Y )

13.1

and a noncentrality parameter m ¼ (13.1.29), we have

217

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m2X þ m2Y . Substituting m in Eqs. (13.1.28) and

rm( cos u cos c þ mY sin u sin c) e exp du r dr s2 0 0 ð 2p 2 2 ð em =s z (1=2)ðr=sÞ2 1 rm( cos u c) e exp ¼ du r dr; z . 0 (13:1:30) s2 2p 0 s2 0 2

em =s FZ (z) ¼ s2

2

ðz

(1=2)ðr=sÞ2

1 2p

ð 2p

The integral under u is the modiﬁed Bessel function of zero order with argument rm=s2 given by ð 2p rm 1 rm( cos uc)=s2 I0 2 ¼ e du s 2p 0 Substituting this value in Eq. (13.1.30) and differentiating with respect to z, we obtain zm z 2 2 2 (13:1:31) fZ (z) ¼ 2 e½(z þm )=s I0 2 ; z . 0 s s This is Rice’s distribution as given in Eq. (7.8.11). With m ¼ 0, we get the Rayleigh distribution. Example 13.1.10 In this example we have Z ¼ g(X,Y ) ¼ XY. The function g(x,y) ¼ xy ¼ z is shown in Fig. 13.1.9 for z . 0. We have to ﬁnd fZ(z) given the joint density fXY(x,y). The distribution function FZ(z) is obtained by integrating fXY(x,y) in the shaded region given by g(x,y) z. Since the region Dxy ¼ fxy zg is bounded by two disjoint curves, we have to integrate fXY (x,y) along the two regions f –1 , x 0g and f0 , x 1g. Hence ðð fXY (x,y)dy dx FZ (z) ¼ xyz

ð0 ð1

ð 1 ð z=x fXY (x,y)dy dx þ

¼ 1

z=x

fXY (x,y)dy dx 0

FIGURE 13.1.9

1

(13:1:32)

218

Functions of Multiple Random Variables

We differentiate Eq. (13.1.32) using the chain rule of differentiation and obtain fZ(z): ð1 z 1 z 1 fXY x, dx þ fXY x, dx x x x x 1 0 ð1 1 z ¼ fXY x, dx; z . 0 x 1 jxj ð0

fZ (z) ¼

(13:1:33)

It can be shown by similar techniques that Eq. (13.1.33) is valid for z 0. If X and Y are independent, then ð1

z 1 fX (x)fY dx x 1 jxj

fZ (z) ¼

for all z

(13:1:34)

We note that Eq. (13.1.34) is not a convolution integral even if the product of independent random variables is involved. The problem in Example 13.1.10 can also be solved using the techniques developed in Chapter 12. We will ﬁrst ﬁnd the conditional density fZ(z j X ¼ x), given the conditional density fY( y j X ¼ x) with Z ¼ xY. Since x is a scaling factor for the random variable Y, we can use Eq. (12.3.7) and obtain fZ (z j X ¼ x) ¼

1 z fY j X ¼ x jxj x

Hence, the unconditional density fZ(z) is given by ð1 fZ (z) ¼

1 z fY j X ¼ x fX (x)dx ¼ x 1 jxj

ð1

z 1 fXY x, dx x 1 jxj

which is the same as before. Example 13.1.11 We will now assume that X and Y are independent random variables with identical Cauchy distributions given by fX (x) ¼

2 2 ; fY (y) ¼ pðx2 þ 4Þ p ð y 2 þ 4Þ

Substituting these values in Eq. (13.1.34), we have ð1 fZ (z) ¼

1 2 2 dx 2 2 þ 4Þ ð jxj p x z 1 p 2þ4 x

Since fZ(z) is an even function, we can rewrite this equation as fZ (z) ¼

42 p2

ð1 0

x dx ðx2 þ 4Þðz2 þ 4x2 Þ

13.1

Function of Two Random Variables, Z 5 g(X,Y )

219

Substituting w ¼ x 2 and dw ¼ 2x dx and expanding into partial fractions, we have 4 p2

ð1

1 dw 2 0 ðw þ 4Þðz þ 4wÞ ð 4 1 1 1 1 ¼ 2 dw p 0 ðz2 16Þ ðw þ 4Þ ðz2 þ 4wÞ

fZ (z) ¼

1

1 4 4 wþ4 2

ln(w þ 4) ln(z þ 4w) 0 ¼ 2 2 ln 2 ¼ 2 2 p ðz 16Þ p ðz 16Þ z þ 4w 0 1

4 wþ4 4 1 4

ln 2 ln ¼ 2 2 ¼ ln 2 p ðz 16Þ z þ 4w 0 p2 ðz2 16Þ 4 z

or fZ (z) ¼

2 4 z ln 16 p2 ðz2 16Þ

and fZ(z) is shown in Fig. 13.1.10. Example 13.1.12 Given Z ¼ g(X,Y ) ¼ Y/X, we have to ﬁnd fZ(z) from the knowledge of the joint density function fXY(x,y). The region Dxy(x,y) ¼ fx,y: y xzg is shown in Fig. 13.1.11. In the two regions of Dxy given by (1) x 0 and (2) x . 0, FZ(z) can be written as follows: ðð FZ (z) ¼

fXY (x,y)dy dx yxz

ð0 ð1

ð 1 ð xz fXY (x,y)dy dx þ

¼ 1 xz

fXY (x,y)dy dx 0

FIGURE 13.1.10

1

(13:1:35)

220

Functions of Multiple Random Variables

FIGURE 13.1.11

Using the chain rule of differentiation in Eq. (13.1.35), we can obtain fZ(z) as ð0

ð1

fZ (z) ¼ ð1

fXY (x,xz)x dx þ

fXY (x,xz)x dx 0

1

jxjfXY (x,xz)dx

¼

(13:1:36)

1

As an illustration, we will ﬁnd fZ(z) given that X and Y are jointly Gaussian random variables with means 0 and variances s2X and s2Y with correlation coefﬁcient r, given by fXY (x,y) ¼

2 1 1 x xy y2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp 2r þ 2(1 r2 ) s2X sX sY s2Y 2psX sY 1 r2

Substituting fXY(x,y) into Eq. (13.1.36) and using the fact that fXY(x,y) is an even function we have, 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fZ (z) ¼ 2psX sY 1 r2

ð1 0

x2 1 z z2 x exp 2r þ dx 2(1 r2 ) s2x sX sY s2Y

1 s2 s2 (1 r2 ) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 X Y 2 psX sY 1 r2 sY 2rsX sY þ z2 sX pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sX sY 1 r2 ¼ sY s2Y 2 2 psX z 2rz þ sX s2X ¼

(13:1:37)

By completing the squares in the denominator of Eq. (13.1.37), we can write pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sX s Y 1 r 2 fZ (z) ¼ 2 p s2X z r ssXY þ s2Y ð1 r2 Þ

(13:1:38)

13.1

Function of Two Random Variables, Z 5 g(X,Y )

221

FIGURE 13.1.12

This is a Cauchy distribution centered at z ¼ r(sY/sX) and is shown in Fig. 13.1.12 with s2X ¼ 4, s2Y ¼ 9, and r ¼ 35 : Integrating Eq. (13.1.38) with respect to z, we obtain the distribution function FZ(z) as " # 1 1 1 sX z rsY pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ FZ (z) ¼ þ tan 2 p s Y 1 r2

(13:1:39)

Example 13.1.13 We are given two independent random variables, X and Y, which are exponentially distributed as fX (x) ¼ 2e2x u(x);

fY (x)ey u( y)

We have to ﬁnd the density function fZ(z) of the random variable given by Z ¼ (Y/X ). From Eq. (13.1.36), we have ð1 fZ (z) ¼

jxjfXY (x,xz)dx 1 ð1

x 2e

¼

2x

e

xz

ð1 dx ¼

0

¼

2xex(2þz) dx

0

2 (2 þ z)2

Integrating fZ (z), we obtain the distribution FZ (z) as FZ (z) ¼ 1 These functions are shown in Fig. 13.1.13.

2 u(z) 2þz

222

Functions of Multiple Random Variables

FIGURE 13.1.13

B 13.2 TWO FUNCTIONS OF TWO RANDOM VARIABLES, Z 5 g(X,Y ), W 5 h(X,Y ) We are given two random variables, X and Y, and their joint density function fXY(x,y). We form the functions Z ¼ g(X,Y ) and W ¼ h(X,Y ) and determine the joint density fZW(z,w) from knowledge of fXY (x,y). The analysis is very similar to the one presented in the previous section. The functions Z ¼ g(X,Y ) and W ¼ h(X,Y ) map the domain space of fZ,Wg given by fCxy, FXY, PXYg into the range space fCzw, FZW, PZWg as shown in Fig. 13.2.1. From Fig. 13.2.1, we obtain Z ¼ g(X,Y), W ¼ h(X,Y);

{Cxy , FXY ,PXY } ! {Czw , FZW , PZW }

(13:2:1)

and the mapping of fX, Y g is from the probability space fS, F, Pg (where S is the product space S ¼ SX SY) into the domain space of fZ,Wg given by fCxy, FXY, PXYg: X,Y : {S, F, P} ! {Cxy , FXY , PXY }

(13:2:2)

Alternately, if Dxy , Cxy and Dzw , Czw, then Dzw is the image of Dxy under the transformation fg,hg and Dxy is the inverse image of Dzw under the transformation {g – 1, h – 1}. Hence, Dzw is the set of all points (x,y) belonging to Dxy such that g(x,y), h(x,y) belongs

FIGURE 13.2.1

13.2

Two Functions of Two Random Variables, Z 5 g(X,Y ), W 5 h(X,Y )

223

to Dzw. This is represented by Dxy ¼ {x,y:g(x,y),h(x,y) [ Dzw }

(13:2:3)

We also have conditions similar to the last section for Z and W to be a random variables. Conditions on Z 5 g(X,Y ) and W 5 h(X,Y ) to Be Random Variables 1. The domain of fg(x,y), h(x,y)g must include the range of the RV Z and W. 2. For each z,w , Dzw, the set fg(x,y) z, h(x,y) wg must form a ﬁeld FZW; that is, it must consist of the unions and intersections of a countable number of intervals. 3. The probability of the events fg(x,y) ¼ +1g and fh(x,y) ¼ +1g is 0. In essence, the transformation fZ ¼ g(X,Y ), W ¼ h(X,Y )g induces a probability measure PZW on FZW, and we have to ﬁnd this probability measure. From Fig. 13.2.1 we can write PZW (Z,W) ¼ P(Z,W [ Dzw ) ¼ PXY g1 (Z,W),h1 (Z,W) [ Dxy ) ¼ PXY X,Y [ Dxy

(13:2:4)

The set Dzw consists of all the countable unions and intersections of the type fa , Z bg, fc , W dg. Without loss of generality, we will assume that Dzw ¼ fZ z,W wg. Thus, substituting for Dxy from Eq. (13.2.3) in Eq. (13.2.4), we have PZW (Z,W) ¼ PXY X,Y [ Dxy ¼ PXY ½ x,y:g(x,y),h(x,y) [ Dzw ¼ PXY ½x,y:g(x,y) z,h(x,y) w ¼ FXY (z,w) ¼ P(Z z,W w) ¼ FZW (z,w)

(13:2:5)

Equation (13.2.5) can also be written as PZW (Z,W) ¼ PXY (X,Y) [ Dxy ¼ FZW (z,w) ¼

ðð fXY (j,h)dj dh

(13:2:6)

Dxy

where fXY(x,y) is the joint density of the random variables X and Y. In conclusion, we can say that the probability of fZ z, W wg is given by the probability of the set of all values of x and y such that fg(x,y) z, h(x,y) wg or by integrating the joint density fXY(x,y) over the domain Dxy. We will use both these interpretations in solving problems. Example 13.2.1 Here we are given that Z ¼ max(X,Y ), W ¼ min(X,Y ) and we have to ﬁnd FZW(z,w) from the knowledge of the joint density FXY(x,y). Both Z and W are shown in Fig. 13.2.2. The regions min(X,Y ) w ¼ fX wg < fY zg and max(X,Y ) z ¼ fX zg > X,Y ) z ¼ fX zg > fY zg are shown in Fig. 13.2.2. There are two regions of interest: w . z and w z. In the ﬁrst region w . z, Dxy equals the semiinﬁnite square deﬁned by the point B ¼ (z, z). Thus FZ,W (z,w) ¼ FXY (z,z) ¼ FZ (z):

w.z

(13:2:7)

224

Functions of Multiple Random Variables

FIGURE 13.2.2

In the second region w z, Dxy equals the semiinﬁnite square deﬁned by the point B ¼ (z,z) minus the square ABCD. Hence FZ,W (z,w) ¼ FXY (z,z) P(ABCD) ¼ FXY (z,z) ½FXY (z,z) FXY (z,w) FXY (w,z) þ FXY (w,w) ¼ FXY (z,w) þ FXY (w,z) FXY (w,w); w z

(13:2:8)

When w ¼ z, Eq. (13.2.8) is identical to Eq. (13.2.7). We can obtain FW(w) by letting z tend to 1 in Eq. (13.2.8), yielding FZ,W (z,w) ¼ FXY (z,w) þ FXY (w,z) FXY (w,w) ¼ FX (w) þ FY (w) FXY (w,w) ¼ FW (w) which is similar to Eq. (13.1.7) derived in the previous section. If X and Y are independent, then Eqs. (13.2.7) and (13.2.8) become w.z FX (z)FY (z): FZ,W (z,w) ¼ FX (z)FY (w) þ FX (w)FY (z) FX (w)FY (w): w z

(13:2:9)

(13:2:10)

Example 13.2.2 To continue the previous example, the density functions of two independent random variables X and Y are given by fX (x) ¼ 2e2x u(x);

fY (y) ¼ ey u(y)

Hence the joint density function fXY(x,y) is fXY (x;y) ¼ 2eð2xþyÞ u(x)u(y) Given random variables Z ¼ max(X,Y ) and W ¼ min(X,Y ), we have to ﬁnd the joint distribution FZW(z,w) and the marginal distributions FZ(z) and FW(w). The joint distribution FXY(x,y) can be obtained by integrating fX(x) and fY( y): FXY (x,y) ¼ 1 e2x ð1 ey Þu(x)u(y) ¼ 1 e2x ey þ e(2xþy) u(x)u(y)

13.2

Two Functions of Two Random Variables, Z 5 g(X,Y ), W 5 h(X,Y )

Using Eq. (13.2.10), we obtain 8 z . 0, 1 e2z ð1 ez Þ; > : 1 e2w ð1 ew Þ; z . 0, 1 e2z ez þ e3z ; z . 0, ¼ 2z z (2zþw) (zþ2w) 3w þe e ; z . 0, 1e e þe

225

w.z wz w.z wz

as the joint distribution function FZW(z,w). The marginal distribution FZ(z) is obtained by letting w ¼ z in the second equation of FZW(z,w) since w cannot exceed z. Thus FZ (z) ¼ 1 e2z ez þ e3z ,

z.0

which is the same as the ﬁrst equation. To obtain FW(w), we let z tend to 1 in the second equation of FZW(z,w) and obtain FW (w) ¼ 1 e3w ,

w.0

Example 13.2.3 We are given Z ¼ X 2 þ Y 2 and W ¼ Y/X, where X and Y are zero mean independent Gaussian random variables with joint density function fXY (x,y) ¼

1 ½(x2 þy2 )=2s2 e 2ps2

The region Dxy is the set of all values of x,y such that x 2 þ y 2z and y/x w and is shown in Fig. 13.2.3. The joint distribution FZW(z,w) is found by integrating FZW(z,w) over the shaded region of the ﬁgure. ðð FZW (z,w) ¼

1 ½(x2 þy2 )=2s2 e dx dy 2 Dxy 2ps

FIGURE 13.2.3

226

Functions of Multiple Random Variables

pﬃﬃ pﬃﬃ pﬃﬃ pﬃﬃ Using polar coordinates, x ¼ z cos (u), y ¼ z sin (u), dx dy ¼ z d z du, we can rewrite the preceding equation as ð pﬃz

ðu FZW (z,u) ¼ 2

(p=2) 0

1 (z=2s2 ) pﬃﬃ pﬃﬃ z d z da e 2ps2

with the factor 2 appearing since there are two shaded regions. This last equation can be simpliﬁed as follows: ð 1 p z 1 (z=2s2 ) pﬃﬃ dz z pﬃﬃ e uþ FZW (z,u) ¼ p 2 0 s2 2 z 1 u p p 2 ¼ þ 1 eðz=2s Þ ; z . 0, , u 2 p 2 2 Since w ¼ ( y/w) ¼ tan(u), we can substitute u ¼ tan – 1(w) and obtain the marginal distributions FZ(z) and FW(w) as follows: 2

FZ (z) ¼ 1 e(z=2s ) ;

z . 0;

FW (w) ¼

1 tan1 (w) þ ; 1 , w , 1 2 p

The corresponding density functions are obtained by differentiation: fZ (z) ¼

1 (z=2s2 ) e ; z . 0; 2s2

fW (w) ¼

1 ; 1 , w , 1 p(1 þ w2 )

The functions FW(w) and fW(w) are Cauchy-distributed and are shown in Figures 13.2.4 and 13.2.5. The examples given above are some of the simple problems to solve with transformations of multiple functions of multiple random variables. In general, this method of solving for distribution functions of two functions of two random variables is more difﬁcult. We now analyze alternate methods of ﬁnding the joint density functions.

FIGURE 13.2.4

13.3

Direct Determination of Joint Density fZW(z,w ) from fXY (x,y )

227

FIGURE 13.2.5

B 13.3 DIRECT DETERMINATION OF JOINT DENSITY fZW(z,w ) FROM fXY (x,y ) It is easier to obtain fZW(z,w) directly from the joint density function fXY(x,y) exactly as we did in the case of a single function of a single random variable. We are given the transformation Z ¼ g(X,Y ) and W ¼ h(X,Y ), and the inverse images g – 1(Z,W ) and h – 1(Z,W ) exist. However, corresponding to the differential parallelogram DDzw in the Z –W P plane, there may be several differential unconnected parallelograms DDxy ¼ dxkdyk, k ¼ 1, . . . , n in the X– Y plane as shown in Fig. 13.3.1. These differential regions are connected by the relation dz dw ¼ dx1 dy1 jJ(x1 ,y1 )j þ þ dxn dyn jJ(xn ,yn )j where jJ(x,y)j is the determinant of the Jacobian matrix J(x,y), deﬁned by 2 3 @z(x,y) @z(x,y) 6 @x @y 7 7 J(x,y) ¼ 6 4 @w(x,y) @w(x,y) 5 @x @x

FIGURE 13.3.1

(13:3:1)

(13:3:2)

228

Functions of Multiple Random Variables

The joint density fZW(z,w) can be obtained from fXY(x,y) as follows: fZW (z,w)dz dw ¼ fXY (x1 ,y1 )dx1 dy1 þ þ fXY (xn ,yn )dxn dyn

(13:3:3)

Substitution of Eq. (13.3.1) in Eq. (13.3.3) results in fZW (z,w)dz dw ¼ ¼

fXY (x1 ,y1 )dz dw fXY (xn ,yn )dz dw þ þ jJ(x1 ,y1 )j jJ(xn ,yn )j n X fXY (xk ,yk )dz dw jJ(xk ,yk )j k¼1

(13:3:4)

n X fXY (xk ,yk ) jJ(x k ,yk )j k¼1

(13:3:5)

or fZW (z,w) ¼

From this derivation it is clear that certain restrictions have to be satisﬁed for g(x,y) and h(x,y) if we have to determine fZW(z,w) directly from fXY(x,y): 1. All the partial derivative of g(x,y) and h(x,y) must exist. 2. There cannot be any planar solutions to g(x,y) ¼ z and h(x,y) ¼ w. We can now formulate the methodology of ﬁnding fZW(z,w) from fXY(x,y). Steps for Direct Determination of fZW(z, w) 1. Check whether the restrictions on g(x,y) and h(x,y) are satisﬁed. 2. Find the n real solutions to the system of equations g(x,y) ¼ z;

h(x,y) ¼ w

and express them in terms of z and w. Let the n solutions be fxk, yk, k ¼ 1, . . . , ng. 3. Find the Jacobian determinant jJj for each xk,yk, and express this also in terms of z and w. 4. Find the density function fZW(z,w) from Eq. (13.3.5): fZW (z,w) ¼ Example 13.3.1

n X fXY (xk ,yk ) jJ(xk ,yk )j k¼1

The functions Z ¼ g(X,Y ) and W ¼ h(X,Y ) are deﬁned by g(X,Y) ¼ X þ Y; h(X,Y) ¼ X Y

where X and Y are independent positive random variables with joint density fXY (x,y) given by 6e – (3xþ2y) . u(x)u( y). We have to ﬁnd fZW(z,w). 1. All partials exist for g(x,y) ¼ x þ y and h(x,y) ¼ x y. 2. In the equations x þ y ¼ z and x y ¼ w, we note that z . 0 and w can have negative values. Solving for x and y results in one real solution: 1 1 1 zþw zw x z ; y¼ ¼ or x ¼ y 1 w 2 1 2 2 Since x and y are positive, we must have z . jwj and z . w. 3. The Jacobian determinant is jJj ¼ 2.

13.3

Direct Determination of Joint Density fZW(z,w ) from fXY (x,y )

229

FIGURE 13.3.2

4. The joint density function fZW(z,w) from Eq. (13.3.5) is given by h zþw 1 z wi þ2 ¼ 3e½(5zþw)=2 ; z . 0, z , w z fZW (z,w) ¼ 6 exp 3 2 2 2 Finding the marginal densities fZ(z) and fW(w) are a little tricky because of ﬁnding the limits of integrations with respect to z and w. ðz fZ (z) ¼ 3e½(5zþw)=2 dw; z . 0, z , w z z

¼ 6 e2z e3z u(z) Function fZ(z) is shown in Fig. 13.3.2. Since w can take positive and negative values, we have the following integrals for fW(w) for the two regions w . 0 and w 0. In the region w . 0, we have ð1 fW (w) ¼ 3e½(5zþw)=2 dz; w . 0 w

6 ¼ e3w u(z) 5 and in the region w 0, we have ð1 fW (w) ¼

3e½(5zþw)=2 dz;

w0

w

¼

6 2w e u(w) 5

The function fW(w) is shown in Fig. 13.3.3. The distribution function FW(w) is obtained by integrating fW(w) and is given by 8 3 > < (e2w ) w0 FW (w) ¼ 5 > : 3 þ 2 (1 e3w ) w . 0 5 5 It is shown in Fig. 13.3.4.

230

Functions of Multiple Random Variables

FIGURE 13.3.3

Example 13.3.2 Two independent positive random variables X and Y are jointly exponentially distributed with density function fXY(x,y) ¼ 6e 2(3x þ 2y) u(x)u( y), exactly as in the previous example. Two other random variables are Z and W, given by, Z ¼ 2X=(X þ Y) and W ¼ X þ Y. We have to ﬁnd the joint density fZW(z,w) and the marginal densities fZ(z) and fW(w). 1. All partials exist for g(x,y) ¼ 2x=(x þ y) and h(x,y) ¼ x þ y. 2. Solving for 2x=(x þ y) ¼ z and x þ y ¼ w, we have x ¼ zw=2 y ¼ ½1 (z=2)w. Since x and y are positive, w . 0 and 0 , z 2. 3. The Jacobian matrix is 2

2y 4 J ¼ (x þ y)2 1

3 2x (x þ y)2 5 1

and the determinant is j Jj ¼

2 2 ¼ xþy w

FIGURE 13.3.4

and

13.3

Direct Determination of Joint Density fZW(z,w ) from fXY (x,y )

231

4. The joint density function fZW(z,w) from Eq. (13.3.5) is given by nh zw w z io fZW (z,w) ¼ 6 exp 3 þ 2(1 )w ¼ 3wew½2þ(z=2) ; w . 0, 0 , z 2 2 2 2 Integrating over all w, we obtain fZ(z) ð1 fZ (z) ¼ 3wew½2þðz=2Þ dw ¼ 0

12 ; (z þ 4)2

0,z2

and the corresponding distribution function FZ(z) is 8 0: > > z0 < 4 : 0,z2 FZ (z) ¼ 3 1 > zþ4 > : 1: z.2 The functions fZ(z) and FZ(z) are shown in Fig. 13.3.5. Integrating over all z, we obtain fW(w) ð2 fW (w) ¼ 3w ew½2þðz=2Þ dz ¼ 6(e2w e3w );

w.0

0

and this result is the same one obtained in the previous example and shown in Fig. 13.3.2. Integrating fW(w), we obtain FW(w): FW (w) ¼ 1 þ 2e3w 3e2w ;

w.0

Example 13.3.3 We will ﬁnd fZW(z,w) from fXY(x,y) for Example 13.2.3, where Z ¼ 2 2 2 X 2 þ Y 2 and W ¼ Y=X and the joint density fXY (x,y) ¼ (1=2ps2 )e½(x þy )=2s . 1. All partials exist for g(x,y) ¼ x2 þ y2 and h(x,y) ¼ y=x. 2. We solve x2 þ y2 ¼ z and y=x ¼ w. Substituting y ¼ wx in x2 þ y2 ¼ z, we obtain rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ z w2 z 2 2 x 1 þ w ¼ z or x ¼ + and y ¼ + 1 þ w2 1 þ w2

FIGURE 13.3.5

232

Functions of Multiple Random Variables

or

2 0rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 3 2 z w z @ A; z . 0, 1 , w , 1 7 , 6 6 7 1 þ w2 1 þ w2 (x1 , y1 ) 6 7 ¼6 7 ! ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ r rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (x2 , y2 ) 6 7 2 z w z 4 5 ; z . 0, 1 , w , 1 , 2 2 1þw 1þw

3. The Jacobian matrix is given by

2

2x J¼4 y 2 x

3 2y 15 x

and the Jacobian determinant

y2 jJj ¼ 2 1 þ 2 ¼ 2(1 þ w2 ) x

4. The joint density fZW(z,w) is given by 1 1 (z=2s2 ) 1 ðz=2s2 Þ fZW (z,w) ¼ e þ e 2(1 þ w2 ) 2ps2 2ps2 or

1 1 (z=2s2 ) e ; fZW (z,w) ¼ p(1 þ w2 ) 2s2

z . 0, 1 , w 1

Integrating this equation with respect to w, the marginal density fZ(z) is ð1 1 dw 2 fZ (z) ¼ 2 e(z=2s ) 2 2s 1 p(1 þ w )

1 1 1 (w)

1 2 tan 2 ¼ 2 e1(z=2s ) ; z . 0 ¼ 2 e(z=2s )

2s p 2s 1 and by a similar integration with respect to z we obtain fW(w) as 1 fW (w) ¼ p(1 þ w2 ) which is a Cauchy density. These results correspond to the results obtained in Example 13.2.3. Example 13.3.4 Two independent zero mean Gaussian random variables X and Y has joint density function fXY(x,y) given by 1 (1=2)½(x2 þy2 Þ=s2 fXY (x,y) ¼ e 2ps2 Two other random variables Z and W are deﬁned by 8 1 Y > > < tan pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X Z ¼ X2 þ Y 2; W ¼ > Y > : p þ tan1 X

if

X.0

if

X0

13.4

Solving Z 5 g(X,Y ) Using an Auxiliary Random Variable

233

The angle W has been deﬁned in this manner so that it covers the region ((p=2), (3p=2) since tan1 (Y=X) covers only the region ((p=2),(p=2)]. We have to ﬁnd the joint density fZW(z,w) and the marginal densities fZ(z) and fW(w). We apply the usual four steps to ﬁnd a solution to this problem. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1. All partials exist for g(x,y) ¼ x2 þ y2 and h(x,y) ¼ tan1 (y=x). pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2. We solve for x2 þ y2 ¼ z and tan1 (y=x) ¼ w for x . 0 and p þ tan1 (y=x) ¼ w ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ for x 0. We note that tan(w) ¼ y/x for both x 0 and x . 0. Squaring p x2 þ y2 ¼ z and substituting y=x ¼ tan (w), we can write y2 2 x 1 þ 2 ¼ z2 or x2 1 þ tan2 (w) ¼ z2 or x2 ¼ z2 cos2 (w) x However, for x . 0; (p=2) , w (p=2) and cos (w) . 0 and for x 0; (p=2) , w (3p=2) and cos(w) 0. Hence, the only solutions are x ¼ z cos(w) and y ¼ z sin(w) in the range (p=2) , w (3p=2). 3. The Jacobian matrix and its determinant are given by 2 3 x y pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 6 x2 þ y2 1 1 x2 þ y2 7 7 J¼6 ¼ 4 y y 5 and jJj ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 z x þy x2 þ y2 x2 þ y2 4. The joint density fZW(z,w) is given by fZW (z,w) ¼

1 z (1=2)(z2 =s2 ) e ; z.0 2p s2

Integrating with respect to w in the range (p=2) , w (3p=2), we obtain z 2 2 fZ (z) ¼ 2 e(1=2)(z =s ) ; z . 0 s which is a Rayleigh distribution, and the angle w is uniformly distributed in (p=2) , w (3p=2), or fW (w) ¼

1 p 3p ; ,w 2p 2 2

and Z and W are also independent random variables.

B 13.4 SOLVING Z 5 g(X,Y ) USING AN AUXILIARY RANDOM VARIABLE The problem of ﬁnding the density function fZ(z) of a function of two random variables Z ¼ g(X,Y ) can be considerably simpliﬁed by converting to two functions of two random variables with the use of an auxiliary random variable W ¼ X or Y, provided g(x,y) satisﬁes the two restrictions of differentiability and no straight-line solutions. Example 13.4.1 We will now solve Example 13.1.10 using an auxiliary random variable. Here g(X,Y ) ¼ Z ¼ XY and we introduce an auxiliary random variable W ¼ h(X,Y ) ¼ X.

234

Functions of Multiple Random Variables

1. The restrictions on g(x,y) ¼ xy and h(x,y) ¼ x are satisﬁed. 2. Solving for xy ¼ z and x ¼ w, we obtain the real solution, x ¼ w and y ¼ z=w. 3. The Jacobian matrix and its absolute determinant are y x J¼ or kJk ¼ jxj ¼ jwj 1 0 4. The joint density fZW(z,w) is given by fZW (z,w) ¼

z 1 fXY w, jwj w

The function fZ(z) is obtained by integrating this equation with respect to w ð1 z 1 FZ (z) ¼ fXY w, dw w 1 jwj which is the same as Eq. (13.1.33). pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Example 13.4.2 This is the same as Example 13.1.7, where Z ¼ g(X,Y) ¼ X 2 þ Y 2 . We introduce the auxiliary random variable W ¼ h(X,Y ) ¼ X and proceed to solve it using the techniques of the previous section. The joint density fXY(x,y) is given by fXY (x,y) ¼

1 (1=2)½(x2 þy2 )=s2 e 2ps2

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1. The restrictions on g(x,y) ¼ x2 þ y2 and h(x,y) ¼ x are satisﬁed. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ obtain x2 þ y2 ¼ z2 or y2 p ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ z2 x2 , and 2. Solving for x2 þ y2 ¼ z and x ¼ w,pwe ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ the two solutions are x1 ¼ w, y1 ¼ z2 w2 and x2 ¼ w, y2 ¼ z2 w2 with 2z , w z for real y1 and y2. 3. The Jacobian matrix and its absolute determinant are 2 3 x y pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ z2 w2 y 2 2 2 2 4 5 ; or kJk ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ J¼ x þy x þy z x2 þ y2 1 0 4. The joint density function fZW(z,w) is given by z 2 (1=2)(z2 =s2 ) e ; z . 0, z , w z fZW (z,w) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ z2 w2 2ps2 Integrating with respect to the variable w, we obtain ðz z dw 2 2 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; z . 0 fZ (z) ¼ 2 e(1=2)(z =s ) s p z z2 w2 Since the integrand is even, we can write this equation as ð z (1=2)(z2 =s2 ) 2 z dw pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; z . 0 fZ (z) ¼ 2 e s p 0 z2 w2 z z 2 2 2 p 2 2 ¼ 2 e(1=2)(z =s ) ; z . 0 ¼ 2 e(1=2)(z =s ) s p2 s which is the same as the result obtained in Eq. (13.1.22).

jwj z

13.4

Solving Z 5 g(X,Y ) Using an Auxiliary Random Variable

235

Example 13.4.3 This is the same as Example 13.1.8, where Z ¼ g(X,Y) ¼ X 2 þ Y 2 . We introduce the auxiliary random variable W ¼ h(X,Y ) ¼ X and proceed to solve it using the techniques of the previous section. The joint density fXY (x,y) is given by fXY (x,y) ¼

1 (1=2)½(x2 þy2 )=s2 e 2ps2

1. The restrictions on g(x,y) ¼ x2 þ y2 and h(x,y) ¼ x are satisﬁed. 2. Solving for x2pþﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y2 ¼ z and x ¼ w, we obtain from Y 2 ¼ z x2 , the two solutions pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃ pﬃﬃ x1 ¼ w, y1 ¼ z w2 and x2 ¼ w, y2 ¼ z w2 , with z , w z for real y1 and y2. 3. The Jacobian matrix and its absolute determinant are pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃ 2x 2y J¼ or kJk ¼ 2y ¼ 2 z w2 ; z . 0, jwj z 1 0 4. The joint density function fZW(z,w) corresponding to the two solutions is given by, 1 2 (1=2)(z=s2 ) e ; fZW (z,w) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2ps 2 zw

pﬃﬃ pﬃﬃ z . 0, z , w z

Integrating with respect to the variable w, we obtain ð pﬃ 1 (1=2)(z=s2 ) z dw pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; z . 0 fZ (z) ¼ 2 e pﬃ 2 2s zp zw pﬃﬃ pﬃﬃ Substituting w ¼ z sin (u) and dw ¼ z cos (u)du, we obtain ð p pﬃﬃ 2 z cos (u)du 1 2 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; z . 0 fZ (z) ¼ 2 e(1=2)(z=s ) 2s p p2 z cos2 (u) ¼

1 (1=2)(z=s2 ) e ; z.0 2s2

which is the same result obtained in Eq. (13.1.24). Example 13.4.4 The random variable Z is given by Z ¼ (X þ Y )2, where X and Y are zero mean independent Gaussian random variables given by the joint density function fXY (x,y) ¼

1 (1=2)½(x2 þy2 )=s2 e 2ps2

We have to ﬁnd the density function fZ(z) using auxiliary random variable W ¼ X. 1. The restrictions on g(x,y) ¼ (x þ y)2 and h(x,y) ¼ x are satisﬁed. pﬃﬃ 2. Solving for (x þ y)2 ¼ z and x ¼ w, we obtain two solutions: x1 ¼ w, y1 ¼ z w pﬃﬃ and x2 ¼ w, y2 ¼ z w. 3. The Jacobian matrix and its absolute determinant are pﬃﬃ 2(x þ y) 2(x þ y) or kJk ¼ 2 z; J¼ 1 0

z.0

236

Functions of Multiple Random Variables

4. The joint density fZW(z,w) corresponding to the two solutions is 1 1 (1=2s2 )½w2 þ(pﬃzw)2 e fZW (z,w) ¼ pﬃﬃ 2 z 2ps2 1 1 (1=2s2 )½w2 þ(pﬃzw)2 þ pﬃﬃ e ; z.0 2 z 2ps2 We will now simplify this expression. Completing the squares in the ﬁrst and the second terms on the righthand side of the equation, we obtain pﬃﬃ 1 1 1 2 2 exp ½w þ ( z w) pﬃﬃ 2 z 2ps2 2s2 ( " pﬃﬃ2 #) z 1 1 2 z exp 2 w þ ¼ pﬃﬃ 2 2 z 2ps2 2s 4 pﬃﬃ 1 1 1 2 2 exp ½w þ ( z w) pﬃﬃ 2 z 2ps2 2s2 ( " pﬃﬃ2 #) z 1 1 2 z exp 2 w þ þ ¼ pﬃﬃ 2 2 z 2ps2 2s 4 Hence the joint density fZW(z,w) is given by 1 1 (z=4s2 ) n (1=s2 )½w(pﬃz=2)2 o e e fZW (z,w) ¼ pﬃﬃ 2 z 2ps2 pﬃ (1=s2 )½wþ( z=2)2 þe ; z . 0, 1 , w , 1 We integrate this equation over all w to obtain fZ(z): ð o pﬃ 1 1 (z=4s2 ) 1 n (1=s2 )½w(pﬃz=2)2 (1=s2 )½wþ( z=2)2 fZ (z) ¼ pﬃﬃ e e þ e dw 2 z 2ps2 1 2

pﬃﬃﬃﬃ 1 1 (z=4s2 ) pﬃﬃﬃﬃ 1 e(z=4s ) pﬃﬃﬃﬃ pﬃﬃ ; z . 0 ¼ pﬃﬃ e ½s p þ s p ¼ z 2 z 2ps2 2s p The distribution function FZ(z) is obtained by integrating fZ(z) over z: pﬃﬃ ð z (z=4s2 ) z 1 e pﬃﬃﬃ dz ¼ erf FZ (z) ¼ pﬃﬃﬃﬃ 2s 2s p 0 z Ðz 2 where erf(z) is deﬁned by erf(z) ¼ p2ﬃﬃpﬃ 0 ej dj and ð pﬃﬃ 1 z ej erf( z) ¼ pﬃﬃﬃﬃ pﬃﬃﬃ dj p 0 j Functions fZ(z) and FZ(z) are shown in Fig.13.4.1. Example 13.4.5 In this example X is a zero mean unit variance random variable given by the density function 1 2 fX (x) ¼ pﬃﬃﬃﬃﬃﬃ e(x =2) 2p

13.4

Solving Z 5 g(X,Y ) Using an Auxiliary Random Variable

237

FIGURE 13.4.1

and Y is an independent chi-square random variable with n degrees of freedom whose density is given by Eq. (7.7.2): y(v=2)1 e(y=2) v u(y) 2v=2 G 2 pﬃﬃﬃﬃﬃﬃﬃﬃ The density function of the random variable Z ¼ X Y=v is called the Student-t distribution (Section 7.9) and ﬁnds extensive application in statistical analyses. We will use an auxiliary variable W ¼ Y: fY (y) ¼

pﬃﬃﬃﬃﬃﬃﬃ 1. The restrictions on gðx;yÞ ¼ x= y=v and h(x,y) ¼ y are satisﬁed. pﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2. Solving for x= y=v ¼ z and y ¼ w, we obtain the solution x ¼ z w=v and y ¼ w. 3. The Jacobian matrix and its absolute determinant are 2

3 1 x p ﬃﬃﬃﬃﬃﬃﬃ J ¼ 4 y=v 2v(y=v3=2 ) 5 0 1

or

1 1 kJk ¼ pﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ ; y=v w=v

w.0

4. The density function fZW(z,w) is given by pﬃﬃﬃﬃﬃﬃﬃﬃ w=v w(v=2)1 e(w=2) 2 v u(w) fZW (z,w) ¼ pﬃﬃﬃﬃﬃﬃ e(z w=v2) 2p 2v=2 G 2 w(v1)=2 2 e(w=2)½(z =vÞþ1 2 ¼ u(w) pﬃﬃﬃﬃﬃﬃ v 2 pv G 2 The density function fZ(z) is found by integrating this equation over w, 1 fZ (z) ¼ pﬃﬃﬃﬃﬃﬃ v 2 pv G 2

ð 1 (v1)=2 w 2 e(w=2)½(z =v)þ1 dw 2 0

238

Functions of Multiple Random Variables

This equation can be rewritten by substituting j ¼ ðw=2Þ½(z2 =nÞ þ 1, resulting in ð1 1 fZ (z) ¼ j(v1)=2 ej dj 2 (vþ1)=2 pﬃﬃﬃﬃﬃﬃ v z 0 þ1 pv G 2 v The integral in this equation is a gamma function given by G½(v þ 1)=2. Thus vþ1 (vþ1)=2 G z2 2 fZ (z) ¼ pﬃﬃﬃﬃﬃﬃ v 1 þ v pv G 2 which is a Student-t density [Eq. (7.9.2)].

B 13.5 MULTIPLE FUNCTIONS OF RANDOM VARIABLES We can now generalize the results of the previous section to n functions of n random variables. Let {X1 , . . . , Xn } be the n random variables deﬁned by the joint density function fX1 , . . . , Xn (x1 , . . . , xn ). Let the n functions be Z1 ¼ g1 (X1 , . . . , Xn ), Z2 ¼ g2 (X1 , . . . , Xn ), . . . , Zn ¼ gn (X1 , . . . , Xn )

(13:5:1)

We assume that these functions satisfy the restrictions of existence of partial derivatives and no planar solutions. We now solve for g1 (x1 , . . . , xn ) ¼ z1 , g2 (x1 , . . . , xn ) ¼ z2 , . . . , gn (x1 , . . . , xn ) ¼ zn

(13:5:2)

Let the solutions be {x1 , . . . , xn } be expressed as functions of {z1 , . . . , zn }: x1 ¼ h1 (z1 , . . . , zn ), x2 ¼ h2 (z1 , . . . , zn ), . . . , xn ¼ hn (z1 , . . . , zn )

(13:5:3)

Let the Jacobian matrix be given by 2

@g1 6 @x1 J(x1 , . . . , xn ) ¼ 6 4 @gn @x1

3 @g1 @xn 7 7 @gn 5 ... @xn ...

(13:5:4)

and its determinant be expressed in terms of {z1 , . . . , zn }: kJk ¼ kJ(z1 , . . . , zn )k

(13:5:5)

The joint density function fZ1 , . . . , Zn (z1 , . . . , zn ) can now be given in terms of Eqs. (13.5.3) and (13.5.5): fZ1 , . . . , Zn (z1 , . . . , zn ) ¼

fX1 , . . . , Xn (x1 , . . . , xn ) kJ(z1 , . . . , zn )k ¼

fX1 , . . . , Xn ½h1 (z1 , . . . , zn )1 , . . . , hn (z1 , . . . , zn ) kJ(z1 , . . . , zn )k

(13:5:6)

For general functions {gk (x1 , . . . , xn , k ¼ 1, . . . , n}, {hk (z1 , . . . , zn , k ¼ 1, . . . , n} and kJ(z1 , . . . , zn )k are not that easy to solve. However, for linear functions

13.5

Multiple Functions of Random Variables

239

{gk (x1 , . . . , xn , k ¼ 1, . . . , n}, the solutions are not that difﬁcult, as shown by the following example. Example 13.5.1 Three independent zero mean unit variance Gaussian random variables X, Y, and Z are given by the joint density function fXYZ (x,y,z) ¼

1 2 2 2 pﬃﬃﬃﬃﬃﬃ e(1=2)(x þy þz ) 2p 2p

Three other random variables U, V, and W are given by 2 3 2 32 3 U 2 2 1 X 4 V 5 ¼ 4 5 3 3 54 Y 5 W 3 2 2 Z We have to ﬁnd the joint density fUVW (u, v, w) and the marginal densities fU(u), fV(v), and fW(w). The solution to 2 3 2 32 3 u 2 2 1 x 4 v 5 ¼ 4 5 3 3 54 y 5 w 3 2 2 z is 2 3 2 x 0 4y5 ¼ 4 1 z 1

2 1 2

32 3 2 3 3 2v 3w u 1 54 v 5 ¼ 4 u v þ w 5 4 w u 2v þ 4w

The Jacobian matrix and its determinant are given by 2 3 2 2 1 J ¼ 4 5 3 3 5; kJk ¼ 1 3 2 2 Hence, using Eq. (13.5.6), we obtain the joint density function fUVW(u, v, w) fUVW (u, v, w) ¼

2 2 2 1 1 pﬃﬃﬃﬃﬃﬃ e(1=2)½(2v3w) þ(uvþw) þ(u2vþ4w) 1 2p 2p

1 2 2 2 pﬃﬃﬃﬃﬃﬃ e(1=2)½(2u 9v 426w þ2uv30vw6wu) 2p 2p 8 2 319 0 u > > < = 1 1B 1 6 7C ½ u v w ½C ¼ exp v @ A 4 5 > > (2p)3=2 jCj1=2 : 2 ; w 2 3 9 19 12 where the matrix C ¼ 4 19 43 27 5and jCj ¼ 1. 12 27 17 ¼

The random variables X, Y, Z are independent, whereas the transformed random variables U, V, W are not independent. Their correlation coefﬁcients can be calculated as follows: 19 rUV ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:966, 9 43

27 rVW ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:999, 43 17

12 rWU ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:970 17 9

240

Functions of Multiple Random Variables

By successively integrating with respect to (v,w), (w,u) and (u,v), we obtain the marginal density functions fU(u), fV(v), and fW(w) as ð1 ð1 1 2 2 2 fU (u) ¼ pﬃﬃﬃﬃﬃﬃ e½(1=2)(2u þ9v 426w þ2uv30vw6wu) dv dw 2p 2p 1 1 1 2 ¼ pﬃﬃﬃﬃﬃﬃ e½(1=2)(u =9) 3 2p ð1 ð1 1 2 2 2 fV (v) ¼ pﬃﬃﬃﬃﬃﬃ e½(1=2)(2u þ9v þ26w þ2uv30vw6wu) dw du 2p 2p 1 1 1 2 ¼ pﬃﬃﬃﬃﬃﬃ e½(1=2)(v =43) 43 2p ð1 ð1 1 2 2 2 fW (w) ¼ pﬃﬃﬃﬃﬃﬃ e½(1=2)(2u þ9v 426w þ2uv30vw6wu) du dv 2p 2p 1 1 1 2 ¼ pﬃﬃﬃﬃﬃﬃ e½(1=2)(w =17) 17 2p These results can be veriﬁed from the results of Example 13.1.5. Since var(ax) ¼ a 2 var(X ), we have var(U ) ¼ 22 þ 22 þ 12 ¼ 9, var(V ) ¼ 52 þ 32 þ 32 ¼ 43, and var(W ) ¼ 32 þ 22 þ 22 ¼ 17, which checks with the results above.

CHAPTER

14

Inequalities, Convergences, and Limit Theorems

B 14.1

DEGENERATE RANDOM VARIABLES

In Section 10.2 it was stated that if the variance is zero, then the random variable will no longer be random. We can state this concept more fully as follows. If E(X 2 a)2¼0, where a is a constant, then the random variable X equals a with probability 1. We shall show this result as follows. We construct a small hole of nonzero width 21 with 1 . 0 about the value a in the region of integration of x from 21 to 1 as shown in Fig. 14.1.1. We integrate (x2a)2 fX(x) from 21 to 1 except for the hole about a and write ð1 ð E(X a)2 ¼ (x a)2 fX (x)dx (x a)2 fX (x)dx (14:1:1) jxaj.1

1

By substituting the lowest value of 1 for jx –aj in Eq. (14.1.1), we can write ð ð 2 2 2 E(X a) 1 fX (x)dx ¼ 1 fX (x)dx ¼ 12 P(jX aj . 1) jxaj.1

jxaj.1

2

Since E(X a) ¼ 0 by assumption, we must have 12 P(jX aj . 1) ¼ 0: Since 1 is a nonzero constant, we must have P(jX aj . 1 ¼ 0: As a consequence P(jX aj ¼ 0) ¼ 1

and

X ¼ a with probability 1:

(14:1:2)

2

What this result conveys is that if E(X2a) ¼ 0, then almost all realizations of X will be equal to a constant a. The random variable X is called a degenerate random variable. This method of analysis will be useful in showing some of the inequalities that will follow. Example 14.1.1 If the variance s2X ¼ 0, then the random variable X ¼ mX almost certainly. Since variance of X equals E(X mX )2 ¼ 0, then, by Eq. (14.1.2) X ¼ mX with probability 1.

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

241

242

Inequalities, Convergences, and Limit Theorems

FIGURE 14.1.1

B 14.2

CHEBYSHEV AND ALLIED INEQUALITIES

In many problems in communications we are interested in ﬁnding bounds on certain performance measures. The idea in using a bound is to substitute a simpler expression for something more complicated, thus gaining some insight into the problem at the expense of accuracy. This process usually requires a fair amount of experience with the results that are to be expected. One of the bounds that we are interested in is the tails probabilities of the type P(X . 1) or P(jXj . 1) (Fig. 14.2.1). Even if we cannot ﬁnd these probabilities exactly, we should be able to ﬁnd bounds on these quantities. These bounds are based on partial information about the random variable, such as mean and variance. Chebyshev Bound

The tails probability bound of the random variable X of the form P(jX –mXj , 1) is based on the knowledge of only the mean and the variance of X and is known as the Chebyshev inequality, given by P{jX mX j 1}

s2X 12

or

FIGURE 14.2.1

CB(1) ¼

s2X 12

(14:2:1)

14.2

Chebyshev and Allied Inequalities

243

where 1 is a positive constant .0 and CB(1) is the Chebyshev bound (this result can be shown using an analysis similar to that in the previous section): ð1 ð (x mX )2 fX (x)dx (x mX )2 fX (x)dx (14:2:2) s2X ¼ jxmX j.1

1

By substituting the lowest value of 1 in Eq. (14.2.2), we obtain ð fX (x)dx ¼ 12 P(jx mX j 1) s2X 12

(14:2:3)

jxmX j.1

and hence Eq. (14.2.1). Geometric Derivation of the Chebyshev Bound We will now derive the Chebyshev bound from a different perspective. We will deﬁne a random variable Y ¼ X – mX. We will make use of the indicator functions deﬁned in Eq. (12.5.7). Let A be a set on the real line such that A ¼ f –1, – 1g < f1, 1g. We deﬁne an indicator function IA as follows: 1 if Y [ A IA ¼ (14:2:4) 0 if Y A

The indicator function IA( y) is shown in Fig. 14.2.2. Taking the expected value of IA, we have E½IA ¼ 1 P(Y [ A) þ 0 P(Y A) ¼ P(Y [ A) ¼ P(jYj 1) ¼ P(jX mX j 1)

(14:2:5)

We will now bound IA by another random variable Y 2/12 as shown in Fig. 14.2.2. From the ﬁgure we can see that IA

Y 2 jX mX j2 ¼ 12 12

FIGURE 14.2.2

(14:2:6)

244

Inequalities, Convergences, and Limit Theorems

Taking expectation on both sides of Eq. (14.2.6), we have E½IA ¼ P(jX mX j 1)

E½Y 2 EjX mX j2 s2X ¼ ¼ 2 12 12 1

(14:2:7)

which is Chebyshev inequality. The bound is E½Y 2 =12 is called the Chebyshev bound. We will use this technique for deriving the Chernoff bound. Alternate Forms of the Chebyshev Bound By rearranging the terms in Eq. (14.2.1), we can rewrite Chebyshev’s inequality in terms of the center probability:

P{jX mX j , 1} 1

s2X 12

(14:2:8)

By substituting 1 ¼ nsX, we can write Eq. (14.2.1) as follows: P{jX mX j nsX }

1 n2

(14:2:9)

Thus, for any random variable X, the probability of a deviation from the mean mX of more than n standard deviations is 1/n 2. For example, in a standard Gaussian if n ¼ 3, 1=n2 ¼ 19 ¼ 0:1111. Chebyshev inequality tells us that the probability of deviation by more than 3 from 0 in a standard Gaussian is 0.1111. In actual cases this probability is 0.0027. We can also rewrite Eq. (14.2.5) analogous to Eq. (14.2.4) as P{jX mX j , nsX } 1

1 n2

(14:2:10)

Generalized Chebyshev Inequality

If f(x) is a convex function (Section 14.6) in the sense that its second derivative is positive for all x, then the inequality in Eq. (14.2.1) can be generalized to P{jXj . 1}

E½f(X) f(1)

(14:2:11)

This can be shown by using techniques similar to those used in demonstrating Chebyshev inequality, applying the fact that f(x) is convex: ð ð1 f(x)fX (x)dx f(x)fX (x)dx (14:2:12) E½f(X) ¼ 1

jxj.1

Using the least value for f(x), namely, f(1) in Eq. (14.2.12), we have ð f(1)fX (x)dx ¼ f(1)P{jXj . 1} E½f(X)

(14:2:9)

jxj.1

and hence Eq. (14.2.11). If f(X ) ¼ (X– mX)2 in Eq. (14.2.11), we get Chebyshev inequality. Example 14.2.1 We will apply the Chebyshev bounds to two different distributions, a Gaussian distribution and a Laplace distribution with zero means and unit variances.

14.2

Chebyshev and Allied Inequalities

245

FIGURE 14.2.3

They are given by 1 2 fN (x) ¼ pﬃﬃﬃﬃﬃﬃ eðx =2Þ ; 2p

pﬃﬃ 1 fL (x) ¼ pﬃﬃﬃ e 2jxj 2

and shown in Fig. 14.2.3. We will ﬁnd the tails probabilities P(jXj 1) for both the distributions and the Chebyshev bound, which will be the same for both since they have the same means and variances. Applying Eq. (14.2.1), the Chebyshev bounds are P{jXj 1}

1 12

Thus the Chebyshev bound CB(1) ¼ 1/12. Since we know both the density functions, we can calculate the exact tails probabilities P (jXj 1), which are given by ð ð 1 1 x2 =2 1 1 x2 =2 1 FN (1) ¼ pﬃﬃﬃﬃﬃﬃ e dx þ pﬃﬃﬃﬃﬃﬃ e dx ¼ 1 erf pﬃﬃﬃ 2 2p 1 2p 1 ð 1 pﬃﬃﬃﬃ ð 1 pﬃﬃﬃﬃ p ﬃﬃ 1 1 e 2x dx þ pﬃﬃﬃ e 2x dx ¼ e 2j1j FL (1) ¼ pﬃﬃﬃ 2 1 2 1 where

rﬃﬃﬃﬃ ð 1 1 2 2 ex =2 dx erf pﬃﬃﬃ ¼ p 0 2

The one-sided tails probabilities FN (1) and FL(1) for the Gaussian density and the Laplace density, and the Chebyshev bound CB(1) are shown in Fig. 14.2.4 and tabulated in Table 14.2.1 for 1 ranging from 1.5 to 5. Conclusions

It is clear from Fig. 14.2.4 and Table 14.2.1 that the Chebyshev bound is a very loose one. As 1 increases, the bound becomes somewhat tighter (Fig. 14.2.4). Table 14.2.1 also shows that the Chebyshev bound is the same for both density functions, which have the same mean and variance. In fact, the Chebyshev bound is applicable to all density

246

Inequalities, Convergences, and Limit Theorems

FIGURE 14.2.4

TABLE 14.2.1 1 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5

Normal FN(1)

Laplace FL(1)

Chebyshev CB(1) ¼ 1/12

0.133614 0.080118 0.0455 0.024449 0.012419 0.00596 0.0027 0.001154 0.000465 0.000177 0.000063 0.000021 0.000007 0.000002 0.000001

0.119873 0.084174 0.059106 0.041503 0.029143 0.020464 0.01437 0.01009 0.007085 0.004975 0.003493 0.002453 0.001723 0.00121 0.000849

0.444444 0.326531 0.25 0.197531 0.16 0.132231 0.111111 0.094675 0.081633 0.071111 0.0625 0.055363 0.049383 0.044321 0.04

functions having the same means and variances. It is useful as a theoretical tool to determine the bounds of the tails probabilities, where only means and variances are known about the density functions.

B 14.3

MARKOV INEQUALITY

There are several different versions of Markov inequality. The usual version is deﬁned for a positive random variable X with a ﬁnite mean mX. It states that for a positive 1 . 0 m P(X 1) X (14:3:1) 1

14.3

Markov Inequality

247

This result can be shown by the usual techniques developed in the previous section ð1 ð1 ð1 xfX (x)dx xfX (x)dx 1 fX (x)dx ¼ 1P(X 1) (14:3:2) mX ¼ 0

1

1

and hence Eq. (14.3.1). In Eq. (14.3.1), the quantity mX =1 is called the Markov bound. Equation (14.3.1) can be slightly generalized to include both positive and negative random variables P(jX aj 1)

E(jX aj) 1

(14:3:3)

where a is an arbitrary constant and 1 . 0. Equation (14.3.3) can be further generalized to P(jX aj 1)

E(jX ajk ) 1k

(14:3:4)

where k is a positive integer. This inequality is called the Bienayme inequality. We will derive the value of a for which P(jX– aj 1) is a minimum in the equation P(jX aj 1)

E½jX aj2 12

(14:3:5)

for which we have to minimize E[jX – aj]2 with respect to a: E½jX aj2 ¼ E½X 2 2Xa þ a2 ¼ E½X 2 2aE½X þ a2

(14:3:6)

Differentiating Eq. (14.3.6) with respect to a and setting equal to 0, we have d {E½X 2 2aE½X þ a2 } ¼ 2a 2E½X ¼ 0 da from which we obtain a ¼ E[X ]. Thus we obtain Chebyshev inequality as follows: P(jX mX j 1)

E½jX mX j2 12

(14:2:1)

Example 14.3.1 A materials manager purchases 10,000 memory chips from a company where 3% of the chips are faulty. We have to ﬁnd the probability that the number of good chips lies between 9675 and 9775. Without loss in generality, we will assume that the chips are independent. Let Xk be the zero – one random variable 1 if the kth chip is good Xk ¼ 0 if the kth chip is bad with the probability of Xk ¼ 1 is 0.97. Thus E[Xk] ¼ 1 0.97 þ 0 0.03 ¼ 0.97 and E[X2k ] ¼ 1 0.97 þ 0 0.03 ¼ 0.97. Hence var[Xk] ¼ 0.97 2 (0.97)2 ¼ 0.0291. If Y is the sum of the random variables fXk, k ¼ 1, . . . , 10,000), then the mean of Y is " # 10,000 10,000 X X E½Y ¼ mY ¼ E Xk ¼ E½Xk ¼ 9700 k¼1

k¼1

With the assumed independence of Xk, the variance of Y is " # 10,000 10,000 X X 2 var½Y ¼ sY ¼ var Xk ¼ var½Xk ¼ 291 k¼1

k¼1

248

Inequalities, Convergences, and Limit Theorems

With a ¼ ð9675 þ 9775Þ=2 ¼ 9725, 1 ¼ 50, and E½Y a2 given by E½Y a2 ¼ E½Y 2 2Ya þ a2 ¼ EY 2 2mY a þ a2 ¼ s2Y þ m2Y 2mY a þ a2 ¼ 291 þ 97002 2 9700 9725 þ 97252 ¼ 916 we can use Eq. (14.3.5) and write P(jY 9725j 50

916 ¼ 0:3664 502

What we want is the complementary event P(jY 9725j 50), and this is given by 1 2 0.3664 ¼ 0.6336. Thus, the probability that the number of good chips will lie between 9675 and 9775 is 63.36%. On the other hand, if we want the number of good chips to lie in the region 9700 + 50, namely, 9650 and 9750, then the probability using Chebyshev inequality is P(jY 9700j 50) ¼ 1 P(jY 9700j 50) ¼ 1

291 ¼ 0:8836 ¼ 88:36% 2500

Example 14.3.2 In a binomially distributed random variable b(k; n, p) ¼ nk pk qnk , we have n ¼ 50 and p ¼ 0.1. Using Markov inequality, we want to ﬁnd the probability of X . 10 and compare it with the true value. The mean value is np ¼ 5 and 1 ¼ 10. Thus, from Eq. (14.3.1), we have the Markov bound P(X 10)

5 ¼ 0:5 10

and the true value is P(X 10) ¼

50 X

50! 0:1k 0:950k ¼ 0:02454 k!(50 k)! k¼10

This Markov bound is very loose! We will now ﬁnd the Chebyshev bound from Eq. (14.2.7) with f(x) ¼ x 2 and compare the results. Equation (14.2.7) becomes P{jXj . 1} E½X 2 =12 and E[X 2] ¼ npq þ (np)2 ¼ 29.5. The Chebyshev bound is {X . 10} (29:5=100) ¼ 0:295, which is better than the Markov bound of 0.5, but nowhere close to the true value of 0.02454.

B 14.4

CHERNOFF BOUND

In determining the Chebyshev bound, the indicator function was bounded by a quadratic. In an effort to improve the tightness of the bound, we will use a positive exponential bound the indicator function as shown in Fig. 14.4.1. The expectation of the indicator function yields E½IA ¼ P(jYj 1) ¼ P(jX mX j 1) l(Y – 1)

(14:4:1)

, l . 0, 1 . 0, as shown in Fig. 14.4.1. From The indicator function is bounded by e the ﬁgure, IA e l(Y21) and taking expectations on both sides, we obtain E½IA ¼ P(Y 1) E½el(Y1)

14.4

Chernoff Bound

249

FIGURE 14.4.1

or P(Y 1) el1 E½elY ¼ el1 MY (l)

(14:4:2)

where jYj has been replaced with Y since we are considering only the positive bound and MY(l) is the moment generating function of Y. The bound given above will hold for any l . 0. However, we would like to choose l so that e 2l1E[e lY] is a minimum. Differentiating e 2l1E[e lY] with respect to l and setting it equal to zero, we obtain d d l(Y1) ¼E E½el(Y1) e dl dl l¼l0 l¼l0 or

E (Y 1)el0 (Y1) ¼ el0 1 E (Y 1)el0 Y ¼ 0

(14:4:3)

Rearranging terms in Eq. (14.4.3), we obtain an implicit relation for the minimum l0 in terms of the moment generating functions (MGFs):

1¼

E½Yel0 Y MY0 (l0 ) ¼ E½el0 Y MY (l0 )

(14:4:4)

The Chernoff bound is given by P(Y 1) el0 1 MY (l0 )

(14:4:5)

where l0 is given by Eq. (14.4.4). A similar result holds for the bound on the negative side. Example 14.4.1 We will ﬁnd the bounds on tails probability for the standard Gaussian random variable Y using Chernoff’s formula and compare it to the Chebyshev bound found in Example 14.2.1. The MGF for any Gaussian random variable N is given by MN (t) ¼ etmþ(t MY (t) ¼ et

2

=2

2 2

s =2)

and for a standard Gaussian, Y N(0,1)

250

Inequalities, Convergences, and Limit Theorems

FIGURE 14.4.2

We will now ﬁnd the optimum l from Eq. (14.4.4): 2

M 0 (l0 ) l0 e(l0 =2) ¼ 1¼ Y 2 MY (l0 ) e(l0 =2)

or l0 ¼ 1

The Chernoff bound from Eq. (14.4.5) is 2

2

P(Y 1) el0 1 el0 =2 ¼ e1 e1

2

=2

¼ e 1

2

=2

¼ ChG(1)

The Chernoff bound, ChG(1), the one-sided Ð 1 x2 =2 Chebyshev bound, CBO(1), and the one-sided e dx are shown in Fig. 14.4.2 and in Table 14.4.1 normal distribution, FNo (1) ¼ p1ﬃﬃﬃﬃ 2p e for 1 ¼ 1.5, . . . , 3. TABLE 14.4.1 1 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

Normal FNo(1)

Chernoff ChG(1)

Chebyshev CBO(1) ¼ 1/212

0.066807 0.0548 0.044566 0.035931 0.028717 0.02275 0.017864 0.013903 0.010724 0.008197 0.00621 0.004661 0.003467 0.002555 0.001866 0.00135

0.324652 0.278037 0.235746 0.197899 0.164474 0.135335 0.110251 0.088922 0.071005 0.056135 0.043937 0.034047 0.026121 0.019841 0.014921 0.011109

0.222222 0.195312 0.17301 0.154321 0.138504 0.125 0.113379 0.103306 0.094518 0.086806 0.08 0.073964 0.068587 0.063776 0.059453 0.055556

14.5

Cauchy–Schwartz Inequality

251

From the ﬁgure and the table we see that for 1 . 2 the Chernoff bound is better than the Chebyshev bound. The crossover point occurs when 1 ¼ 2.075.

B 14.5

CAUCHY–SCHWARTZ INEQUALITY

An important inequality used in communication problems is the Cauchy – Schwartz inequality. Let g(x) and h(x) be two complex functions with ﬁnite norm deﬁned by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð 1

g(x)g (x)dx , 1

kg(x), g(x)l ¼ kg(x)k ¼ 1

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð

(14:5:1)

1

h(x)h (x)dx , 1

kh(x), h(x)l ¼ kh(x)k ¼ 1

The Cauchy–Schwartz inequality is given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð 1 ð1 ð1 2 g(x)h (x)dx jg(x)j dx jh(x)j2 dx 1

1

(14:5:2)

1

We can also state this inequality in terms of the inner product deﬁned by ð1 kg(x), h(x)l ¼ g(x)h (x)dx ¼ kh(x), g(x)l

(14:5:3)

1

and the inequality becomes kg(x), h(x)l kg(x)k kh(x)k

(14:5:4)

If we deﬁne two real-valued functions, a(x) and b(x), as a(x) ¼ jg(x)j,

b(x) ¼ jh(x)j

then we have to show that sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð ð

ð1

1

1

1

a2 (x)dx

a(x)b(x)dx 1

b2 (x)dx

(14:5:5)

1

Let us represent a(x) and b(x) in a function space spanned by the basis vectors f1(x) and f2(x). These vectors have the following orthonormal property: ð1 ð1 ð1 f21 (x)dx ¼ f22 (x)dx ¼ 1; f1 (x)f2 (x)dx ¼ 0 (14:5:6) 1

1

1

Thus a(x) ¼ a1 f1 (x) þ a2 f2 (x);

b(x) ¼ b1 f1 (x) þ b2 f2 (x)

(14:5:7)

where a1,a2,b1,b2 are coordinates in the function space of ff1(x), f2(x)g and the vectors, a ¼ ½a1 ,a2 ; b ¼ ½b1 ,b2 as shown in Fig. 14.5.1.

252

Inequalities, Convergences, and Limit Theorems

FIGURE 14.5.1

We will ﬁnd the inner product, ka(x), b(x)l ¼ can write

Ð1 1

a(x)b(x)dx. From Eq. (14.5.7) we

ð1

ð1

½a1 f1 (x) þ a2 f2 (x)½b1 f1 (x) þ b2 f2 (x)dx

a(x)b(x)dx ¼ 1

1

½using orthonormal properties of f1 (x) and f2 (x) ð1 ½a1 b1 f21 (x) þ a2 b2 f22 (x) dx ¼ a1 b1 þ a2 b2 ¼

(14:5:8)

1

From Fig. 14.5.1 we can write cos(u) ¼ cos(ua ub ) ¼ cos(ua ) cos(ub ) þ sin(ua ) sin(ub ) ¼

a1 b1 a2 b2 ab þ ¼ jaj jbj jaj jbj jajjbj

(14:5:9)

Since jcos(u)j 1, we have ja bj 1 and ja bj jajjbj (14:5:10) jajjbj ﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ Ð1 Ð1 Substituting for jaj ¼ 1 a2 (x)dx; jbj ¼ 1 b2 (x)dx in Eq. (14.5.10), we obtain Eq. (14.5.5) and hence Eq. (14.5.2). The equality is obtained when we substitute a ¼ Kb in Eq. (14.5.10) or a(x) ¼ Kb(x) and hence g(x) ¼ Kh(x). If g(X ) and h(X ) are real random variables, then we deﬁne the norms and the inner product given by Eqs. (14.5.1) and (14.5.3) as kg(X), g(X)l ¼ E½g2 (X) ¼

ð1

g2 (x)f (x)dx , 1

1

2

ð1

kh(X), h(X)l ¼ E½h (X) ¼

(14:5:11) 2

h (x)f (x)dx , 1

1

ð1 g(x)h(x)f (x)dx ¼ kh(X), g(X)l

kg(X), h(X)l ¼ E½g(X)h(X) ¼ 1

(14:5:12)

14.5

Cauchy–Schwartz Inequality

253

FIGURE 14.5.2

and Cauchy–Schwartz inequality of Eq. (14.5.2) becomes ﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð 1 ð1 ð1 2 2 g(x)h(x)f (x)dx g (x)f (x)dx h (x)f (x)dx 1

1

(14:5:13)

1

or pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E½g(X)h(X) E½g2 (X)E½h2 (X) Example 14.5.1 A communication system is shown in Fig. 14.5.2, where s(t) is the input signal and n(t) is a random variable representing input noise, so(t) and no(t) are the output signal and output noise, H( f ) is the transfer function. gi(t) the input signal-to-noise ratio (SNR), and go(t) the output signal-to-noise ratio are given by gi (t) ¼

s2 (t) ; E½n2 (t)

go (t) ¼

s2o (t) E½n2o (t)

The output so(t) is given by the inverse Fourier transform of S( f )H( f ), or ð1 ð1 s(t)h(t t)dt ¼ S( f )H( f )e j2pf df so (t) ¼ 1

(14:5:14)

(14:5:15)

1

and the output noise is characterized by E½n2o (t) ¼

ð1

jH( f )j2 N( f )df

(14:5:16)

1

where N( f ) is the frequency-domain characterization of the input noise n(t). With these characterizations of Eqs. (14.5.15) and (14.5.16), the output SNR go(t) can be given by Ð 1 2 j2pf df s2o (t) 1 S( f )H( f )e ¼ Ð1 go (t) ¼ 2 E½n2o (t) 1 jH( f )j N( f )df

(14:5:17)

The Cauchy–Schwartz inequality of Eq. (14.5.2) is modiﬁed to suit the frequency-domain characterization as follows 2 ð 1 ð1 ð1 2 A( f )B ( f )df jA( f )j df jB( f )j2 df 1

1

1

with the equality holding for A( f ) ¼ KB ( f ). We substitute pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A( f ) ¼ H( f ) N( f )

(14:5:18)

254

Inequalities, Convergences, and Limit Theorems

and S( f )e j2pft B( f ) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N( f )

(14:5:19)

A( f )B( f ) ¼ S( f )H( f )e j2pft

(14:5:20)

Thus

Applying Eqs. (14.5.19) and (14.5.20) in Eq. (14.5.18), the Cauchy–Schwartz inequality becomes 2 ð 1 ð 1 ð1 jS( f )j2 j2pft 2 df (14:5:21) S( f )H( f )e df jH( f )j N( f )df 1 1 1 N( f ) Substituting Eq. (14.5.21) in Eq. (14.5.17), the output SNR is given by ð1 ð1 jS( f )j2 2 df jH( f )j N( f )df 2 so (t) 1 1 N( f ) Ð ¼ g (t) o 1 2 E½n2o (t) 1 jH( f )j N( f )df or ð1 go (t)

jS( f )j2 df 1 N( f )

(14:5:22)

The equality will hold for A( f ) ¼ KB ( f ), or pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ S ( f )ej2pfT H( f ) N( f ) ¼ K pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N( f )

or Hopt ( f ) ¼ K

S ( f )ej2pfT jN( f )j

(14:5:23)

where we have substituted T the time of observation for t. The maximum output SNR is ð1 jS( f )j2 df (14:5:24) go, max 1 jN( f )j

B 14.6

JENSEN’S INEQUALITY

Convex Functions

Before we enunciate Jensen’s inequality, we want to make clear the idea of a convex function. We are interested in a convex function because convexity guarantees a unique minimum over the interval of convexity. A function h(x) is convex in the interval (a, b) if for every x1 belonging to the interval (a,b), there exists a constant m such that the following inequality is satisﬁed for all x: h(x) h(x1 ) þ m(x x1 )

(14:6:1)

As seen in Fig. 14.6.1, h(x1 ) þ m(x x1 ) is the equation to a straight line passing tangentially to the point h(x1) and m is the slope of the straight line at the point x1. It is clear from the ﬁgure that the inequality is satisﬁed geometrically. For the sake of completeness, we will give a more formal deﬁnition of convexity.

14.6

Jensen’s Inequality

255

FIGURE 14.6.1

A function h(x) is convex over an interval (a,b), if every x1 and x2 belonging to (a,b) and for 0 l 1, the following equation holds. h½lx1 þ (1 l)x2 lh(x1 ) þ (1 l)h(x2 )

(14:6:2)

If this equation is a strict inequality, then h(x) is strictly convex. Geometrically, this equation conveys that if the function h(x) lies below the line segment connecting two points in the interval (a,b), then it is convex. It can also be shown that it is necessary and sufﬁcient that the second derivative d2 h(x)=dx2 0 for all x in the interval (a,b), for Eq. (14.6.2) to be satisﬁed, in which case h(x) is convex. Example 14.6.1 is nonnegative:

The following functions h(x) are convex because the second derivative

1. x2 , since h00 (x) ¼ 2, positive for all x 2. jxj, since h00 (x) ¼ 0, nonnegative for all x 3. ex , since h00 (x) ¼ ex , positive for all x 4. ln(x), for 0 , x , 1, since h00 (x) ¼ 1=x2 , positive in the range of x 5. x ln(x), for 0 , x , 1, since h00 (x) ¼ 1=x, positive in the range of x Jensen’s Inequality

Jensen’s inequality states that if h(x) is convex and if X is a random variable with EjXj , 1, then E½h(X) h E½X (14:6:3) Substituting E[X ] ¼ x1 in the convexity [Eq. (14.6.1)], we obtain h(x) h E½X þ m x E½X Taking expectations on both sides of Eq. (14.6.4) gives E½h(X) h E½X þ m E½X E½X ¼0 and hence Jensen’s inequality [Eq. (14.6.3)].

(14:6:4)

256

Inequalities, Convergences, and Limit Theorems

Example 14.6.2 Since E[X 2n]¼ E[(X n)2] is a convex function from Example 14.6.1, it follows from Jensen’s inequality that 2 (14:6:5) E½X 2n E½X n and as a consequence 2 E½X 2 E½X

(14:6:6)

which is a restatement of the fact that the variance of a random variable is nonnegative.

B 14.7

CONVERGENCE CONCEPTS

We saw in Section 14.1 that if E(X a)2 ¼ 0, then X ¼ a with probability 1 or X converges to a with probability 1. There are a few other convergence concepts, and we shall make these a little clearer. A machinist wants to make piston rings of diameter a inches to a tolerance limit of +1. Let us assume that the variability of the measurement is s2X. According to the Chebyshev inequality of Eq. (14.2.8), we have P{jX aj , 1} 1

s2X 12

(14:7:1)

If sX is very much smaller than 1, then the observed random variable X is between a 2 1 and a þ 1 is almost certain, and one measurement of X is sufﬁcient. However, if sX is not sufﬁciently small compared to 1, then the estimate a will not give results of sufﬁcient accuracy. To improve the accuracy of the estimate a, we take n measurements corresponding to n random variables fXi, i ¼ 1, . . . , ng with mean a and noise random variables Wi given by Xi ¼ a þ W i ,

i ¼ 1, . . . , n

where we will assume that the random variables Wi are independent with zero mean and variance s2. We will represent the average of these n random variables by X 1 þ X2 þ þ X n b X¼ (14:7:2) n where the mean of b X is a and the variance is ns2X and the corresponding Chebyshev inequality is given by P{jX aj , 1} 1

s2X n12

(14:7:3)

We can see from Eq. (14.7.3) that as the number of samples n becomes very large, the probability that the jX aj is less than 1 becomes almost certain even though s2X may not be small compared to 1. Pointwise Convergence

A discrete sequence of random variables {X1 , X2 , . . . , Xn , . . .}

(14:7:4)

converges to a limiting random variable X if and only if for any 1 . 0, however small, we can ﬁnd a number n0 such that jXn (j) X(j)j , 1

(14:7:5)

14.7

Convergence Concepts

257

for every n . n0 and every j. This type of convergence is also called pointwise or everywhere convergent. This may be all right for a real variable, but it is highly restrictive for a random variable. It is possible that the sequence fXng may converge for j points belonging to a subset of the sample space S. Thus, convergences using probability measures are more useful than this type of convergence. Almost Sure Convergence (a.s.)

A sequence of random variables fXng converges almost surely (a.s.), or almost everywhere (a.e.), or strongly to the random variable X, if for every j point in the sample space S satisﬁes the following criterion: lim jXn (j) X(j)j ! 0 with probability 1

n!1

(14:7:6)

The criterion can also be written as P(Xn ! X) ¼ 1

as

n ! 1

(14:7:7)

In Eq. (14.7.7) the quantity Xn ! X is to be interpreted as the set of all outcomes j such that the number Xn(j) tends to the number X(j). In some cases the limit X may not be known a priori. In this case the Cauchy criterion of convergence comes handy to deﬁne mutual convergence. Cauchy Convergence Criterion

A sequence of random variables fXng on the probability space {S, F, P} is convergent almost surely if the set of points j for which the real sequence {Xn (j)} is convergent with probability 1. Or, in other words P{j [ S : Xnþm (j) Xn (j) ! 0}

as

n ! 1

(14:7:8)

for every m . 0. Using this criterion, the sequence fXng converges mutually almost surely if P{Xnþm Xn } ! 0

as

n ! 1

(14:7:9)

Convergence in Probability (p)

A sequence of random variables fXng converges in probability or weakly converges to the random variable X if for every 1 . 0, however small lim P{jXn Xj 1} ¼ 0

(14:7:10)

lim P{jXn Xj 1} ¼ 1

(14:7:11)

n!1

or equivalently n!1

Using the Cauchy criterion, the sequence fXng converges mutually in probability to X if lim P{jXnþm Xn j 1} ! 0

n!1

(14:7:12)

Convergence in probability is widely used in probability theory. In particular, it is used to determine the weak law of large numbers and the consistency of estimators.

258

Inequalities, Convergences, and Limit Theorems

Convergence in Mean Square (m.s.)

A sequence of random variables fXng converges in the mean square or limit in the mean to the random variable X if lim E{jXnþm Xj2 } ! 0

n!1

(14:7:13)

This is also called quadratic mean convergence. Mutual convergence in the mean square is deﬁned by lim E{jXnþm Xn j2 } ! 0

n!1

(14:7:14)

for all m . 0. Convergence in Distribution

The widest form of convergence is the convergence in distribution. If we denote the distribution functions of Xn and X by Fn(x) and F(x), respectively, then the sequence fXng converges in distribution to the random variable X if lim Fn (x) ! F(x)

n!1

(14:7:15)

for every continuity point x of F(x). However, the sequence fXn(j)g need not converge for any j. Properties of Convergences 1. Almost-sure convergence implies convergence in probability. The converse is not true. 2. Mean-square convergence also implies convergence in probability. Again the converse is not true. 3. If fXng converges in probability, then there exists a subsequence fXnkg of fXng that converges almost surely. 4. Convergences almost surely, probability, and mean square all imply convergence in distribution. Of course, the converse is not true. All the four convergences are shown schematically in Fig. 14.7.1.

FIGURE 14.7.1

14.8

B 14.8

Limit Theorems

259

LIMIT THEOREMS

Weak Law of Large Numbers (WLLN)

Armed with the convergence concepts, we will now explore some limit theorems. Let fXng be a sequence of independent identically distributed random variables with ﬁnite means m and ﬁnite variances s2. Let Sn be the sum of the n random variables given by S n ¼ X 1 þ X2 þ þ X n

(14:8:1)

Let the random variable Zn, called the sample mean, be deﬁned by X 1 þ X2 þ þ X n n

Zn ¼

Clearly the expected value of Zn is X 1 þ X2 þ þ X n nm E½Zn ¼ E ¼ ¼m n n and because of the independence of the sequence fXng, the variance of Zn is X 1 þ X2 þ þ X n ns2 s2 var½Zn ¼ var ¼ 2 ¼ n n n

(14:8:2)

(14:8:3)

(14:8:4)

The weak law of large numbers states that for every 1 . 0 lim P½jZn mj . 1 ! 0

n!1

(14:8:5)

To show this result, we apply Chebyshev’s inequality [Eq. (14.2.1)] P½jZn mj . 1

s2 n12

(14:8:6)

and since 1 . 0, the limit of Eq. (14.8.6) as n tends to 1 is indeed 0. In other words, the probability of the sample mean Zn for a speciﬁc n of a sequence fXng of random variables deviating from the true mean m by an arbitrary 1 . 0 becomes vanishingly small as n tends to 1. Or the sample mean Zn converges in probability to the true mean. Example 14.8.1 In the design of a gyroscope for a navigation system we want the probability of the sample drift rate Zn calculated for n production gyros differing from the unknown true drift rate m by more than 0.1 be less than 2%. We want to ﬁnd the number n of the gyros needed to establish this criterion expressed in terms of the unknown variance s2. Applying Chebyshev inequality given by Eq. (14.2.8), we can write 1 s2 100 . 0:98 P jZn mj , 1 10 n From 1 ð100s2 =nÞ . 0:98 we have 1 0:98 n

100s2 n 100s2 ¼ 5000s2 0:02

If the quality control is very good, s will be small, in which case the number n will be small. However, since we are using Chebyshev inequality that gives very loose bounds,

260

Inequalities, Convergences, and Limit Theorems

we expect n to be overly conservative. We will revisit this example after discussing the central limit theorem in the next section. Example 14.8.2 We will assume that there are a sequence of independent events fAig that will be denoted by the sequence of indicator functions IAi (j), given by

1 if j [ Ai IAi (j) ¼ 0 if j [ Ai Zn ¼

n 1X IA n i¼1 i

We will assume that P(Ai ) ¼ p and P(Ai ) ¼ 1 p for i ¼ 1, . . . , n. Thus E½IAi ¼ p

and

E½IA2i ¼ p

and

E½Zn ¼ p

and

var½Zn ¼

p(1 p) n

var½IAi ¼ p(1 p), i ¼ 1, . . . , n

We can now apply WLLN and write P{jZn pj , 1} 1

p(1 p) n1 2

Assuming p ¼ 0.4, we will estimate p within 0.1. If n ¼ 500, then this equation becomes P{jZn pj , 0:1} 1

0:4 0:6 8 117 ¼1 ¼ 500 0:01 125 125

What this equation tells us is that if we perform 500 trials, then 468 of these will result in an error of fZn 2 pg of less than 0.1. Strong Law of Large Numbers

The weak law of large numbers given by Eq. (14.8.5) states that the sample mean Zn converges in probability to the true mean m as n tends to 1. The strong law of large numbers states that this convergence is not only in probability but also with probability 1, which is almost surely (a.s.). Or n o (14:8:7) P lim (Zn m) ¼ 0 ¼ 1 (a.s.) n!1

Unlike the weak law of large numbers, which is easy to show with Chebyshev’s inequality, the strong law of large numbers is difﬁcult to show and will not be attempted in this book. Even though Eqs. (14.8.5) and (14.8.7) look similar, they are dramatically different. Equation (14.8.7) states that the sequence of the sample means Zn for every n will tend to the true mean m and will stay close to it. The strong law gives us conﬁdence that sample means calculated from a number of realizations will always tend to the true mean as the number increases and isolated discrepancies become unimportant. Central Limit Theorem (CLT)

Perhaps the most important theorem in statistics is the central limit theorem, which in its most general form states that if we sum a sufﬁcient number of random variables, the resulting probability is a Gaussian. Let fXng be a sequence of n independent

14.8

Limit Theorems

261

identically distributed random variables with mean mX and variance s2X. Their sum Sn is given by S n ¼ X 1 þ X2 þ þ X n

(14:8:8)

The mean m and the variance s2 of Sn are E½Sn ¼ m ¼ nmX var½Sn ¼ s2 ¼ ns2X

(14:8:9)

We will now deﬁne a normalized random variable Zn with mean 0 and variance 1 as follows: Zn ¼

Sn m Sn nmX pﬃﬃﬃ ¼ s sX n

(14:8:10)

The central limit theorem states that as n tends to 1, Zn tends to a standard normal distribution: ðz 1 2 pﬃﬃﬃﬃﬃﬃ e(x =2) dx (14:8:11) lim P(Zn z) ¼ n!1 2p 1 We can show this result heuristically by using moment generating functions (MGFs) in Zn ¼

n Sn nmX 1 X pﬃﬃﬃ ¼ pﬃﬃﬃ (Xi mX ) sX n i¼1 sX n

(14:8:12)

Taking MGF on both sides, we obtain pﬃﬃ Pn MZn (t) ¼ E(etZn ) ¼ E e(t=sX n) i¼1 (Xi mX ) " ¼E

n Y

e

pﬃﬃ (t=sX n)(Xi mX )

#

i¼1

( )n n Y pﬃﬃ pﬃﬃ (t=sX n)(Xi mX ) (t=sX n)(XmX ) ¼ E e ¼ E e

(14:8:13)

i¼1

In Eq. (14.8.13) we have used the independence property of fXng and X has the same pdf as Xn. We can now expand the last exponential in the above equation and write h i pﬃﬃ 1 t2 (t=sX n)(XmX ) 2 E e (X mX ) þ hn (X; t) ¼ E 1 þ pﬃﬃﬃ (X mX ) þ sX n 2ns2X ¼1þ0þ

t2 þ E½hn (X; t) 2n

where hn(X; t) represents the higher-order terms. Hence

n t2 MZn (t) ¼ 1 þ 0 þ þ E½hn (X; t) 2n

n t2 2 lim MZn (t) ¼ 1 þ ¼ et =2 n!1 2n

(14:8:14)

262

Inequalities, Convergences, and Limit Theorems 2

Since E[hn(X; t)] is of the order of 1/n 2, it is zero in the limit as n tends to 1. The term et =2 is the MGF of a standard Gaussian, and we have shown the central limit theorem. The number n is usually assumed to be 30 for CLT to hold, but as we shall see in the following example, it holds for n much less than 30. Example 14.8.3 Random variables Xi, i ¼ 1,2,3,4 are independent and identically distributed with a uniform distribution in the interval (0,1] with fXi(x) ¼ u(x) 2 u(x 2 1). 1 . We form the random The mean values are mXi ¼ 12 and the variances are sXi2 ¼ 12 variables Y1 ¼ X1 , Y2 ¼ X1 þ X2 , Y3 ¼ X1 þ X2 þ X3 , Y4 ¼ X1 þ X2 þ X3 þ X4 , and their density functions are fY1 (y) ¼ fX1 (y), fY2 (y) ¼ fX1 (y) fX2 (y), fY3 (y) ¼ fY2 (y) fX1 (y), fY4 (y) ¼ fY3 (y) fX1 (y). These densities are given by

fY1 ( y) ¼

1 0

if 0 , y 1 otherwise

8 if 0 , y 1 > : 0 otherwise

8 2 y > > > if 0 , y 1 > > 2 > > > 3 < 2 y þ 3y if 1 , y 2 fY3 ( y) ¼ 2 > > y2 > 9 > > 3y þ if 2 , y 3 > > 2 > :2 0 otherwise 8 3 y > > if 0 , y 1 > > 6 > > > 3 > y 2 > > > if 1 , y 2 þ 2y2 2y þ > < 2 3 3 fY2 ( y) ¼ y 22 > 4y2 þ 10y if 2 , y 3 > > 2 3 > > > > y3 32 > > > if 3 , y 4 þ 2y2 8y þ > > 6 3 : 0 otherwise These densities are compared to the corresponding normal densities given by 1 12 fNi ( y) N(mNi , s2Ni ) ¼ N i , i , i ¼ 1,2,3,4 2 12 or 2 1 fN1 ( y) ¼ rﬃﬃﬃﬃ e½x(1=2) =(1=6) p 6

2 1 fN2 ( y) ¼ rﬃﬃﬃﬃ e(x1) =(1=3) p 3

2 1 fN3 ( y) ¼ rﬃﬃﬃﬃ e½x(3=2) =(1=2) p 2

2 1 fN4 ( y) ¼ rﬃﬃﬃﬃﬃﬃ e(x2) =(2=3) 2p 3

These densities are shown in all four panels of Fig. 14.8.1. We see that the approximation progressively improves as n increases from 1 to 4, and it is very good when n ¼ 4 (Fig. 14.8.1d).

14.8

Limit Theorems

263

FIGURE 14.8.1

Example 14.8.4 We will revisit Example 14.8.1 in light of the CLT and determine whether we can get a better bound on the number n of gyros needed. In order to use the Gaussian tables, we have to convert the random variable Zn into the standard Gaussian form by substituting pﬃﬃﬃ Zn m n(Zn m) Wn ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ s var(Zn ) 1 0:98 and arrive at We can substitute Wn in P½jZn mj , 10 pﬃﬃﬃ sjWn j 1 n P pﬃﬃﬃ , 0:98 or P jWn j , 0:98 10 10s n

Referring pﬃﬃﬃ to Gaussian tables, we have the value for a probability of 0.98 is x ¼ 2.3263. Thus, n=10s 2:3263 or n 2:32632 100s2 ¼ 541:17s2 and n 542 s2. This is a much better number than the 5000s2 that we obtained in Example 14.8.1.

CHAPTER

15

Computer Methods for Generating Random Variates

If we have an actual probabilistic system, we can obtain the pdf by analyzing the data from the system. On the other hand, if we are designing a system such as transmission of packets through a communication channel, we must get some idea as to how the system will behave before it is actually implemented. We resort to simulation of the system on a computer. The probability distributions may be known a priori, and we have to generate random data for the given probability distributions. We call such generated values random variates. It is not feasible to develop techniques for generating random variates for every possible distribution function. As it turns out, generating a uniform distribution in the interval (0,1] is fast on a computer. Starting with the uniform distribution, we can generate random variates for any given distribution function by a number of techniques that will be discussed.

B 15.1

UNIFORM-DISTRIBUTION RANDOM VARIATES

Almost every computer will have a uniform number generator built into the system using some form of the modulus function. One such generator uses the linear congruential or power residue method that produces a sequence of integers fxi, i ¼ 0, . . . , m 2 2g. It is given by the recursive relationship xiþ1 ¼ (a xi þ c)mod m

(15:1:1)

where a is a constant multiplier, c is the increment, and m is the modulus function. The initial value x0 is called the “seed.” Equation (15.1.1) can be recursively solved to yield n a 1 x n ¼ an x 0 þ c mod m (15:1:2) a1 All these parameters affect the mean, variance, and cycle length of the sequences. It is called a mixed congruential generator if c = 0 and multiplicative congruential generator if c ¼ 0. The sequence repeats after xm22 and the period of repetition is less than or equal to m 2 1 depending on the parameters.

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

264

15.1

265

Uniform-Distribution Random Variates

FIGURE 15.1.1

We will form two sequences with

Example 15.1.1

1. fxig : x0 ¼ 1, a ¼ 7, c ¼ 0, m ¼ 13 2. fyig : y0 ¼ 1, a ¼ 7, c ¼ 3, m ¼ 13 Using the recursive relationship of Eq. (15.1.1), these two sequences fxig and fyig are shown in Fig. 15.1.1 and Table 15.1.1. As we can see from Fig. 15.1.1 and Table 15.1.1, the repetition period is 13 2 1 ¼ 12. Also changing c from 0 to 3 has radically changed the sequence. We can conclude that almost all random-number generators are not truly random but pseudorandom sequences that have the appearance in the statistical sense of being random. Thus, when we refer to random numbers we really mean pseudorandom sequences where the period is very high. A random-number generator must have the following properties: 1. The numbers generated must have good statistical properties. They must be independent and identically distributed. In other words, the values may not be correlated. 2. The repetition period must be long for any simulation to be useful. 3. Random numbers must be repeatable. For each speciﬁed x0, a, c, and m, the generator must produce the same sequence of numbers. 4. Many simulations will require thousands of random numbers, and hence the generation of these numbers must be fast and thus must also have computational efﬁciency. 5. It must be easy to generate separate sequences of random numbers.

TABLE 15.1.1 i

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

2

1

7

10

5

9

1

10

8

7

Series 1: x0 ¼ 1, a ¼ 7, c ¼ 0, m ¼ 13 xi

1

7

10

5

9

11

12

6

3

8

4

Series 2: x0 ¼ 1, a ¼ 7, c ¼ 3, m ¼ 13 yi

1

10

8

7

0

3

11

2

4

5

12

266

Computer Methods for Generating Random Variates

The linear congruential generator may satisfy all these conditions provided a and m are carefully chosen. Good statistical properties have been obtained with a ¼ 75 ¼ 16,807 and m ¼ 231 2 1 ¼ 2,147,483,647. Under these conditions the period of the sequence is 231 2 2 ¼ 2,147,483,646. Sometimes a ¼ 742,938,285 and m ¼ 248 are also used.

B 15.2

HISTOGRAMS

Having generated random variates, we have to see how well they approximate the desired probability density. From the deﬁnition of the density function fX (x), we have fX (x)

P(X x þ Dx) P(X x) Dx

or fX (x)Dx P(x , X x þ Dx)

(15:2:1)

which gives the value of the probability mass at the point x and width Dx. Hence with the generated random variates fxig we will be able to construct a histogram that will approximate the given mass function fX (x)Dx. The purpose of the histogram is to construct the model of the desired distribution from the generated data. The usual form of the histogram is to divide the range of data points N into equal-size bins or subintervals Dx. Or the necessary number of bins K needed will be given by Number of bins K ¼

range of data N bin width Dx

(15:2:2)

Next, the number of data points fni, i ¼ 1, . . . , Kg falling into each bin is counted. The vertical axis (ordinate, y) represents the counts in each bin, and the horizontal axis (abscissa, x ) represents K . Dx, the product of the number of bins K and the bin interval Dx. If fX (x) is ﬁnite, then the product K . Dx must cover the entire range of x. If it is inﬁnite or semiinﬁnite, then the product can cover a reasonable range. Finally, the histogram must be normalized so that it can be compared to the desired density function. The normalization proceeds as follows. The normalized count is the count in each bin fni, i ¼ 1, . . . , Kg, divided by the total number N of observations multiplied by the bin width Dx so that the area under the histogram equals one. This type of normalization results in a histogram that is similar to the probability density function. Even though this may be less intuitive, it is the one to be used if the histogram is to model the desired probability density function. We will now model a uniform distribution U(0,1) from the random variates generated by the linear congruential generator of a computer. For this model, N ¼ 10,000 points were generated and K ¼ 100 bins were chosen with each bin interval Dx ¼ 0.01. Note that K . Dx ¼ 1 covers the x axis of the U(0,1) distribution. The histogram subroutine is called that places the counts fni, i ¼ 1, . . . , Kg in each of the K bins. The number of points falling in the ﬁrst 10 bins is shown in Table 15.2.1. The expected number in each bin is 100. The resulting histogram is divided by N . Dx ¼ 10,000 0.01 ¼ 100 to yield the model for U(0,1) distribution. The normalized histogram is shown in Fig. 15.2.1. Superimposed over the histogram is the actual uniform distribution. By “eyeballing” the two plots in Fig. 15.2.1, we see that the modeling is indeed good. More formally, we can also test the goodness of ﬁt by a number of mathematical tests, and we will use the chisquare test to ascertain whether the histogram ﬁts the true pdf sufﬁciently well. We will now enumerate the procedure for the chi-square test as applied to this case.

15.2 Histograms

267

TABLE 15.2.1 Bins Count Expected

0

1

2

3

4

5

6

7

8

9

102 100

100 100

109 100

100 100

117 100

107 100

86 100

93 100

86 100

81 100

Chi-Square Test 1. From the histogram ﬁnd the number of observations ni in each of the i ¼ 1, . . . , K bins. 2. From the probability density function fX (x) calculate the occupancy mi ¼ fX(xi)NDx in each of the i ¼ 1, . . . , K bins. 3. Calculate the deviation Di from Di ¼ ni mi ; i ¼ 0; . . . ; K 1: PK1 4. Form the chi-square statistic from x ¼ i¼0 (Di =mi )2 . This will have K22 degrees of freedom. 5. Set the probability level pg or the signiﬁcance level a ¼ 1 2 pg. Usually pg is set at 95% or 99% or a is set at 5% or 1%. The higher the signiﬁcance level, the tighter will be the bounds. 6. From the chi-square tables ﬁnd the threshold value ta for the given signiﬁcance level and K22 degrees of freedom. 7. If the x value is less than ta, the hypothesis that the ﬁt is good is accepted. Otherwise it is rejected. We can use the x2 test to determine the goodness of the ﬁt to the uniform distribution that has been achieved. The x value is found to be x ¼ 91.39 at the 5% signiﬁcance level. The corresponding ta value for 100 2 2 ¼ 98 degrees of freedom at the 5% signiﬁcance level is obtained from Table 15.2.2 as ta ¼ 122.10773. The chi-square density and distribution for 98 degrees of freedom is shown in Figs. 15.2.2 and 15.2.3 with the probability and signiﬁcance levels. The hypothesis that the histogram is a good ﬁt to the true normal distribution is accepted since 91.39 , 122.10773.

FIGURE 15.2.1

268

Computer Methods for Generating Random Variates

TABLE 15.2.2 K

pg 95%

pg 97.5%

pg 99%

K

pg 95%

pg 97.5%

pg 99%

1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22 24 26 28 30

3.84146 5.99146 7.81473 9.48773 11.07050 12.59159 14.06714 15.50731 16.91898 18.30704 21.02601 23.68475 26.29620 28.86928 31.41042 33.92443 36.41502 38.88513 41.33713 43.77297

5.02389 7.37776 9.34840 11.14329 12.83250 14.44938 16.01276 17.53455 19.02277 20.48306 23.33660 26.11891 28.84532 31.52636 34.16959 36.78070 39.36407 41.92316 44.46079 46.97924

6.63490 9.21034 11.34487 13.27670 15.08627 16.81189 18.47531 20.09024 21.66599 23.20918 26.21693 29.14122 31.99992 34.80530 37.56623 40.28936 42.97982 45.64168 48.27823 50.89218

35 40 45 50 55 60 65 70 75 80 82 84 86 88 90 92 94 96 98 100

49.80185 55.75848 61.65623 67.50481 73.31149 79.08194 84.82064 90.53122 96.21667 101.87947 104.13874 106.39484 108.64789 110.89800 113.14527 115.38979 117.63165 119.87094 122.10773 124.34211

53.20335 59.34170 65.41016 71.42019 77.38046 83.29767 89.17714 95.02318 100.83934 106.62857 108.93729 111.24226 113.54360 115.84143 118.13589 120.42708 122.71511 125.00007 127.28207 129.56120

57.34207 63.69074 69.95683 76.15389 82.29212 88.37942 94.42208 100.42518 106.39292 112.32879 114.69489 117.05654 119.41390 121.76711 124.11632 126.46165 128.80325 131.14122 133.47567 135.80672

Starting from uniformly distributed random variates, we can generate other random variates given the desired probability distribution. We will discuss three widely used techniques for generating these variates: 1. Inverse transformation 2. Convolution 3. Acceptance –rejection In all the simulations that follow we have used ui U(0,1) distribution shown in Section 15.2 with N ¼ 10,001 points.

FIGURE 15.2.2

15.3

Inverse Transformation Techniques

269

FIGURE 15.2.3

B 15.3

INVERSE TRANSFORMATION TECHNIQUES

One of the more useful ways of generating random variates is through the inverse transformation technique, which is based on the following result. If U is a random variable uniformly distributed in (0,1] with distribution function FU (u) ¼ u,u 1(0,1], and X is a random variable with distribution function FX(x), then the random variates for X can be 21 generated from X ¼ F21 X (U ). We will show that FX (U ) is a random variable with distri21 bution FX(x). Assuming that X ¼ FX (U ), we have P{X x} ¼ P{FX1 (U) x}

(15:3:1)

Since FX (x) is monotonically increasing, we can write Eq. (15.3.1) as follows: P{FX1 (U) x} ¼ P{U FX (x)} ¼ FU {FX (x)}

(15:3:2)

However, U is uniformly distributed in (0,1] and hence FU(u) ¼ u, 0 , u 1. Applying this fact in Eq. (15.3.2), we have FU {FX (x)} ¼ FX (x)

(15:3:3)

This result can be used to obtain the random variates for any distribution function FX(x) only if F21 X can be solved in closed form or FX can be accurately and efﬁciently inverted numerically. The methodology of ﬁnding random variates fXig by the inversion technique can be enumerated as follows: 1. FX(x) is known a priori. 2. Uniform random variates fui, i ¼ 1, . . . , Ng in the interval (0,1] are generated. 3. The inverse F21 X (u) of the cdf FX(x) can be analytically or numerically determined. 4. The variates fxig are generated from xi ¼ F21 X (ui), i ¼ 1, . . . , N. 5. The histogram for fxig is constructed and tested using chi-square goodness of ﬁt. We will use this procedure to ﬁnd random variates for a number of discrete and continuous distributions starting from 10,001 points of the uniform random variates in (0,1] generated in the previous section.

270

Computer Methods for Generating Random Variates

Discrete Distributions

Since FX(x) is a monotonically increasing function, the inverse for a ﬁnitedimensional integer-valued discrete distribution functions can be obtained in a simple manner as shown in the following two examples. Example 15.3.1 given by

The distribution of packets in bytes in a communication system is 8 0, > > > > 0:2, > > < 0:4, FX (x) ¼ 0:6, > > > > 0:8, > > : 1,

0 , x 64 64 , x 128 128 , x 256 256 , x 320 320 , x 384 x . 384

and is shown in Fig. 15.3.1. It is not difﬁcult to ﬁnd the inverse function F21 X (u). From Fig. 15.3.1, for values of the uniform random variate ui given by 0.4 , ui 0.6, F21 X (u) ¼ 256 bytes. Similarly, we can obtain the complete inverse function as 8 64, 0 , ui 0:2 > > > > 128, 0:2 , x 0:4 < xi ¼ FX1 (ui ) ¼ 256, 0:4 , x 0:6 > > 320, 0:6 , x 0:8 > > : 384, 0:8 , x 1 and the sequence fxig represents the random variates for this discrete distribution. The generated uniform random variates ui are assigned values 64, 128, 256, 320, and 384 depending on the ranges in which they fall along the vertical axis. They are then sorted in ascending order and plotted with x ¼ F21 X (u) along the horizontal axis and i scaled by 1/10,000 along the vertical axis so that the range is from 0 to 1. The plot is shown in Fig. 15.3.2. As an example, the uniform random variates the falling in the shaded region in the ﬁgure between 0.4 and 0.6 represent 256 bytes.

FIGURE 15.3.1

15.3

Inverse Transformation Techniques

271

FIGURE 15.3.2

Out of the 10,001 uniform random variates generated, 2017 fall in the range 0– 0.2, 1912 fall in the range 0.2– 0.4, 1976 fall in the range 0.4– 0.6, 2017 fall in the range 0.6 –0.8, and 2079 fall in the range 0.8 –1.0 as shown in Fig. 15.3.2, which is exactly the same as Fig. 15.3.1. Example 15.3.2 In this example, we will simulate a binomial distribution FB(x) with n ¼ 5 and p ¼ 0.4 given by FB (x) ¼

x X

5! (0:4)k (0:6)nk k!(5 k)! k¼0

and shown below and graphed in Fig. 15.3.3:

FIGURE 15.3.3

272

Computer Methods for Generating Random Variates

FIGURE 15.3.4

8 0:07776, > > > > 0:33696, > > < 0:68256, FB (x) ¼ 0:91296, > > > > 0:98976, > > : 1,

0,x1 1,x2 2,x3 3,x4 4,x5 x.5

The inverse function F 21 B (U) can be found in a manner similar to that in the previous example. In Fig. 15.3.3, for values of the uniform random variate ui between 0.33696 and 0.68256, F21 B (u) ¼ 2. Similarly, we can obtain the complete inverse function as 8 0, 0 , ui 0:07776 > > > > 1, 0:07776 , ui 0:33696 > > < 2, 0:33696 , ui 0:68256 FB1 (ui ) ¼ 3, 0:68256 , ui 0:91296 > > > > 4, 0:91296 , ui 0:98976 > > : 1, 0:98976 , ui 1 which is shown in Fig. 15.3.4. 1 The vertical axis has been scaled by 10,000 for the range to correspond to (0,1]. Out of the 10,001 uniform random variates generated, 797 fall in the range 0 – 0.07776, 2555 fall in the range 0.07776 –0.33696, 3404 fall in the range 0.33696 – 0.68256, 2312 fall in the range 0.68256 –0.91296, 801 fall in the range 0.91296 – 0.98976, and 132 fall in the range 0.98976 –1. Figure 15.3.4 is the same as Fig. 15.3.3. Continuous Distributions

If the cumulative distribution function of a random variable can be expressed in a closed form, then it is not difﬁcult to ﬁnd the inverse in a closed form. We will give several examples to clarify the inverse transformation method. Example 15.3.3 (Exponential Distribution) In this example we will obtain random variates for an exponential distribution given by FX (x) ¼ 1 elx ,

x0

(15:3:4)

15.3

Inverse Transformation Techniques

273

FIGURE 15.3.5

The inverse is given by 1 elx ¼ u

or elx ¼ 1 u

and

1 x ¼ FX1 (u) ¼ ln (1 u), l

x0

The exponential random variates are generated from this equation with l ¼ 13, resulting in the following equation: xi ¼ 3 ln (1 ui ), x 0, i ¼ 0, 1, . . . , 10,000 The histogram subroutine hist(K,Dx,x) is called with K ¼ 100 bins and bin width Dx ¼ 0.1. The resulting histogram is divided by N . Dx ¼ 1000 to yield the normalized histogram fE(xk), k ¼ 1, . . . , 100. This is shown in Fig. 15.3.5. Superimposed on top of the histogram is the true density function, fX (x) ¼ 13 e(x=3) , x . 0. We can see from ﬁgure that the ﬁt is very good. We can also apply the chi-square goodness of ﬁt, which it easily passes. Example 15.3.4 (Weibull Distribution)

The cdf of a Weibull distribution is given by a

FX (x) ¼ 1 e½l(xm) ,

x.m

(15:3:5)

This distribution, as mentioned in Chapter 7, is a generalization of the exponential with parameters, scale l, shape a, and location m. Since this is in a closed form, the inverse is given by a a 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 e½l(xm) ¼ u or e½l(xm) ¼ 1 u and FX1 (u) ¼ m þ a ln(1 u) l with u , 1. The Weibull random variates are generated with l ¼ 13 , a ¼ 3 and m ¼ 2 from the following equation: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ xi ¼ 2 þ 3 3 ln(1 mi ), i ¼ 0, . . . , 10,000 The histogram subroutine hist(K, Dx, x) is called with K ¼ 100 bins and bin width Dx ¼ 0.1. The resulting histogram is divided by N . Dx ¼1000 to yield the normalized histogram fW (xk), k ¼ 1, . . . , 100. This is shown in Fig. 15.3.6. 3Superimposed on 3top of the histogram is the true Weibull density function, fX (x) ¼ 3 13 (x 2)2 e½(x2)=3 , x . 2.

274

Computer Methods for Generating Random Variates

FIGURE 15.3.6

We can see from the ﬁgure that the ﬁt is very good. We can also apply the chi-square goodness of ﬁt, which it readily passes. Example 15.3.5 (Rayleigh Distribution) The random variates corresponding to a Rayleigh distribution can be generated by taking (1) the direct inverse of the distribution or (2) the square root of the sum of the squares of two independent Gaussian distributions. The cdf of a Rayleigh distribution is given by FZ (z) ¼ 1 e(z

2

=2s2 )

, z.0

(15:3:6)

Since this is in a closed form, the inverse is found as follows: 2

1 e(z

=2s2 )

¼u

or e(z

2

=2s2 )

¼ 1 u and

z ¼ FZ1 (u) ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2s2 ln(1 u)

The Rayleigh random variates are generated from this equation with s2 ¼ 1, resulting in the following equation: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ zi ¼ 2s2 ln(1 ui ), z . 0, i ¼ 0, 1, . . . , 10,000 The histogram subroutine hist(K, Dx, x) is called with K ¼ 200 bins and bin width Dx ¼ 0.05. The resulting histogram is divided by N . Dx ¼ 500 to yield the normalized histogram fR(xk), k ¼ 1, . . . , 200. This is shown in Fig. 15.3.7. Superimposed on top of the 2 histogram is the true density function, fZ (z) ¼ ze(z =2) , z . 0. We can see from ﬁgure that the ﬁt is very good. We can also apply the chi-square goodness of ﬁt, which it easily passes. Special Methods

Even though some distributions may not have a closed form, random variates can be obtained by special transformations. The very important Gaussian random variates can be generated by these special transformation techniques. Two independent random variables U and V are both uniformly distributed in (0,1] and whose joint density is given by 1, 0 , u 1, 0 , v 1 fUV (u,v) ¼ (15:3:7) 0, otherwise

15.3

Inverse Transformation Techniques

275

FIGURE 15.3.7

The transformation used to generate standard Gaussian random variables is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X ¼ 2 ln(U) cos(2pV) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Y ¼ 2 ln(U) sin(2pV)

(15:3:8)

We can ﬁnd the joint density fXY (x,y) from the transformation. Solving for u and v in Eq. (15.3.8), we obtain y

1 2 2 tan1 (15:3:9) u ¼ e½(x þy )=2 : v ¼ 2p x The Jacobian matrix of transformation is 2 3 2 cos (2pv) dx dx pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u 2 ln(u) 6 du dv 7 6 6 ¼ J¼4 dy dy 5 4 sin (2pv) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ du dv u 2 ln(u)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2p sin(2pv) 2 ln(u)

3

7 7 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 5 2p cos(2pv) 2 ln(u)

and the absolute value of the Jacobian determinant is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos (2pv) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2p sin (2pv) 2 ln(u) 2p u 2 ln(u) 2 2 ¼ kJk ¼ ¼ 2pe(x þy )=2 sin (2pv) p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ u pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2p cos (2pv) 2 ln(u) u 2 ln(u) Hence, from Eqs. (15.3.7) and (15.3.11) we have fUV (u,v) 1 (x2 þy2 )=2 1 (x2 =2) 1 (y2 =2) pﬃﬃﬃﬃﬃﬃ e ¼ e ¼ pﬃﬃﬃﬃﬃﬃ e fXY (x,y) ¼ kJk 2p 2p 2p

(15:3:10)

(15:3:11)

(15:3:12)

which yields two independent standard Gaussian random variables. This transformation is called the Box– Mueller transformation. Example 15.3.6 (Gaussian Distribution) We will now use the Box – Mueller transformation to generate two independent standard Gaussian random variates. In this example, 10,001 points of two independent uniformly distributed random variates fuig and fvig are generated. They are passed through the transformation of Eq. (15.3.8) to form the

276

Computer Methods for Generating Random Variates

Gaussian random variates fzig and fwig. Theoretically, these must be zero mean and unit variance. If they are not, they can be transformed into exactly zero mean and unit variance by another transformation wi w ; i ¼ 0; . . . , 10;000 sw qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ P P where z ¼ (1=N) Ni¼0 zi and sz ¼ (1=N) Ni¼0 (zi z)2 , N ¼ 10,000, with similar deﬁnitions for w and sw . The histogram subroutines are called with K ¼ 100 bins and bin width Dx ¼ Dy ¼ 0.08. The resulting histogram is normalized by NDx ¼ NDy ¼ 800 to plots yield histograms fXk and fYk, k ¼ 1, . . . , 100 for the Gaussian random variates. pﬃﬃﬃﬃﬃﬃ Both 2 are shown p inﬃﬃﬃﬃﬃﬃ Fig. 15.3.8 with the true standard Gaussian fX (x) ¼ (1= 2p)e(x =2) and 2 fY (y) ¼ ð1= 2pÞe(y =2) superimposed on them. From these ﬁgures we ﬁnd that the ﬁt is very good. These random variates are stored for later use in simulating stochastic systems. xi ¼

zi z ; sz

yi ¼

FIGURE 15.3.8

15.3

Inverse Transformation Techniques

277

Example 15.3.7 (Rayleigh Distribution with Phase) We have already found random variates for a Rayleigh distribution using a direct transformation. We will now apply the relationship to two independent standard Gaussian random variables by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z ¼ X2 þ Y 2 to generate random variates for the Rayleigh distribution. The Gaussian random variates are generated by the Box – Mueller transformation as explained in the previous section. The Rayleigh random variates and the corresponding phase variates are given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 9 zi ¼ x2i þ y2i > = , i ¼ 0, . . . , N y i > wi ¼ tan1 ; xi The sequences fzig and fwig are both passed through histogram subroutines with the number of bins K ¼ 200 and the bin width Dz ¼ Dw ¼ 0.05. Both these histograms are normalized with division by NDz ¼ NDw ¼ 500. The resulting histograms fZk and fWk 2 are shown in Fig. 15.3.9 with the true Rayleigh density fZ (z) ¼ ze(z =2) , z . 0 and uniform density U(– p/2, p/2] superimposed on them. We can see from both plots that the ﬁt is very good. Table 15.3.1 shows the densities with their closed-form distribution functions and their inverses.

FIGURE 15.3.9

278

a b aþ1 , x.0 b xþb

b p½(x a)2 þ b2

1 (x2 þy2 )=2 e 2p

Pareto

Cauchy

Gaussian

z.0

z (z2 =s2 ) e , s2

ala (x m)a1 e½l(xm)

Rayleigh

Weibull

a

x.m

=2s2 )

,

b a , xþb

2

1 2p

1 1

ðx ðy

e(j

2

þh2 )=2

dhdj

x.0

z.0

x.m

1 1 1 x a þ tan 2 p b

1

1 e(z

a

1 e½l(xm) ,

1 lx 1 e , x 0 1 elx , 2 2

l ljxj , e 2

Laplace

l0

(1 elx )u(x)

x.0

1 x a 2 a,xb 2 ba 1 x þ a 2b 2 1 b , x 2b a 2 ba

xa ba

cdf F(x)

1 lx e u(x) l

1 ba

pdf f (x)

Exponential

Triangular

Uniform (a,b)

Type

TABLE 15.3.1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1p a ln (1 u), l

u,1

u,1

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ln (u1 ) cos (2pu2 ) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y ¼ 2 ln (u1 ) sin (2pu2 ) x¼

1 a þ b tan p u 2

1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 , b p a 1u

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2s2 ln (1 u), u , 1

mþ

1 1 1 ln (2u), u ln½2(1 u), l 2 l

1 ln (1 u), u , 1 l u.

pﬃﬃﬃﬃﬃ 1 a þ (b a) 2u, u 2 h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃi 1 b þ (b a) 1 2(1 u) u . 2

a þ u(b a)

Inverse F 21(u)

1 2

15.4

B 15.4

Convolution Techniques

279

CONVOLUTION TECHNIQUES

In cases where it may not be possible to obtain inverses, we can use the convolution technique. From Eq. (13.1.15) we recall that the density of the sum Z of two independent random variables X and Y is the convolution of the densities of X and Y: ð1 fZ (z) ¼

fX (x)fY (z x)dx

(15:4:1)

1

We can generate random variates from independent random variates that have already been generated. We will give two examples. Example 15.4.1 (Triangular Distribution) We want to generate random variates for a triangular distribution. Two independent uniformly distributed random variates, faig and fbig, are generated. We form the sum x i ¼ a i þ bi ,

i ¼ 0, . . . , 10,000

The sequence fxig is passed through the histogram subroutine with the number of bins K ¼ 100 and the bin width Dx ¼ 0.02. The resulting histogram is normalized with NDx ¼ 10,000 0.02 ¼ 200. The histogram fXk is graphed in Fig. 15.4.1. The true triangular density given by fX (x) ¼

x, 2 x,

0,x1 1,x2

is superimposed on the histogram. The ﬁt is very close and passes the chi-square test easily. Example 15.4.2 (Erlang Distribution) The Erlang distribution is obtained by the n-fold convolution of n independent exponential distributions as given in Eq. (7.4.3). Hence we can generate Erlang random variates with n degrees of freedom by summing n independent exponential random variates. We will now generate Erlang random variates with 2 degrees

FIGURE 15.4.1

280

Computer Methods for Generating Random Variates

FIGURE 15.4.2

of freedom. From two sets of uniform random variates faig and fbig, we generate two identically distributed exponential random variates fxig and fyig, with l ¼ 13 given by xi ¼ 3 ln (ai );

yi ¼ 3 ln (bi )

Erlang random variates fzig are formed by summing fxig and fyig. The variates fzig are then passed through the histogram subroutine with number of bins K ¼ 200 and bin width Dz ¼ 0.1. The resulting histogram is normalized by division with NDx ¼ 1000. The histogram fZk is graphed for k ¼ 0, . . . , 200 and shown in Fig. 15.4.2. The true Erlang density for 2 degrees of freedom given by 2 1 ze(z=3) , fZ (z) ¼ 3

z0

is superimposed on the histogram as shown in the ﬁgure. The ﬁt is very good as seen in the diagram.

B 15.5

ACCEPTANCE – REJECTION TECHNIQUES

The rejection method does not require that the cumulative distribution function [indeﬁnite integral of p(x)] be readily computable, much less the inverse of that function—which was required for the transformation method in the previous section. Usually the inverse transformation technique uses the cumulative distribution function in closed form, in which case it is the preferred method of generating random variates. However, in many cases the cdf may not be expressible in closed form or it may be very inefﬁcient to calculate it numerically. On the other hand, the density functions may be available in closed form such as the beta or the gamma density. The rejection method is a technique for generating random deviates fxig whose density function fX(x), if known in closed form, is amenable directly to determine these variates. We will now outline the acceptance– rejection method of generating variates. Consider a function g(x) that upper-bounds the density fX(x) or g(x) fX(x) for all x. This is called the bounding or comparison function g(x) which, is not a density function

15.5 Acceptance – Rejection Techniques

281

Ð1 Ð1 since 1 g(x)dx 1 fX (x)dx ¼ 1. We deﬁne a normalization constant c ¼ Ð1 1 g(x)dx 1, and another function wX (x) ¼ g(x)=c for all x, thus rendering wX(x) a proper density function. We can now state the acceptance – rejection algorithm: 1. Generate random variates fyi, i ¼ 0, . . . , Ng having the density function wY( y). 2. Generate random variates fui, i ¼ 0, . . . , Ng uniformly distributed in (0,1] and independent of fyig. 3. If ui ½ fX ( yi )=½g( yi ) 1, accept the point yi and set fxi ¼ yig. Otherwise, reject yi. The number of accepted points fxig will be equal to N/c. 4. The sequence fxig are the desired random variates with density function fX(x). We will now show from conditional probability considerations that the generated variates fxig do indeed have the distribution FX(x). From the deﬁnition of conditional probability, we obtain P(generated X x) ¼ P(Y x j accept) ¼

P(Y x, acceptÞ P(accept)

(15:5:1)

We need the following result for any y:

fX (y) P(accept j Y ¼ y) ¼ P U g(y)

(15:5:2)

Since U is uniformly distributed in (0,1], we have FU(u) ¼ u, and it is independent of Y. Hence Eq. (15.5.2) can be written as follows: fX (y) fX (y) fX (y) P U ¼ FU ¼ (15:5:3) g(y) g(y) g(y) Incorporating Eq. (15.5.3) into Eq. (15.5.2), we obtain P(accept j Y ¼ y) ¼

fX (y) g(y)

(15:5:4)

We can evaluate the denominator in Eq. (15.5.1) from Eq. (15.5.4) with substitution of wY (y) ¼ g(y)=c. ð1 P(accept j Y ¼ y)wY (y)dy P(accept) ¼ 1

ð1 ¼

1

fX (y) g(y) 1 dy ¼ g(y) c c

We will now evaluate the numerator of Eq. (15.5.1): ð1 P(Y x, accept) ¼ P(accept, Y x j Y ¼ y)wY (y)dy

(15:5:5)

(15:5:6)

1

Since Y ¼ y in Eq. (15.5.6) we can rewrite it as ðx P(Y x, accept) ¼ P(accept, y x j Y ¼ y)wY (y)dy 1

(15:5:7)

282

Computer Methods for Generating Random Variates

where the integration is terminated at x because y x. Applying Eq. (15.5.4) and substituting wY (y) ¼ g(y)=c in Eq. (15.5.7), we obtain ðx fX (y) g(y) P(Y x, accept) ¼ dy 1 g(y) c ¼

FX (x) c

(15:5:8)

Combining Eqs. (15.5.5) and (15.5.8), we obtain the result that P(generated X x) ¼

FX (x) c ¼ FX (x) c

(15:5:9)

thus showing that the acceptance –rejection method yields the desired probability distribution. Example 15.5.1 We will give an example of generating beta density variates by the method described above. The general beta density is given by fb (x) ¼

G(a þ b) a1 x (1 x)b1 , G(a)G(b)

0x1

and there is no closed-form solution for the cdf. For a and b integers, we have fb (x) ¼

(a þ b 1)! a1 x (1 x)b1 , 0 x 1 (a 1)!(b 1)!

Random variates will be generated for a beta(3,6) density (Fig. 15.5.1) given by fb (x) ¼ 168x2 (1 x)5 ,

0x1

The maximum of the density function occurs at x ¼ 27 ¼ 0.2857 and fb(27) ¼ 2.55. We can bound this density with a rectangle of width 2.55 so that g(x) ¼ 2.55, 0 ,x 1. Hence the normalizing constant c ¼ 2.55 and wX (x) ¼ g(x)=c is a U(0,1] density function. Algorithm for Acceptance – Rejection 1. Generate a sequence fyi, i ¼ 0, . . . , Ng with uniform density wY (y) ¼ g(y)=c. 2. Generate uniform random variates fui, i ¼ 0, . . . , Ng independent of fyig. 3. If ui ½168y2i (1 yi )5 =2:55; return xi ¼ yi, i ¼ 0, . . . , N. Sequences fyi, i ¼ 0, . . . , 10,000g and fui, i ¼ 0, . . . , 10,000g are generated, and if the condition 3 (above) is satisﬁed, xi was set equal to yi. The sequence fxig has 3936 points out of the original total of 10,001 points. This corresponds to the expected 10,001/2.55 ¼ 3922 points. The sequence fxig consisting of 3936 points was passed through a histogram subroutine with K ¼ 100 bins and bin width Dx ¼ 0.01. The histogram was normalized by division with 3936 0.01 ¼ 39.36. The normalized histogram is graphed in Fig. 15.5.1 with the true beta(3,6) density superimposed for comparison. The bounding function g(x) is also shown. From the ﬁgure it is seen that the ﬁt is good. Discussion

We have discussed three methods for generation of random variates for a given distribution function. The inversion method is used where the cumulative distribution

15.5 Acceptance – Rejection Techniques

283

FIGURE 15.5.1

function can be expressed in a closed form or can be computed efﬁciently. The convolution method can be used when the density function is expressible as the resultant of a sum of independent random variates. The acceptance –rejection method is used directly on the probability density function. In generating histograms the number of bins K and the bin width Dx have to be manipulated ﬁrst by “eyeballing” and later conﬁrming that the x value is less than the ta value in the chi-square test at the signiﬁcance level a. Note also that higher signiﬁcance levels yield tighter bounds.

CHAPTER

16

Elements of Matrix Algebra

A number of results depend on an understanding of matrices and manipulations of matrices. We will now discuss relevant aspects of matrix algebra. We will assume familiarity with linear vectors spaces [56].

B 16.1

BASIC THEORY OF MATRICES

Deﬁnitions

A matrix X is deﬁned as an array of m column vectors {xj , j ¼ 1, . . . , m} with each column vector x.j having n real or complex elements x.j ¼ fx1j, x2j, . . . , xi j, . . . , xn jgT. Thus the matrix X is an n m rectangular array of real or complex numbers and written as 2 3 x1 x2 xj . . . xm 2 36 x11 x12 x1j x1m 7 7 x1 6 6 7 6 x2 76 x21 x22 x2j x2m 7 7 76 X¼6 (16:1:1) 7 4 xi 56 6 xi1 xi2 xij xim 7 7 xn 6 4 5 xn1 xn2 xnj xnm Notation: In the double subscript (i j ) the ﬁrst subscript denotes the row, and the second subscript denotes the column. Thus, an n m array consists of n rows and m columns. Sometimes the matrix X is also written as fxijg. A square matrix is a matrix with equal number of rows and columns. It is denoted as n n square matrix or an n matrix. If the column vectors {xj ¼ ½x1j , x2j , . . . , xnj T , j ¼ 1, 2, . . . , m} are linearly independent, then the matrix X is column-independent. On the other hand, if the row vectors denoted by {xi ¼ ½xi1 , xi2 , . . . , xim , i ¼ 1, 2, . . . , n} are linearly independent, then the matrix X is row-independent. The dots in the column-vector x j and the row-vector x i represent the integers 1, . . . , n. An arbitrary rectangular matrix cannot be both column- and rowindependent unless it is a square matrix. However, not all square matrices need be row-independent and column-independent.

Parts of this chapter are based, with permission, on Chapters 30, 31, and 33 from the book Linear Systems Properties—a Quick Reference, by Venkatarama Krishnan, published by CRC Press, 1988.

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

284

16.1

Basic Theory of Matrices

285

Matrix Addition

Addition is deﬁned for two matrices X and Y of similar dimensions: X þ Y ¼ {xij } þ {yij } ¼ {xij þ yij }

(16:1:2)

The addition rule is commutative: X þ Y ¼ Y þ X. Example 16.1.1 "

#

"

1

5 2

3

4 7

3

8

6

11

10

16

,

X¼ "

3 4

8

6 9

#

" ,

Y¼

4 6

#

Z¼ 3 2

# ,

XþY¼

2

X þ Z and Y þ Z are not defined

Matrix Multiplication

Matrix multiplication is a little complex. If we denote X Y ¼ Z, then the number of columns of X must be equal to the number of rows of Y. Otherwise, multiplication is not deﬁned. If X is ? n, then Y has to be n ? In this case the dimension of Z is ? ?. The product X Y is deﬁned by Z¼XY¼

n X

xik ykj ¼ {zij}

(16:1:3)

k¼1

Example 16.1.2 "

3

1 2

2

# ,

X¼ 4 "

5 1

4

6 6 Y ¼ 63 4 1

1

3

7 7 27 5 5

34þ13þ21 31þ12þ25

#

XY¼

"

17 15

#

¼ 44þ53þ11 41þ52þ15 2

32 19 3

2

3

43þ14

41þ15

42þ11

16

9

9

6 6 Y X ¼ 63 3 þ 2 4 4 13þ54

31þ25

7 6 7 6 3 2 þ 2 1 7 ¼ 6 17 5 4 12þ51 23

13

7 7 18 7 5 7

11þ55

26

In general, matrix multiplication is not commutative, that is, XY = YX as shown above. As an example, consider the following symmetric matrices: a b e f , Y¼ X¼ f h b d ae þ bf af þ bh ae þ bf be þ df XY ¼ , YX ¼ be þ df bf þ dh af þ bh bf þ dh

286

Elements of Matrix Algebra

The matrices X and Y commute only if af þ bh ¼ be þ df. We usually denote in the product XY that Y is premultiplied by X or X is postmultiplied by Y. Unlike real numbers, division of one matrix by another matrix is not deﬁned.

Transpose of Matrix

The transpose of a matrix X ¼ fxijg is obtained by interchanging the rows and columns and is denoted by XT ¼ {xij }T ¼ {x ji }

(16:1:4)

The transpose of a matrix product (XY) T is given by Y TX T whereas (X þ Y)T ¼ X T þ Y T. If X ¼ X T, then we say that the matrix X is symmetric.

Example 16.1.3

2 X¼ 1

5 3

27 (XY) ¼ 21 T

4 6

2

3 6 Y ¼ 41 4

30 ; 23

3 2 7 1 5;

2 6 XT ¼ 4 5 4

3

27 30 Y X ¼ 21 23 T

T

2

3 1 3 7 3 5 YT ¼ 2 6

1 1

4 3

Determinant of a Matrix

The nonzero determinant of a square matrix is one of the two most important invariants of a matrix. It is usually denoted by det X or jXj. It is computed as det X ¼

n X

xij Dij ¼ x1j D1j þ x2j D2j þ þ xnj Dnj

(16:1:5)

i¼1

with no implied summation over j and Dij is the cofactor of the matrix X deﬁned by Dij ¼ (1)iþj Mij

(16:1:6)

where Mij is the minor of the matrix X obtained by taking the determinant after deleting the ith row, i ¼ 1, . . . , n, and the ﬁxed jth column. Equation (16.1.5) is called a cofactor expansion about the jth column. The difference between the cofactor and the minor is the sign as shown in Eq. (16.1.6).

Example 16.1.4

The determinant of the following matrix is evaluated as follows: 2 3 2 5 3 2 6 1 2 4 57 7 X¼6 4 4 1 3 2 5 3 4 2 1

16.1

Basic Theory of Matrices

287

Expanding in terms of minors about the second column, the expression for the determinant jXj is 2 3 1 4 2 5 jXj ¼ 5 4 3 2 þ 2 4 3 2 3 3 2 1 2 1 2 2 3 2 3 2 5 þ 1 1 4 5 þ 4 1 4 3 4 3 2 2 1 Each determinants in the above equation can again be expanded in terms of minors about the second column as shown by the expansion of the ﬁrst term: 1 4 5 1 5 4 2 5 4 3 2 ¼ 5 4 5 3 3 3 1 1 3 2 1 1 5 5 (2) ¼ 350 4 2 Evaluating the determinant 42 þ 376 ¼ 632.

by

this

expansion,

we

have

jXj ¼ 350 252 2

If only the determinant of the matrix is zero, we call that it a singular or simply degenerate matrix. In addition to the determinant, if the ﬁrst- and higher-order minors are also zero, then we call the matrix multiply degenerate. Properties of Determinants (X and Y Nonsingular) 1. det X ¼ det X T: The following properties (2,3,4,5,6) are very useful in the manipulation of matrices. 2. If any row (or column) of X is multiplied by a constant a to yield a new matrix Y, then det X ¼ a . det Y. 3. If any row (or column) of X is proportional to the other one, then the rows (or columns) are linearly dependent and det X ¼ 0. 4. If any two rows (or columns) of X are interchanged an odd number of times to yield Y, then det Y ¼ 2det X. If they are interchanged an even number of times to yield Z, then det Z ¼ det X. 5. If any row (or column) of X is modiﬁed by adding to it a times, the corresponding elements of another row (or column) to yield Y, then det Y ¼ det X. (The determinant remains unchanged.) 6. The determinant of a triangular matrix X is the product of its diagonal terms det X ¼ x11.x22 . . . . .xnn. 7. If the matrices X and Y are both n n, then det ( XY) ¼ (det X)(det Y). 8. det (aX) ¼ an det X. 9. Even if X and Y are n n, det (X þ Y) = det X þ det Y in general.

288

Elements of Matrix Algebra

Trace of a Matrix

The trace of a square matrix Tr(X) is the second most important invariant of a matrix. The trace is the sum of the diagonal terms of a matrix. The trace of the matrix given in Example 16.1.4 is 2 þ 2 þ 3 þ 1 ¼ 8.

Rank of a Matrix

The column rank of a rectangular matrix is the number of linearly independent column vectors. Similarly, the row rank of a matrix is the number of linearly independent rows. If the column rank and the row rank are the same, then we call that number the rank of the matrix. Thus in a nonsingular square matrix the column rank equals the row rank and the rank is the dimension of the matrix. In general, rank of any n m matrix can be deﬁned as the number of elements 2, 3, . . . , n or m required to form a nonzero determinant. Forming all possible combinations of determinants from any given matrix is a difﬁcult job. Finding a rank is not always easy, but some methods are easier than others. We show one method for ﬁnding the rank and the determinant of any matrix using pivotal 2 2 determinants. Example 16.1.5 2

a11

6 6 a21 6 6 X¼6 6 a31 6 6 a41 4 a51

Find the rank and determinant of the following nonsingular matrix: a12

a13

a14

a22

a23

a24

a32

a33

a34

a42

a43

a44

a52

a53

a54

a15

3

2

1

7 6 6 a25 7 7 6 1 7 6 6 a35 7 7¼6 2 7 6 6 a45 7 5 4 1 a55

2

2

3

2

1

1

2

1

2

2

2

2

1

1

1

2

3

3

7 87 7 7 1 7 7 7 07 5

(16:1:7)

5

Step 1. Remove a11 ¼ 1 outside the matrix in Eq. (16.1.7) using property 2 as shown in the previous section. We form a 4 4 matrix X1 from 2 2 determinants of the form fa11 . ajk 2 aj1 . a1k, j ¼ 2, . . . , 5 : k ¼ 2, . . . , 5) as shown below: 2

3

6 6 3 6 X1 ¼ 1 6 6 0 4 5

4

4

8

6

1

1

5

2

11

3

7 7 7 7 7 3 7 5 1

(16:1:8)

Step 2. Normalize the ﬁrst element ¼ 3 as in step 1, and the resultant matrix is shown below. 2

1

6 6 3 6 X1 ¼ 1:3 6 6 0 4 5

4 3 8

4 3 6

1

1

5

2

11 3 3 7 7 7 7 7 3 7 5 1

(16:1:9)

16.1

Basic Theory of Matrices

289

Step 3. Form a 3 3 matrix X2 in an exactly analogous fashion by forming 2 2 determinants as in step 1 and normalize the ﬁrst element in X2 as shown in Eq. (16.1.10) 2 3 2 3 1 4 2 4 1 1 6 7 2 6 7 7 1 1 3 7 ¼ 1:3 4 6 X2 ¼ 1:3 6 (16:1:10) 1 1 3 6 7 4 5 14 4 5 14 52 5 52 5 3 3 3 3 3 3 Step 4. Form a 2 2 matrix X3 in an exactly analogous fashion by forming 2 2 determinants as in step 1 and normalize the ﬁrst element in X3 as shown in Eq. (16.1.11): 2 3 1 " # 4 1 8 1 6 2 7 23 (16:1:11) X3 ¼ 1:3 4 4 5 ¼ 1:3 4 23 19 2 19 6 6 Step 5. Continue this process of reducing the order of the matrix until we end up with either a 1 1 matrix or a matrix whose elements are all zero. Finally we form a 1 1 matrix X4 in an exactly analogous fashion by forming 2 2 determinants as in step 1: X4 ¼ 1:3 4

1 35 ¼ 70 2 3

(16:1:12)

Since we have come to a scalar, the operations stop at this point. The number of cycles is 5 and hence the rank of this matrix is 5. The determinant is given by X4 ¼ 70. Thus, we can obtain both the rank and the determinant by this method. Example 16.1.6 In this example we will determine the rank and the maximum possible determinant of a singular matrix given by 2 3 6 11 5 4 1 6 5 3 2 3 47 6 7 7 X¼6 8 12 4 2 5 6 7 4 6 7 1 0 35 3 0 3 1 2 We form the various matrices as in the previous example, and the results are shown below: 2 3 2 3 11 5 4 1 6 11 5 4 1 6 1 6 6 6 67 6 6 7 3 2 3 47 6 5 7 6 5 3 2 3 47 6 7 6 7 X¼6 2 5 7 6 8 12 4 7¼66 8 12 4 2 5 7 6 7 6 7 6 7 7 1 0 35 4 6 4 6 7 1 0 35 3 0 3 1 2 3 0 3 1 2 2

37 6 6 6 6 8 6 6 X1 ¼ 6 6 3 6 6 4 6 4 11 2

37 6 8 3 4

19 3 22 3 4

11 2

3

2 3 19 1 6 6 7 6 7 6 8 11 7 7 37 6 6 7 6 3 3 7¼6 7 6 6 6 4 2 7 6 7 4 11 5 3 2

2

1 8 3 4 11 2

38 37 22 3 4 3

3 19 37 7 7 11 7 7 7 3 7 7 27 7 35 2

290

Elements of Matrix Algebra

We now form a 3 3 matrix using 2 2 pivotal determinants. This will yield the ﬁrst column to be all zeros. Interchanging two columns of the resulting matrix is also shown below: 2

2 3 170 85 170 6 6 7 37 37 7 6 37 6 37 37 4 2 7 6 6 4 16 0 16 X2 ¼ 6 7¼6 6 6 37 7 6 6 37 37 4 4 98 98 49 5 0 37 37 37 2 3 1 0 1 6 7 2 6 7 37 170 6 4 2 7 ¼6 07 6 7 6 37 6 37 37 4 98 49 5 0 37 37 0

3 85 0 7 37 7 2 7 07 7 37 5 49 0 37

We now form a 2 2 matrix " 37 170 0 X3 ¼ 6 6 37 0

0

#

"

0

0

0

0

#

¼ 170 0

After three cycles we have a premature termination with a 2 2 zero matrix. Hence 170 the rank of this matrix is 3 and the maximum determinant is 6 37 6 37 ¼ 170:

Inverse of a Matrix

Since division by a matrix is not deﬁned, we deﬁne the inverse of a nonsingular matrix as the matrix X 21 such that XX 21 ¼ X 21X ¼ I, an identity matrix. Thus inverse matrices commute with their direct counterparts. The cofactor expansion of the determinant of the matrix X from Eq. (16.1.5) is det X ¼

n X

xij Dij ¼ x1j D1j þ x2j D2j þ þ xnj Dnj

(16:1:5)

i¼1

where the expansion is carried about the n elements in any column j. Equation (16.1.5) can be rewritten as ( n X det X, k ¼ j xij Dik ¼ (16:1:13) det X ¼ 0, k=j i¼1 In Eq. (16.1.13), when k ¼ j, we obtain det X by Eq. (16.1.5), but when k = j, then we have cofactor expansion about the n elements in the jth column but with cofactors obtained from the kth column. This is equivalent to ﬁnding the determinant by replacing the kth column with the jth column in the matrix X. Thus X will contain two identical columns and hence from property 3 in the previous section, det X ¼ 0. We divide Eq. (16.1.13) by det X and obtain ( n X 1kn Dik (16:1:14) xij ¼ dkj , det X 1jn i¼1

16.1

Basic Theory of Matrices

291

where dkj is the Kro¨necker delta, which equals 1 if k ¼ j and 0 if k = j. Equation (16.1.14) is a scalar statement of the matrix equation X21X ¼ I, and the inverse is given by 2 3 D11 D21 Dn1 7 Dik 1 6 6 D12 D22 Dn2 7 xij ¼ X1 ¼ {aki } ¼ (16:1:15) 6 .. .. .. 7 det X det X 4 . . . 5 Dnn D1n D1n Note that the cofactors are in reverse order of the sequence of elements of the matrix X. The matrix fDjig of the cofactors is sometimes called the adjoint matrix. Finding the inverse using Eq. (16.1.15) is very inefﬁcient and is of only theoretical interest since it gives an explicit expression for the inverse. Solution of Linear Equations

We can use matrix analysis to solve linear equations of the form Ax ¼ y where x and y are n-vectors: 2 32 3 2 3 y1 x1 a11 a12 a1n 6a 6 6 7 7 7 6 21 a22 a2n 7 6 x2 7 6 y2 7 6 . 7 6 7 6 (16:1:16) .. 7 6 .. 7 ¼ 6 .. 7 .. 6 . 7 . 54 . 5 4 . 5 . 4 . an1

an2

ann A

xn x

yn y

We will assume that A is a nonsingular n n matrix and x and y are n 1 vectors. We want to solve for x in terms of y. Since A is invertible, we can write the solution as x ¼ A1 y This equation can be expanded as 2 3 2 2 32 3 3 D j1 y1 D11 D21 Dn1 x1 n 6 D j2 7 6 x2 7 76 7 1 6 1 X 6 7 6 D12 D22 Dn2 7 6 y2 7 6 7 ¼ 6 .. 7 ¼ 6 .. 6 6 . 7yj 7 7 .. .. .. 5 det A . 4 . 5 det A 4 . 4 4 5 . . . . 5 j¼1 D1n D1n xn Dnn yn D jn x A1 y

(16:1:17)

(16:1:18)

Note again that the order of the cofactors is transposed from that of the original matrix A, that is, if aij is an element in the ith row and jth column, then Dji is in the jth row and ith column. Example 16.1.7 x2, x3, x4:

We shall now solve the following set of linear equations for x1, x1 3x3 2x4 ¼ 4 5 1 15 23 x1 þ x2 þ x3 þ 6x4 ¼ 2 2 2 2 2x1 7x3 6x4 ¼ 11 7 1 25 39 x1 þ x2 þ x3 þ 10x4 ¼ 2 2 2 2

292

Elements of Matrix Algebra

or in matrix form

2

1

0

3

6 5 6 6 2 6 6 6 2 6 4 7 2

1 2 0

15 2 7

1 2

25 2

2 3 2 3 4 72 x 3 6 23 7 6 7 1 6 7 76 7 6 2 7 76 x2 7 6 7 74 5 ¼ 6 7 x3 6 11 7 6 7 6 7 7 4 5 5 x4 39 10 2

The determinant of the matrix A is found as follows: 2 1 0 3 2 3 2 2 3 1 15 5 6 5 1 7 6 7 15 6 6 6 7 2 6 7 6 2 62 7 6 2 2 2 6 6 7 6 7 jAj ¼ 6 6 7 3 6 2 7 ¼ 1 6 0 7 6 6 7 6 2 0 7 7 6 4 7 41 5 6 7 25 4 5 10 7 1 25 2 2 2 10 2 2 2 2 3 5 1 15 6 2 2 2 7 6 7 6 7 þ 2 6 2 0 7 7 6 7 4 7 1 25 5 2

2

1 2 0 1 2

2

Or jAj ¼ 1 þ 3 3 ¼ 1. The inverse of matrix A is 2

A1

1

0

3

6 5 6 6 2 6 ¼6 6 2 6 4 7 2

1 2 0

15 2 7

1 2

25 2

2 31 2 1 7 6 7 6 7 16 2 7 7 ¼ 6 1 14 6 7 3 7 5 2 10

2 1 1 1 2

1 2 1 1

The solution to the original set of equations is given by 2 3 2 3 4 2 3 1 2 1 2 6 23 7 2 3 x1 1 6 7 6 2 1 2 17 7 6 7 6 x2 7 6 76 2 17 6 7 6 7 ¼ 6 1 1 1 17 7¼6 4 x3 5 6 76 4 15 6 7 4 3 1 1 56 11 7 x4 1 4 5 1 39 2 2 2 2 Properties of Inverses (X, Y, Z Same Order and Nonsingular) 1. X X21 ¼ X21X matrix commutes with its inverse 2. Xm Xn ¼ X m1n 3. (XY)21 ¼ Y21X21 4. XY ¼ XZ ) Y ¼ Z 5. (X T)21 ¼ (X21)T

3 2 17 7 17 5 1 2

3 6 7 7 7 6 7 7 5 10

16.2 Eigenvalues and Eigenvectors of Matrices

1 det X (1)iþj Mij , where Mij is the minor of X ¼ det X

293

6. det X21 ¼ 7. X1

(16.1.19)

Orthogonal Matrices

The matrix X is orthonormal if X TX ¼ I. Since X21X is also equal to the identity matrix I, we have XT X ¼ I ¼ X1 X

or XT ¼ X1

(16:1:20)

It immediately follows from Eq. (16.1.20) that the determinant of an orthogonal matrix is equal to +1. Note that the converse is not true. In any orthogonal matrix X, real or complex, individual row or column vectors x are orthonormal: 1, j ¼ i H (16:1:21) x i xj ¼ 0, j = i where H stands for conjugate transpose. An example of a 4 4 orthogonal matrix used in discrete Fourier transform analysis is shown below: 2 3 1 1 1 1 6 1 ej(2p=4) ej(4p=4) ej(6p=4) 7 7 (16:1:22) W¼6 4 1 ej(4p=4) ej(8p=4) ej(12p=4) 5 1 ej(6p=4) ej(12p=4) ej(18p=4) It can be veriﬁed that the rows and columns of W are orthonormal and det W ¼ 1.

B 16.2 EIGENVALUES AND EIGENVECTORS OF MATRICES Eigenvalues

The concept of eigenvalues and eigenvectors plays an important role. We motivate this concept by investigating solutions to a linear system of equations given by Ax ¼ lIx ¼ lx where A is an n n matrix and x is an n 1 vector, l is a scalar constant, and I is an n n identity matrix. Simpliﬁcation of the equation above yields (lI A)x ¼ 0

(16:2:1)

A trivial solution to Eq. (16.2.1) is x ¼ 0. Nontrivial solutions can exist only if det(lI A) ¼ jlI Aj ¼ 0

(16:2:2)

Equation (16.2.2) is called the characteristic equation of the matrix A. Expansion of the determinant in Eq. (16.2.2) yields a polynomial in l of degree n called the characteristic polynomial of the matrix A, given by ln þ an1 ln1 þ an2 ln2 þ þ a0 ¼ 0

(16:2:3)

294

Elements of Matrix Algebra

The characteristic equation has a fundamental signiﬁcance in linear systems. The n roots of the characteristic equation are called the eigenvalues of the matrix A, and in linear systems they are also known as the “poles” of the transfer function. The eigenvalues can be real, repeated, or complex. From the well-known result for the roots of polynomials, we also have the sum of the eigenvalues as 2an21, and the product of the eigenvalues, which is also the determinant of the matrix A, is given by (21)n a0. Example 16.2.1 determined:

The eigenvalues of the following nonsingular matrix have to be 2

3

1:5625

4:5

2:0625

5:25

6 0:4375 6 6 A¼6 6 0:4375 6 4 0:0625

2:5

1:9375

6:25

2:5

1:9375

7:25

3:5

0:4375

6:25

1:25 7 7 7 1:25 7 7 7 0:25 5

1:5

3

1

0:5

0

0:75

The characteristic equation of A is det (lI 2 A)¼0 is given by 2 l þ 1:5625 4:5 6 6 0:4375 l þ 2:5 6 6 jlI Aj ¼ 6 0:4375 2:5 6 6 6 3:5 4 0:0625 0:5 0

2:0625

5:25

1:9375

6:25

l þ 1:9375

7:25

0:4375 1:5

l 6:25 3

3 0:75 7 1:25 7 7 7 1:25 7 7 7 0:25 7 5 l1

¼ l5 1:25l4 2:375l3 þ l2 þ 1:375l þ 0:25 ¼ 0 The solutions of the ﬁfth-order polynomial of this equation are the eigenvalues of A, given by l1 ¼ 2, l2 ¼ 1, l3 ¼ 1, l4 ¼ 0:5, l5 ¼ 0:25 The product of the eigenvalues is the determinant given by (21)5 . (a0) ¼ 20.25 and the sum of the eigenvalues is a4 ¼ 21.25. We can also ﬁnd the eigenvalues of a square matrix that is singular. In this case the number of nonzero eigenvalues represents the rank of the matrix and the number of zero eigenvalues, the degree of singularity.

Cayley– Hamilton Theorem

The Cayley– Hamilton theorem states that every square matrix satisﬁes its own characteristic equation. In Eq. (16.2.3), if we substitute the matrix A for l, we obtain the Cayley –Hamilton equation An þ an1 An1 þ an2 An2 þ þ a0 A0 ¼ 0 where 0 represents a zero matrix.

(16:2:4)

16.2 Eigenvalues and Eigenvectors of Matrices

Example 16.2.2

295

Given the matrix 2

0 A¼4 6 6

1 0 11

3 0 15 6

equation is l3 þ 6l2 þ 11l þ 6 ¼ 0. According to the Cayley– we should have 2 32 2 31 33 0 0 1 0 0 1 0 6 7 6 7 7 1 5 þ 64 0 0 1 5 þ 114 0 0 15 6 6 11 6 6 11 6 30 2 3 2 3 1 0 6 11 6 0 0 6 7 6 7 6 7 0 1 5 ¼ 4 36 60 25 5 þ 4 36 66 36 5 11 6 150 239 90 216 360 150 3 2 3 11 0 6 0 0 7 6 7 0 11 5 þ 4 0 6 0 5 121 66 0 0 6

the characteristic Hamilton theorem, 2 0 1 6 0 4 0 6

11 2 0 6 þ 64 0 6 2 0 6 þ4 0 66

or

2

0 ¼ 40 0

0 0 0

3 0 05 0

showing the validity of the theorem. Deﬁniteness of Matrices

A matrix A is positive deﬁnite if all its eigenvalues are either positive or have positive real parts. For example, the matrix A in Example 16.2.2 is positive deﬁnite, whereas the matrix A in Example 16.2.1 is nondeﬁnite. Similarly, matrices having eigenvalues that are negative or have negative real parts are known as negative deﬁnite. If some of the eigenvalues are zero, then the matrix is called positive semideﬁnite or negative semideﬁnite accordingly. Since the determinants are (21)n times the product of eigenvalues, deﬁniteness can also be deﬁned by the sign of the determinants of all principal minors of a matrix. Principal minors are those minors obtained by successively deleting the rows and columns of a matrix starting from the ﬁrst row and ﬁrst column. If all the determinants of principal minors are positive, then the matrix is positive-deﬁnite: if they are negative then they are negative deﬁnite. Example 16.2.3 eigenvalues:

We will check the deﬁniteness of the following matrix by ﬁnding the 2

1

3

7

7

5

6

63 6 6 A¼6 63 6 4 2:5

9

11

4

9

11

3

8

9:5

4

87 7 7 87 7 7 75

0:5

0

1:5

3

1

296

Elements of Matrix Algebra

As in the previous example, setting det (lI 2 A) ¼ 0 gives 3 2 7 5 6 l 1 7 6 3 l9 11 4 8 7 6 7 6 7 2 3 2 jlI Aj ¼ 6 9 l þ 11 3 8 7 6 3 7 ¼ l (l þ 4l þ l 6) ¼ 0 6 7 4 2:5 8 9:5 l þ 4 7 5 0:5 0 1:5 3 l1 Thus the rank of the matrix A is 3 since there are two zero eigenvalues. The eigenvalues are given by l1 ¼ 23, l2 ¼ 22, l3 ¼ 1, l4 ¼ 0, l5 ¼ 0. Here the matrix A is nondeﬁnite since it has positive, negative, and zero eigenvalues.

Eigenvectors

Eigenvectors are solutions to Eq. (16.2.1) for any given eigenvalue. We shall assume that all eigenvalues are distinct and there are no repeated eigenvalues. If we substitute any eigenvalue li in det(lI 2 A) ¼ 0, then, by the very deﬁnition, we have det(liI– A) ¼ 0 and hence solutions to Eq. (16.2.1) are possible, and these solutions are the unnormalized eigenvectors ci. However, these solutions are unique only upto a multiplicative constant. To make the solutions unique we can normalize these eigenvectors and the normalized eigenvectors are denoted by fi ¼ ci/jcij. Example 16.2.4 repeated below:

Find the eigenvectors of the nonsingular matrix of Example 16.2.1, 2

2:0625

6 0:4375 2:5 6 6 A¼6 6 0:4375 2:5 6 4 0:0625 3:5

1:9375

6:25

1:9375

7:25

0:4375

6:25

1:25 7 7 7 1:25 7 7 7 0:25 5

1:5

3

1

0:5

0

5:25 0:75

3

1:5625 4:5

Substituting l1 ¼ 2 in this equation, we can solve for the unnormalized eigenvectors: 2 32 3 3:5625 4:5 2:0625 5:25 0:75 c11 6 6 0:4375 4:5 7 1:9375 6:25 1:25 7 6 76 c12 7 6 76 7 6 7 ðl1 I AÞC1 ¼ 6 3:9375 7:25 1:25 7 6 0:4375 2:5 76 c13 7 ¼ 0 6 76 7 4 0:0625 3:5 0:4375 4:25 0:25 54 c14 5 0:5

0

1:5

3

1

c15

As mentioned earlier, the equation above can be solved for only four of the unknowns in terms of the ﬁfth. Let us assume that c11 ¼ 1 and solve for c12,c13,c14,c15 in terms of c11. The preceding equation can be rewritten by substituting c11 ¼ 1: 2 3 2 32 3 4:5 1:9375 6:25 1:25 c12 0:4375 6 7 6 76 7 6 2:5 3:9375 7:25 1:25 76 c13 7 6 0:4375 7 6 7¼6 76 7 6 3:5 0:4375 4:25 0:25 76 c 7 6 0:0625 7 4 54 14 5 4 5 0 1:5 3 1 c15 0:5

16.2 Eigenvalues and Eigenvectors of Matrices

297

Solving for c12 ,c13 ,c14 ,c15 and adjoining c11 ¼ 1, we obtain the unnormalized eigenvector corresponding to the eigenvalue l1 ¼ 2: 2 3 2 3 1 c11 2 3 2 3 c12 1:333333 6 7 6 7 6 c12 7 6 1:333333 7 6 c 7 6 1:952381 7 6 7 6 7 6 13 7 6 7 c13 7 ¼6 1:952381 7 6 7¼6 7 and 6 6 7 6 7 4 c14 5 4 1:238095 5 6 7 6 7 4 c14 5 4 1:238095 5 c15 1:285714 c15 1:285714 Normalization of this equation yields the eigenvector f1: 2 2 3 2 3 3 c11 1 0:319838 6 6 7 6 7 7 6 1:333333 7 6 0:426451 7 6 c12 7 6 7 6 7 7 c1 6 1 6 1:952381 7 ¼ 6 0:624446 7 6 c13 7 ¼ f1 ¼ 6 7 6 7 6 7 jc1 j 6 7 6 7 7 3:126581 6 4 1:238095 5 4 0:395990 5 4 c14 5 c15 1:285714 0:411220 l1 ¼2

In a similar manner, the other eigenvectors corresponding to l2 ¼ 1, l3 ¼ 21, l4 ¼ 20.5, and l5 ¼ 20.25 are shown below: 2 3 2 3 2 3 0:275010 0:774597 0:679189 6 6 6 7 7 7 6 0:366679 7 6 0:516398 7 6 0:549819 7 6 7 6 7 6 7 7 6 7 6 7 f2 ¼ 6 6 0:641681 7 , f3 ¼ 6 0:258199 7 , f4 ¼ 6 0:032342 7 , 6 7 6 7 6 7 4 0:275010 5 4 0:258199 5 4 0:291081 5 0:550019 0 0:388108 2

l2 ¼1

3

l3 ¼1

l4 ¼0:5

0:679104 6 7 6 0:325871 7 6 7 7 f5 ¼ 6 6 0:340796 7 6 7 4 0:166667 5 0:531373 l5 ¼0:25

Modal Matrix

A matrix F consisting of the linearly independent eigenvectors of a matrix A is called the modal matrix of A. In the previous example the modal matrix of A was 2 3 0:319838 0:319838 0:774597 0:679189 0:679104 6 0:516398 0:549819 0:325871 7 6 0:426451 0:426451 7 6 7 7 0:624446 0:624446 0:258199 0:032342 0:340796 F¼6 6 7 6 7 0:258199 0:291081 0:166667 5 4 0:395990 0:395990 0:411220 0:411220 0 0:388108 0:531373 The determinant of F ¼ 20.00141.

298

Elements of Matrix Algebra

Diagonalization of Matrices

An n n square matrix A is diagonalizable if there exists another nonsingular n n matrix T such that T21AT ¼ D, where D is a diagonal matrix. The conditions of diagonalizability reduce to the existence of n linearly independent eigenvectors of A or the modal matrix F of A is invertible, in which case the jth diagonal element of D is the jth eigenvalue of A. In this case the modal matrix F is the desired matrix T. The modal matrix F is not the only matrix that will diagonalize A. There are other matrices that may also diagonalize A, but all the diagonal elements obtained from the modal matrix only are the eigenvalues of A. As a consequence of the preceding discussion, we can write the matrix equivalent of the eigenequation (16.2.1) as follows: or F1 AF ¼ L

AF ¼ FL Example 16.2.5

Diagonalize the following matrix: 2 0 1 0 6 4 0 0 A¼6 4 0 2 0 5 1 1

(16:2:5)

3 0 07 7 15 0

The eigenvalues of A are obtained by solving the determinant equation jlI 2 Aj ¼ 0, or 2 3 l 1 0 0 6 4 l 0 07 7 ¼ l4 5l2 þ 4 ¼ 0 jlI Aj ¼ 6 4 0 2 5 l 1 5 1 1 l The characteristic equation can be factored as (l 2 2)(l þ 2)(l 2 1)(l þ 1) ¼ 0, yielding the eigenvalues as l1 ¼ 2, l2 ¼ 22, l3 ¼ 1, l4 ¼ 21. We ﬁnd the eigenvectors corresponding to these eigenvalues. The eigenvectors following the Example 16.2.4 are 2 3 2 3 3 3 2 3 2 3 ﬃ7 6 pﬃﬃﬃﬃﬃ 7 6 pﬃﬃﬃﬃﬃﬃﬃ 0 0 6 74 7 6 146 7 6 6 7 6 7 6 7 6 6 7 6 pﬃﬃﬃﬃﬃ 7 6 0 7 6 0 7 6 pﬃﬃﬃﬃﬃﬃﬃﬃ 7 6 6 7 6 7 7 6 7 6 74 7 6 1 7 6 1 7 6 146 7 f1 ¼ 6 , f2 ¼ 6 , f3 ¼ 6 pﬃﬃﬃ 7, f4 ¼ 6 pﬃﬃﬃ 7 7 7 6 5 7 6 27 6 27 6 1 7 6 pﬃﬃﬃﬃﬃ 7 6 7 6 7 6 pﬃﬃﬃﬃﬃﬃﬃﬃ 7 4 1 5 4 1 5 6 74 7 6 146 7 6 7 6 7 pﬃﬃﬃ pﬃﬃﬃ 4 2 5 4 10 5 2 2 pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ 146 74 or 2

0:348743

3

2

0:248282

3

6 0:697486 7 6 0:496564 7 6 7 6 7 f1 ¼ 6 7 , f2 ¼ 6 7, 4 0:581238 5 4 0:082761 5 0:827606

0:232495 l1 ¼2

2

l2 ¼2

3

0 6 7 0 6 7 f4 ¼ 6 7 4 0:707107 5 0:707107 l4 ¼1

2

0

3

6 7 0 6 7 f3 ¼ 6 7, 4 0:707107 5 0:707107 l3 ¼1

16.2 Eigenvalues and Eigenvectors of Matrices

299

The modal matrix F is given by 2

F ¼ ½f1

f2

f3

3 6 pﬃﬃﬃﬃﬃ 6 74 6 6 6 pﬃﬃﬃﬃﬃ 6 6 74 f4 ¼ 6 6 5 6 pﬃﬃﬃﬃﬃ 6 74 6 4 2 pﬃﬃﬃﬃﬃ 74

3 pﬃﬃﬃﬃﬃﬃﬃﬃ 0 146 6 pﬃﬃﬃﬃﬃﬃﬃﬃ 0 146 1 1 pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 146 2 10 1 pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 146 2

3 0 7 7 7 0 7 7 7 1 7 7 pﬃﬃﬃ 7 27 7 1 5 pﬃﬃﬃ 2

and 2

2 60 1 6 F AF ¼ 4 0 0

0 2 0 0

0 0 1 0

3 0 07 7 05 1

Symmetric Matrices

Real symmetric matrices are those matrices that satisfy the condition AT ¼ A

(16:2:6)

Complex symmetric matrices satisfy the Hermitian symmetry condition AH ¼ (A )T ¼ A

(16:2:7)

where the symbol H deﬁned earlier stands for the conjugate transpose. Symmetric matrices play a very important role in probability; for example, the covariance matrices are all real symmetric matrices. They have some important properties that come in handy in simplifying algebraic manipulations. Some of the more important properties of symmetric matrices are 1. All eigenvalues are real. 2. The eigenvectors corresponding to distinct eigenvalues are orthogonal. Hence the eigenvectors can form the basis vectors of an n-dimensional coordinate system. 3. The modal matrix F is an orthogonal matrix, and hence F21 ¼ F T. The diagonalization of a symmetric matrix can be accomplished by the transformation F TAF ¼ L. Example 16.2.6 The eigenvectors and eigenvalues of the symmetric matrix given below are to be determined and their properties studied: 2 3 1 2 1 1 6 2 1 1 1 7 7 A¼6 4 1 1 2 2 5 1 1 2 0 The characteristic equation is det [lI 2 A] ¼ 0: l 1 2 1 1 2 l 1 1 1 jlI Aj ¼ ¼ l4 4l3 7l2 þ 22l þ 24 ¼ 0 1 1 l 2 2 1 1 2 l

300

Elements of Matrix Algebra

The eigenvalues are l1 ¼ 3, l2 ¼ 22, l3 ¼ 21, l4 ¼ 4, obtained by factoring the characteristic equation. The four normalized eigenvectors are obtained as outlined in the previous examples and they are given by 2 3 2 3 2 3 2 3 1 1 1 1 7 7 7 7 1 6 1 6 1 6 1 6 6 17 617 6 1 7 6 17 f1 ¼ pﬃﬃﬃ 6 f2 ¼ pﬃﬃﬃ 6 7 , f3 ¼ pﬃﬃﬃﬃﬃ 6 7 , f4 ¼ pﬃﬃﬃﬃﬃ 6 7 7, 3405 10 4 2 5 24 05 15 4 3 5 0

1

l1 ¼3

2

l2 ¼2

2

l3 ¼1

l4 ¼4

The modal matrix F is 2

1 1 0 1

1 1 1 1 1 6 1 F ¼ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ 6 2 3 10 15 4 0 0 It can be veriﬁed that

fi fj ¼

1, 0,

1 1 2 2

3 1 17 7, 35 2

jFj ¼

30 ¼1 30

j¼i j=i

showing that the modal matrix is orthogonal and jFj ¼ 1. The original matrix A can be diagonalized by the modal matrix as shown below: 2

1 6 1 6 1 FT AF ¼ 6 30 4 1

1 1

0 0

1

2

1

1

3

2

1 6 1 6 6 4 0 0

1 1 1 1 0 1

2 2

32 0 1 7 6 1 76 2 76 2 54 1

2 1

1 1

1

2

3 1 1 7 7 7 2 5

2 1 1 2 0 3 2 3 1 3 0 0 0 6 0 2 17 0 07 7 1 6 7 ¼6 7 7 3 5 30 4 0 0 1 0 5 2 0 0 0 4

Similarity Transformations

In the previous section we obtained the diagonal matrix L by transforming the matrix A by the modal matrix F. The resulting diagonal matrix L has all the fundamental properties of the original matrix A. For example these two matrices have the same determinants and the same traces. Such matrices are called similar matrices, and the transformation is known as a similarity transformation. Any nonsingular matrix A can be transformed into another similar matrix B by the following similarity transformation T1 AT ¼ B where T is another nonsingular transforming matrix. Example 16.2.7 With 2 0 1 6 1 0 T¼6 4 1 1 1 1

1 1 1 1

3 0 17 7 05 1

2

and

1 2 6 2 1 A¼6 4 1 1 2 1

3 1 2 2 17 7 1 05 1 2

16.3

Vector and Matrix Differentiation

301

we have 2

7 6 2 1 T AT ¼ B ¼ 6 4 7 2

0 7 2 2 4 6 3 1

3 5 37 7 75 2

In this example the determinants of A and B are 1 and the traces of A and B are 21.

B 16.3

VECTOR AND MATRIX DIFFERENTIATION

The deﬁnitions for differentiation with respect to a real vector and a complex vector are slightly different. In the case of differentiation of a complex vector, the question of whether the derivative is a total or partial also makes a difference. We shall ﬁrst discuss differentiation with respect to real vectors.

Differentiation with Respect to Real Vectors

Let x be an n-vector variable deﬁned by x ¼ fx1, x2, . . . , xngT. The total and partial derivative operators with respect to x are deﬁned by 2 2 3 3 d @ 6 dx1 7 6 @x1 7 6 6 7 7 6 d 7 6 @ 7 d @ 6 6 7 7 ¼ ¼6 (16:3:1) 7, 7 dx 6 6 dx2 7 @x 6 @x2 7 4 d 5 4 @ 5 @xn dxn Just as the (d=dx)x ¼ 1, we want the derivative with respect to the vector x to be equal to the identity matrix I, taking care that the dimensional compatibility conditions are satisﬁed. Thus, (d=dx)x cannot be deﬁned since the compatibility conditions are not met. We can, however, deﬁne (d=dx)xT ¼ I as shown below: 2 3 d 6 dx1 7 2 3 6 7 1 0 0 7

d T 6 d 4 5 7 (16:3:2) x ¼6 6 dx2 7 x1 x2 xn ¼ 0 1 0 ¼ I dx 6 7 0 0 0 1 4 d 5 dxn In an analogous fashion we can deﬁne the following operation: 2 3 d 6 dx1 7 2 3 6 7 2x1 6 7

d T d 7 2 2 2 4 5 x x¼6 6 dx2 7 x1 þ x2 þ þ xn ¼ 2x2 ¼ 2x dx 6 7 2x n 4 d 5 dxn

(16:3:3)

302

Elements of Matrix Algebra

We can also write the following differentiation formulas as follows d T x Ax ¼ (A þ AT )x dx d T d y Ax ¼ xT AT y ¼ AT y dx dx

(16:3:4)

Note that in Eq. (16.3.4) x TAx is a scalar, known as the quadratic form, and y TAx is also a scalar, known as the bilinear form. A scalar is its own transpose, that is, x TAx ¼ x TA Tx and y TAx ¼ x TA Ty. Corresponding to Eq. (16.3.2), we can also deﬁne the following operations: 2 3 x1 T 6 x 7 27 d d d d 6 6 . 7¼n x¼ 7 . dx dx1 dx2 dxn 6 4 . 5 xn

d dx

T

xxT ¼

d dx1 2

d dx2 x21

x1 x2

6x x 2 6 2 1 x2 6 6 . .. 4 .. . xn x1 xn x2 T d d d xyT ¼ dx dx1 dx2 2 x1 y1 x1 y2 6x y x y 6 2 1 2 2 6 .. 6 .. 4 . . xn y1 xn y2

d dxn

3

x1 xn

x2 xn 7 7 T .. 7 7 ¼ (n þ 1)x . 5

d dxn

(16:3:5)

x2n

x1 yn

3

x2 yn 7 7 T .. 7 7 ¼ ny . 5 xn yn

Differentiate f(x) with respect to the vector x: " #" # x1

3 2 T f (x) ¼ x Ax ¼ x1 x2 ¼ 3x21 þ 6x1 x2 þ 5x22 4 5 x2

Example 16.3.1

Differentiating f(x) with respect to x1 and x2, we obtain 9 d " > (3x21 þ 6x1 x2 þ 5x22 ) ¼ 6x1 þ 6x2 > = 6 dx1 ¼ d > 6 > (3x21 þ 6x1 x2 þ 5x22 ) ¼ 6x1 þ 10x2 ; dx2 Using the formula ½d(xT Ax)=dx ¼ ½A þ AT x, we (" # " 3 2 3 d T þ f (x) ¼ ½A þ A x ¼ dx 4 5 2

obtain # )" # 4 x1 5

x2

6

#"

10

"

x1

#

x2

6

6

6

10

¼

#"

x1 x2

#

16.3

Vector and Matrix Differentiation

303

We differentiate f (x, y) with respect to the vector x in the following

Example 16.3.2 example: T

f (x, y) ¼ y Ax ¼ y1

3 y2 4

2 5

x1 x2

¼ 3x1 y1 þ 4x1 y2 þ 2x2 y1 þ 5x2 y2

Using the formula ½d(yT Ax)=dx ¼ AT y, we can write 3 4 y1 3y1 þ 4y2 AT y ¼ ¼ 2y1 þ 5y2 2 5 y2 Complex Differentiation Total Derivatives Differentiation with respect to a complex variable z ¼ x þ jy is a bit more involved. If we have a function of a complex variable

w ¼ f (z) ¼ u(x, y) þ jv(x, y)

(16:3:6)

then the total derivative of f(z) with respect to z is deﬁned by d Dw f (z þ Dz) f (z) w ¼ lim ¼ lim Dz!0 Dz Dz!0 dz Dz

(16:3:7)

If the derivative df (z)=dz is to exist, then the limit must exist independent of the way in which Dz approaches zero. This condition imposes restrictions on the existence of the total derivative. With Dw ¼ Du þ j Dv and Dz ¼ Dx þ j Dy, we have the following from Eq. (16.3.7): lim Dz ¼ lim Dz!0

Dx!0 Dy!0

Du þ jDv Dx þ jDy

(16:3:8)

If we ﬁrst let Dy ! 0 and then let Dx ! 0 in Eq. (16.3.8), we have df (z) @u @v ¼ þj dz @x @x

(16:3:9)

Similarly, if we ﬁrst let Dx ! 0 and then let Dy ! 0, we obtain df (z) @v @u ¼ j dz @y @y

(16:3:10)

If the derivative df (z)=dz is to be unique then Eqs. (16.3.9) and (16.3.10) must match. Hence @u @v ¼ @x @y @v @u ¼ @x @y

(16:3:11)

These conditions are known as the Cauchy –Riemann (C– R) conditions. If C – R conditions are satisﬁed, then the derivative of f (z) is given by either Eq. (16.3.9) or Eq. (16.3.10).

304

Elements of Matrix Algebra

A complex function f (z) is said to be analytic in the neighborhood of z ¼ z0, if C –R conditions are satisﬁed in that neighborhood. In addition, if it is also analytic at z ¼ z0, then the point z0 is a regular point. However, if it is not analytic at z ¼ z0, then z0 is called a singular point. Example 16.3.3 We will ﬁnd the derivative of f (z) ¼ z 2 ¼ (x þ jy)2 ¼ x 2 2 y 2 þ 2jxy. We will determine whether f (z) satisﬁes the Cauchy – Riemann conditions of Eq. (16.3.9). z2 ¼ (x þ jy)2 ¼ x2 y2 þ 2jxy u(x, y) ¼ x2 y2

:

v(x, y) ¼ 2xy

@u @v ¼ 2x ¼ ¼ 2x @x @y and hence

@v @u ¼ 2y ¼ ¼ 2y @x @y

The C – R conditions are satisﬁed and df (z)=dz is given by d 2 @u @v z ¼ þ j ¼ 2(x þ jy) ¼ 2z dz @x @x

Example 16.3.4 On the other hand, we will ﬁnd the derivative of f(z) ¼ z 2 ¼ (x 2 jy)2 ¼ x 2 2 y 2 2 2jxy. We will determine whether f(z) satisﬁes the Cauchy – Riemann conditions of Eq. (16.3.9): z2 ¼ (x jy)2 ¼ x2 y2 2jxy u(x, y) ¼ x2 y2

: v(x, y) ¼ 2xy

@u @v ¼ 2x = ¼ 2x @x @y @v @u ¼ 2y = ¼ 2y @x @y Since the C –R conditions are not satisﬁed at points other than at z ¼ 0, z 2 does not have a total derivative dz2 =dz at points other than at z ¼ 0. Partial Derivatives Many functions occurring in engineering practice are not analytic, and we are interested in optimization problems involving partial derivatives with respect to a complex variable z. With z ¼ x þ jy, we shall deﬁne the partial derivative operator @=@z as

@ @ @ ¼ þj @z @x @y

(16:3:12)

16.3

Vector and Matrix Differentiation

305

From the deﬁnition given in Eq. (16.3.12), we have the following: @ @ @ z¼ þj (x þ jy) ¼ 1 1 ¼ 0 @z @x @y @ @ @ z ¼ þj (x jy) ¼ 1 þ 1 ¼ 2 @z @x @y

(16:3:13)

Note that from Example 16.3.3 the total derivative (d/dz)z does not exist since z is not analytic except at z ¼ 0. From Eq. (16.3.12) we can also write @ @ @ (z z) ¼ þj (x2 þ y2 ) @z @x @y ¼ 2(x þ jy) ¼ 2z @ 2 @ @ z ¼ þj (x2 þ 2jxy y2 ) @z @x @y ¼ 2(x þ jy) 2(x þ jy) ¼ 0 @ @ @ (z w) ¼ þj (xu þ yv þ j(xv yu)Þ @z @x @y

(16:3:14)

¼ 2(u þ jv) ¼ 2w @ @ @ (zw) ¼ þj (xu yv þ j(xv þ yu)Þ @z @x @y ¼ (u þ jv) (u þ jv) ¼ 0 Note that if we used the chain rule of differentiation in Eq. (16.3.14) and apply Eq. (16.3.13), we obtain the same results. This fact gives credibility to the deﬁnition of the partial derivative @=@z. Note also that if z and w were real, then we would get (@=@z)z2 ¼ 2z and (@=@z)zw ¼ w. These differences also manifest themselves in deﬁning vector differentiation with respect to a complex variable.

Vector Partial Differentiation Analogous to Eq. (16.3.12), partial derivatives with respect to a vector complex variable can be deﬁned. With z ¼ [x1 þ jy1, x2 þ jy2, . . . , xn þ jyn]T, partial derivatives @=@z and @=@z are deﬁned as follows:

2

2

x1 þ jy1

3

6 x þ jy 7 27 6 2 7; z¼6 .. 6 7 4 5 . xn þ jyn

3 @ @ þj 6 @x1 @y1 7 6 7 6 @ @ 7 6 7 þj @ 6 @y2 7 ¼ 6 @x2 7; 7 @z 6 .. 6 7 . 6 7 4 @ @ 5 þj @xn @yn

2

3 @ @ j 6 @x1 @y1 7 6 7 6 @ @ 7 6 7 @ 6 @x2 j @y2 7 ¼ 6 7 7 @z 6 .. 6 7 . 6 7 4 @ @ 5 j @xn @yn

(16:3:15)

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Elements of Matrix Algebra

With this deﬁnition of @=@z and @=@z , we can deﬁne the following operations: 2

3 @ @ þj 6 @x1 @y1 7 6 7 6 @ @ 7 6 @ H 6 @x þ j @y 7 7 2 7 x1 jy1 z ¼6 2 6 7 @z .. 6 7 . 6 7 4 @ @ 5 þj @xn @yn 2 3 2 0 0 0 60 2 0 07 6 7 ¼6 7 ¼ 2In 40 0 2 05 0

0 0

x2 jy2

xn jyn

(16:3:16)

2

where z H is the conjugate transpose of z and In is an nth-order identity matrix. In a similar manner, we obtain 2

3 @ @ j 6 @x1 @y1 7 6 7 6 @ @ 7 6 7 j @ T 6 @x @y2 7 2 z ¼ 6 7 x1 þ jy1 6 7 @z .. 6 7 . 6 7 4 @ @ 5 j @xn @yn 2 3 2 0 0 0 60 2 0 07 6 7 ¼6 7 ¼ 2In 40 0 2 05 0

0

0

x2 þ jy2

xn þ jyn

(16:3:17)

2

In addition, we also have 2

3 @ @ þj 6 @x1 @y1 7 6 7 6 @ @ 7 6 7 þ j @ T 6 @x @y2 7 z ¼6 2 7 x1 þ jy1 6 7 @z . 6 7 .. 6 7 4 @ @ 5 þj @xn @yn ¼ 0n ¼

x2 þ jy2

xn þ jyn

(16:3:18)

@ H z @z

where 0n is an nth-order zero matrix. Equations (16.3.16) – (16.3.18) are very useful in complex vector minimization of scalar quadratic error performance criteria, as we will show in the following examples. Example 16.3.5 The terms z and w are two complex 2-vectors as shown below. It is desired to minimize the complex matrix scalar criteria 11 ¼ zH w and 12 ¼ wH z with

16.3

respect to the complex vector z. With a þ jb z¼ ; c þ jd

Vector and Matrix Differentiation

w¼

e þ jf g þ jh

307

and the corresponding error criteria

11 ¼ zH w ¼ ae þ bf þ cg þ dh þ j(af be þ ch dg) 12 ¼ wH z ¼ ae þ bf þ cg þ dh j(af be þ ch dg) we can differentiate 11 and 12 partially with respect to z, and using Eqs. (16.3.16) and (16.3.18), we obtain e þ e þ j( f þ f ) @ @ H z w¼ 11 ¼ ¼ 2w @z @z g þ g þ j(h þ h) e e þ j( f f ) @ @ @ T 12 ¼ w H z ¼ z w ¼ ¼0 @z @z @z g g þ j(h h) The same result can be obtained by the chain rule of differentiation and Eqs. (16.3.16) – (16.3.18). Example 16.3.6 We take an example of minimizing a scalar quadratic error surface with respect to a complex weight vector a p called adaptive coefﬁcients and ﬁnding the optimum value of ap. The quadratic error surface is given by H H E p ¼ s2 þ a H p R p ap þ r p ap þ a p r p

(16:3:19)

We differentiate with respect to a p and set the result to zero and solve for a po using Eqs. (16.3.16) – (16.3.18): @ @ H H Ep ¼ ½s2 þ aH p Rp ap þ rp ap þ ap rp ¼ 0 @ap @ap ¼

@ 2 @ H @ H @ H s þ a R p ap þ r ap þ a rp ¼ 0 @a p @a p p @ap p @ap p

(16:3:20)

¼ 0 þ 2Rp ap þ 2rp ¼ 0, or apo ¼ R1 p rp The solution a po ¼ R1 p rp is called the Wiener solution for the adaptive coefﬁcients a p. Summary of Some Vector Partial Derivatives of Real and Complex Quadratic Forms @ @ @ T @ T AT x ¼ ½A þ AT x : x real J(x) ¼ ½xT Ax ¼ x Ax þ x @x @x @x @x @ @ @ H @ T AT x ¼ 2Ax þ 0 : x complex J(x) ¼ ½xH Ax ¼ x Ax þ x @x @x @x @x (16:3:21) @ @ @ T J(x, y) ¼ ½xT Ay ¼ x Ay ¼ Ay : x real @x @x @x @ @ H @ H J(x, y) ¼ ½x Ay ¼ x Ay ¼ 2Ay : x complex @x @x @x

308

Elements of Matrix Algebra

A ﬁnal point to note in taking the vector partial derivatives is that if x is real, then (@=@x)xT ¼ I, and if x is complex, then (@=@x)xH ¼ 2I and (@=@x)xT ¼ 0. Differentiation with Respect to Real Matrices

In some minimization problems there is a need to differentiate a scalar function of a real matrix A with repect to the real matrix A. We will use the following result in Section 22.3. If B is any square matrix and A is any other matrix, then @ Tr½ABAT ¼ A½B þ BT @A

(16:3:22)

@ Tr½ABAT ¼ 2AB @A

(16:3:23)

If B is symmetric, then

Example 16.3.7

We are given the matrices X and B as x11 x12 b11 b12 X¼ ; B¼ x21 x22 b21 b22

We will show that Eq. (16.3.22) is true. The trace of XBX T is given by Tr½XBXT ¼ x211 b11 þ x222 b22 þ x212 b11 þ x221 b22 þ x11 x21 b12 þ x11 x21 b21 þ x22 x12 b12 þ x22 x12 b21 Taking partial derivatives with respect to x11, x22, x12, and x21, we obtain @ Tr½XBXT ¼ 2x11 b11 þ x21 (b21 þ b12 ) @x11 @ Tr½XBXT ¼ 2x22 b22 þ x12 (b21 þ b12 ) @x22 @ Tr½XBXT ¼ 2x12 b11 þ x22 (b21 þ b12 ) @x12 @ Tr½XBXT ¼ 2x21 b22 þ x11 (b21 þ b12 ) @x21 x11 x12 b11 b12 b11 ¼ þ x21 x22 b21 b22 b12

B 16.4

b21 b22

¼ X{B þ BT }

BLOCK MATRICES

In some applications we will encounter block matrices whose elements are also matrices. An (n m) (n m) matrix Z can be formulated as a block matrix consisting of

16.4

individual blocks A,B,C,D, as 2 x11 6 6 6 6 xn1 6 Z¼6 6 6 xnþ1,1 6 6 4 xnþm,1 2 A 6 6nn ¼6 6 C 4 mn

Block Matrices

309

shown below: x1n

x1, nþ1

xn,n

xn, nþ1

B

xnþ1,n

xnþ1, nþ1

A C

D

xnþm,n xnþm, nþ1 3 B 7 nm7 7 D 7 5 mm

x1, nþm xn, nþm xnþ1, nþm

3 7 7 7 7 7 7 7 7 7 7 5

xnþm, nþm

(16:4:1)

We can enumerate some properties of block matrices that may be useful in later chapters: 1. Multiplication. The compatibility of multiplying blocks must be satisﬁed as shown: 2 3 2 3 F E B A 6 7 6nn 6 nm7 nm7 6 76 n n 7 4 C D 56 H 7 4 G 5 mn mm mm mn 2 3 AF þ BH AE þ BG 6 nn nm 7 6 7 6 7 (16:4:2) ¼6 CF þ DH 7 4 CE þ DG 5 mm mn 2. Determinant. If A is nonsingular and D is square, then " # B A ¼ det A det(D CA1 B) det C D

(16:4:3)

3. Inverse. a. Matrix Inversion Lemma. If A and D are nonsingular n n and m m matrices respectively, then, (A þ BD1 C)1 ¼ A1 A1 B(D þ CA1 B)1 CA1

(16:4:4)

b. If inverses for A and D exist then,

A C

0 D

A 0

B D

"

1 ¼

"

1 ¼

A1

0

D1 CA1

D1

A1 0

A1 BD1 D1

# ; #

(16:4:5)

310

Elements of Matrix Algebra

c. If inverses for A and D exist then, 1 " D1 A B A ¼ 1 C D D1 D CA

A1 BD1 D D1 D

# (16:4:6)

where D A and D D are the Schur components of the matrices A and D deﬁned by DA ¼ A BD1 C DD ¼ D CA1 B

(16:4:7)

d. Using property 3a, the inverses of the Schur component D A and D D are given by 1 1 þ A1 BD1 D1 A ¼A D CA 1 1 D1 þ D1 CD1 D ¼D A BD

(16:4:8)

1 We can substitute for either D1 A or DD from Eq. (16.4.8) and solve Eq. (16.4.6).

CHAPTER

17

Random Vectors and Mean-Square Estimation

B 17.1

DISTRIBUTIONS AND DENSITIES

Joint Distributions and Densities

We have come across sequences of random variables in Chapter 14, and these can be represented by a random n-vectors. We can deﬁne two random vectors X and Y as 2 3 2 3 X1 Y1 6X 7 6Y 7 6 27 6 27 6 7 7 X¼6 (17:1:1) 6 .. 7; Y ¼ 6 .. 7 4 . 5 4 . 5 Xn

Ym

The joint distribution of the random vector X is given by FX (X) ¼ P½X x ¼ P½X1 x1 , . . . , Xn xn

(17:1:2)

with FX(1) ¼ 1 and FX(21) ¼ 0. The corresponding joint density function fX(x) can be deﬁned as P(x , X x þ DxÞ DX!0 Dx

fX (x) ¼ lim ¼ lim

Dx1 !0 ... Dxn !0

Pðx1 , X1 x1 þ Dx1 , . . . , xn , Xn xn þ Dxn Þ Dx1 Dxn

(17:1:3)

which is the nth partial derivative of the distribution function FX(x): fX (x) ¼

@n FX (x) @x1 @xn

(17:1:4)

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

311

312

Random Vectors and Mean-Square Estimation

Given the density function fX(x), the distribution function FX(x) can be obtained by an n-fold integration: ð xn ð x1 fX (x0 )dx01 dx0n (17:1:5) FX (x) ¼ 1

1

T Given an n-vector X ¼ X1 Xn T , the joint distribution of X and Y is given by

Joint Distribution of Two Vectors X and Y

and an m-vector Y ¼ Y1 Ym

FXY (x,y) ¼ PðX x, Y yÞ

(17:1:6)

and the joint density, by Pðx , X x þ Dx, y , Y y þ DyÞ Dx Dy

fXY (x,y) ¼ lim

Dx!0 Dy!0

x1 , X1 x1 þ Dx1 , . . . , xn , Xn xn þ Dxn

!

P ¼ lim

Dx1 !0

.. .

y1 , Y1 y1 þ Dy1 , . . . , ym , Ym ym þ Dym Dx1 Dxn Dy1 Dym

Dxn !0

.. .

Dy1 !0

.. .

Dym !0

or fXY (x,y) ¼

@n @m FXY (x,y) @x1 @xn @y1 @ym

(17:1:7)

The distribution function FXY(x,y) can also be obtained by integrating Eq. (17.1.7): ðx ðy fxy (x0 ,y0 )dy0 dx0 (17:1:8) FXY (x,y) ¼ 1

1

Since Eq. (17.1.8) is a distribution function, we infer the following relationships: FXY (x,1) ¼ FXY (1, y) ¼ 0 FXY (x,þ1) ¼ FX (x), marginal distribution of X FXY (þ1,y) ¼ FY (y), marginal distribution of Y

(17:1:9)

FXY (þ1,þ1) ¼ 1 The marginal density functions are obtained from Eq. (17.1.7) by integration as follows: ð1 fX (x) ¼

ð1 fXY (x,y)dy1 dym

1 ð1

fY (y) ¼

1 ð1

1

(17:1:10) fXY (x,y)dx1 dxn

1

17.1

Distributions and Densities

313

T Example 17.1.1 Thejoint density T function of a random 2-vector X ¼ X1 X2 and a random 2-vector Y ¼ Y1 Y2 is given by 8 < 0 , x1 1, 0 , x2 2 1 fXY (x,y) ¼ (x1 þ 2x2 )(4y1 þ y2 ) , 1 , y1 2, 1 , y2 3 : 80 0 otherwise We have to determine (1) marginal density fX(x), (2) marginal density fY(y), (3) marginal distribution FX(x), (4) marginal distribution FY(y), (5) the distribution FX(x1), and (6) the distribution FX(x2). 1. The marginal density fX(x) is obtained by integrating fXY(x,y) over all y: ð3 ð2 1 1 fX (x) ¼ ½(x1 þ 2x2 )(4y1 þ y2 )dy1 dy2 ¼ (x1 þ 2x2 ) 80 5 1 1 2. The marginal density fY(y) is obtained by integrating fXY(x,y) over all x: ð2 ð1 1 1 fY (y) ¼ ½(x1 þ 2x2 )(4y1 þ y2 )dx1 dx2 ¼ (4y1 þ y2 ) 16 0 0 80 3. The marginal distribution FX(x) is obtained by integrating fX(x) as shown below: ð x2 ð x 1 1 1 2 FX (x) ¼ (j1 þ 2j2 )dj1 dj2 ¼ x1 x2 þ 2x1 x22 10 0 0 5 4. The marginal distribution FY(y) is obtained by integrating fY(y) as shown below: ð y2 ð y1 1 FY (y) ¼ (4h1 þ h2 )dh1 dh2 16 1 1 1 4( y2 1Þy21 þ ( y22 1)y1 y22 4y2 þ 5 ¼ 32 5. The distribution FX(x1) is obtained by substituting the end value x2 ¼ 2 in FX(x): 1 2 1 2 FX (x1 ) ¼ ¼ x21 þ 4x1 x1 x2 þ 2x1 x2 10 5 x2 ¼2 6. The distribution FX(x2) is obtained by substituting the end value x1 ¼ 1 in FX(x): 1 2 1 2 2 FX (x2 ) ¼ x x2 þ 2x1 x2 2x2 þ x2 ¼ 10 1 10 x1 ¼1 The random vectors X and Y are independent since fXY(x,y) ¼ fX(x) . fY(y), but the random variables X1 and X2 are not independent since FX(x1, x2) = FX(x1) . FX(x2).

Conditional Distributions and Densities

The conditional distribution function FXjA (xjA) given that an event A has occurred is deﬁned by FX j A (x j A) ¼ P(X x j A) ¼

P(X x, A) , P(A) = 0 P(A)

(17:1:11)

314

Random Vectors and Mean-Square Estimation

Analogous to the one-dimensional case, we can also deﬁne a total probability. If there are disjoint events fAi, i ¼ 1, . . . , ng such that n [

Ai ¼ S, and

\

Ai Aj ¼ 1

i=j

i¼0

then the total probability is given by FX (x) ¼ P(X x) ¼

n X

P(X x j Ai )P(Ai ) ¼

i¼1

n X

FX j Ai (x j Ai )P(Ai )

(17:1:12)

i¼1

The density functions corresponding to Eqs. (17.1.11) and (17.1.12) are given by @n FX j A (x j A) , P(A) = 0 @x1 @xn n X fX j Ai (x j Ai )P(Ai ) fX (x) ¼

fX j A (x j A) ¼

(17:1:13) (17:1:14)

i¼1

If there are events A and B deﬁned by A: x , X x þ Dx and B : y , Y y þ Dy, then the probability of B occurring conditioned on A is given by P(B j A) ¼ P(y , Y y þ Dy j x , X x þ Dx) ¼ fY j X (y j X)Dy ¼

P(y , Y y þ Dy, x , X x þ Dx) P(x , X x þ Dx)

¼

fXY (x,y)Dx Dy fXY (x,y)Dy ¼ fX (x)Dx fX (x)

(17:1:15)

From Eq. (17.1.15) we can conclude that the conditional density fY j X (y j X) is given by fY j X (y j X) ¼

fXY (x,y) fX (x)

(17:1:16)

The corresponding expression for the conditional distribution FY j X (Y j X x) is FY j X (y j X x) ¼

P(Y y, X x) FYX (y,x) ¼ P(X x) FX (x)

(17:1:17)

Example 17.1.2 In Example 17.1.1 we established that the random variables X and Y are independent and the random variables X1 and X2 are not independent. We will now ﬁnd the conditional density fX2 j X1 (x2 j X1 ¼ x1 ). By Eq. (17.1.16), we have fX2 j X1 (x2 j X1 ¼ x1 ) ¼

fX2 X1 (x2 , x1 ) fX1 (x1 )

(x1 þ 2x2 ) x1 þ 2x2 5 ¼ ¼ 2(x1 þ 2) 2(x1 þ 2) 5 and the corresponding conditional distribution FX2 j X1 (x2 j X1 ¼ x1 ) is obtained by integrating the conditional density and is given by ð x2 x1 þ 2j2 x2 (x1 þ x2 ) dj2 ¼ FX2 j X1 (x2 j X1 ¼ x1 ) ¼ 2(x þ 2) 2(x1 þ 2) 1 0

17.1

Distributions and Densities

315

Note that when the end value of x2 ¼ 2 is substituted in the distribution shown above, we obtain FX2 j X1 (2 j X1 ¼ x1 ) ¼ 1, as we should. We can also ﬁnd the conditional distribution FX2 j X1 (x2 j X1 x1 ) as follows: FX2 j X1 (x2 j X1 x1 ) ¼

P(X2 x2 , X1 x1 ) FX2 X1 (x2 , x1 ) ¼ P(X1 x1 ) FX1 (x1 )

1 2 (x1 x2 þ 2x1 x22 ) x (x þ 2x ) 2 1 2 10 ¼ ¼ 1 2 2(x1 þ 4) (x þ 4x1 ) 5 1 Here also if the end value of x2 ¼ 2 is substituted in the preceding distribution, we obtain FX2 j X1 (2 j X1 x1 ) ¼ 1, as we should. By differentiating the preceding equation with respect to x2, we can ﬁnd the corresponding conditional density function fX2 j X1 (x2 j X1 x1 ). fX2 j X1 (x2 j X1 x1 ) ¼

d x2 (x1 þ 2x2 ) x1 þ 4x2 ¼ dx2 2(x1 þ 4) 2(x1 þ 4)

The difference between these two conditional distributions is to be particularly noted. In the ﬁrst case the conditioning event is fX1 ¼ x1g, and in the second case it is fX1 x1g.

Points of Clarity

In Eq. (8.2.7) we noted some points of clarity regarding conditional densities and distributions. We will elaborate on this point. The joint density function fXY(x,y) is deﬁned in Eq. (9.2.3) but can also be deﬁned as an inﬁnitesimal probability fXY (x,y)Dx Dy ¼ P(x , X x þ Dx, y , Y y þ Dy)

(17:1:18)

and the corresponding distribution function FXY(x,y), by FXY (x,y) ¼ P(X x, Y y) which can also be obtained by integrating fXY(x,y) over x and y: ðx ðy FXY (x,y) ¼ fXY (j,h)dh dj

(9:2:2)

(17:1:19)

1 1

The function fXY(x,y) can also be obtained by differentiating FXY(x,y) as given in Eq. (9.2.4): fXY (x,y) ¼

@2 FXY (x,y) @x @y

(9:2:4)

There is a need for deﬁning another density – distribution function fXY (X x,y) that can be used in formulating directly conditional density problems involving the conditioning event such as fX x): fXY (X x,y) ¼ lim

Dy!0

P(X x,y , Y y þ Dy) Dy

(17:1:20)

316

Random Vectors and Mean-Square Estimation

Equation (17.1.20) can be obtained by integrating fXY(x,y) over x as shown below: ðx fXY (X x,y) ¼ fXY (j,y)dj (17:1:21) 1

We can integrate Eq. (17.1.21) over y and obtain FXY (x,y): ðy FXY (x,y) ¼ fXY (X x,h)dh

(17:1:22)

1

The conditonal density fY j X (y j X x) can be determined as follows: fY j X (y j X x)Dy ¼ P(y , Y y þ Dy) j X x) ¼

P(y , Y y þ Dy, X x) P(X x)

¼

fYX (y, X x)Dy FX (x)

Hence fY j X (y j X x) ¼

fYX (y, X x) FX (x)

(17:1:23)

Example 17.1.3 We will continue the previous example (Example 17.1.2) and ﬁnd the conditional density fX2 j X1 (x2 j X1 x1 ). From Eq. (17.1.23) we have fX2 j X1 (x2 j X1 x1 ) ¼

fX2 X1 (x2 , X1 x1 ) FX1 (x1 )

From Eq. (17.1.21) we have ð x1 j1 þ 2x2 x1 (x1 þ 4x2 ) dj1 ¼ fX2 X1 (x2 ,X1 x1 ) ¼ 5 10 0

and

FX1 (x1 ) ¼

x21 þ 4x1 5

Hence x1 (x1 þ 4x2 ) x1 þ 4x2 fX2 j X1 (x2 j X1 x1 ) ¼ 2 10 ¼ 2(x1 þ 4) x1 þ 4x1 5 By integrating fX2 j X1 (x2 j X1 x1 ) with respect to x2, we obtain the conditional distribution FX2 j X1 (x2 j X1 x1 ) ¼

x2 (x1 þ 2x2 ) 2(x1 þ 4)

Bayes’ Theorem

Similar to the one-dimensional case, we can formulate a vector form of Bayes’ theorem given that the event Ai has occurred. We want to ﬁnd the a posteriori probability P(Ai j X x) given that the event fX xg has occurred. This probability is given by P(Ai j X x) ¼

P(X x, Ai ) P(X x j Ai )P(Ai ) ¼ P(X x) P(X x)

17.1

Distributions and Densities

317

Substituting for P(X x) from Eq. (17.1.12), we obtain the vector form of Bayes’ theorem: P(X x j Ai )P(Ai ) P(Ai j X x) ¼ Pn k¼1 P(X x j Ak )P(Ak )

(17:1:24)

We can also formulate Bayes’ rule for random vector densities. If fXY(x,y) represents the joint density associated with random vectors X and Y, then, from Eq. (17.1.16), we obtain fXY (x,y) ¼ fY j X (y j X ¼ x)fX (x) and Bayes’ theorem can be formulated for fX j Y(x j Y ¼ y) as follows: fX j Y (x j Y ¼ y) ¼

fY j X (y j X ¼ x)fX (x) fY (y) fY j X (y j X ¼ x)fX (x) 0 0 0 1 fY j X (y j X ¼ x )fX (x )dx

¼ Ð1

(17:1:25)

In many engineering estimation problems the random vector X cannot be observed but the random vector Y connected to X can be observed. In this connection fX(x) is called the a priori density and fX j Y(x j Y ¼ y) is called the a posteriori density after observing fY ¼ yg.

The density fX(x) of a two-dimensional random vector X is given by

Example 17.1.4

fX (x) ¼

4x1 þ 7x2 , 0 , x1 1, 0 , x2 2 18

The conditional density fY j X(y j x) of another two-dimensional random vector Y is fY j X (y j x) ¼

4x1 y1 þ 2x1 y2 þ 5x2 y1 þ 5x2 y2 , 5(4x1 þ 7x2 )

0 , x1 1, 0 , x2 2 1 , y1 2, 1 , y2 3

Given this information, we have to ﬁnd the a posteriori density fX j Y(x j y) using Bayes’ theorem of Eq. (17.1.24). The joint density function fYX(x,y) is given by fYX (y,x) ¼ fY j X (y j x)fX (x) 4x1 y1 þ 2x1 y2 þ 5x2 y1 þ 5x2 y2 4x1 þ 7x2 5(4x1 þ 7x2 ) 18 0 , x1 1, 0 , x2 2 1 ¼ (4x1 y1 þ 2x1 y2 þ 5x2 y1 þ 5x2 y2 ), 90 1 , y1 2, 1 , y2 3 ¼

The marginal density function fY(y) is obtained by integrating the preceding equation with respect to x:

1 7y1 þ 6y2 4x1 y1 þ 2x1 y2 þ 5x2 y1 þ 5x2 y2 dx2 dx1 ¼ 45 0 90

ð1 ð2 fY (y) ¼ 0

318

Random Vectors and Mean-Square Estimation

The a posteriori density can be obtained from Eq. (17.1.25) as fX j Y (x j y) ¼

fY j X (y j x)fX (x) fY (y)

4x1 y1 þ 2x1 y2 þ 5x2 y1 þ 5x2 y2 4x1 þ 7x2 5(4x1 þ 7x2 ) 18 ¼ 7y1 þ 6y2 45 0 , x1 , 1, 0 , x2 2 4x1 y1 þ 2x1 y2 þ 5x2 y1 þ 5x2 y2 , ¼ 2(7y1 þ 6y2 ) 1 , y1 2, 1 , y2 3 which, when integrated over all x, indeed gives the result 1. Example 17.1.5

The density fX(x) of a two-dimensional vector X is given by fX (x) ¼ a1 d(x1 m1 , x2 m2 ) þ a2 d(x1 m1 , x2 m2 )

where a1 þ a2 ¼ 1, d(x1,x2) is a two-dimensional Dirac delta function and m1,m2 are constants. The conditional density of a two-dimensional random vector Y correlated with X is 1 (y1 x1 )2 þ (y2 x2 )2 e fY j X (y j x) ¼ p From Eq. (17.1.6) the joint density function fYX(x,y) is given by fYX (y,x) ¼ fY j X (y j x)fX (x)

1 (y1 x1 )2 þ(y2 x2 )2 a1 d(x1 m1 , x2 m2 ) þ a2 d(x1 m1 , x2 m2 ) ¼ e p

1 (y1 m1 )2 þ(y2 m2 )2 a1 d(x1 m1 , x2 m2 ) þ a2 d(x1 m1 , x2 m2 ) e ¼ p The marginal density function fY(y) is obtained by integrating this equation with respect to x: ð1 ð1 a1 d(x1 m1 , x2 m2 ) 1 (y1 m1 )2 þ(y2 m2 )2 fY (y) ¼ e dx1 dx2 þa2 d(x1 m1 , x2 m2 ) 1 1 p 1 (y1 m1 )2 þ(y2 m2 )2 (y1 m1 )2 þ(y2 m2 )2 a1 e ¼ þ a2 e p The a posteriori density can be obtained from Eq. (17.1.25) as fy j x (y j x)fX (x) fY (y)

(y1 m1 )2 þ(y2 m2 )2 a1 d(x1 m1 , x2 m2 ) þ a2 d(x1 m1 , x2 m2 ) e ¼ 2 2 2 2 a1 e (y1 m1 ) þ(y2 m2 ) þ a2 e (y1 m1 ) þ(y2 m2 )

fx j y (x j y) ¼

¼ b1 (y)d(x1 m1 , x2 m2 ) þ b2 (y)d(x1 m1 , x2 m2 ) where 2

2

ai e½(y1 m1 ) þ(y2 m2 ) , bi (y) ¼ 2 2 2 2 a1 e (y1 m1 ) þ(y2 m2 ) þ a2 e (y1 m1 ) þ(y2 m2 )

i ¼ 1,2

17.2

B 17.2

Moments of Random Vectors

319

MOMENTS OF RANDOM VECTORS

Expectation Vector

Analogous to the single-dimensional case, the expected value of a random n-vector X is another n-vector mX deﬁned by ð1 ð1 x f X (x)dx1 dxn (17:2:1) E½X ¼ mX ¼ 1

1

and ð1 E½Xi ¼ mXi ¼ ð1

ð1 ...

1

xi f X (x)dx1 dxi dxn 1

xi f Xi (xi )dxi

¼

(17:2:2)

1

since ð1

ð1 fX (x)dx1 dxi1 dxiþ1 dxn ¼ 1

1

1

Hence the mean value n-vector mX can be written as T mX ¼ mX1 mXi mXn

(17:2:3)

The term X is a 3-vector whose pdf fX(x) is given by 2 2x 3 2e 1 4 fX (x) ¼ 3e3x25, x1 ,x2 ,x3 . 0 4e4x3

Example 17.2.1

The mean value vector mX is: 2 3 1 2 3 627 E(X1 ) 6 7 617 mX ¼ 4E(X2 )5 ¼ 6 7 637 E(X3 ) 415 4 Correlation Matrix

The correlation matrix is a square matrix RX representing the set of all second moments of the components of a real random vector X of the form ð1 xi xj fXi Xj (xi , xj )dxi dxj , i = j, i, j ¼ 1, . . . , n (17:2:4) E½Xi X j ¼ 1

and is deﬁned by 2

E½X12

E½X1 X2

6 6 E½X2 X1 E½X22 E½XX ¼ RX ¼ 6 6 .. .. 4 . . E½Xn X1 E½Xn X2 T

E½X1 Xn

3

7 E½X2 Xn 7 7 7 .. 5 . 2 E½Xn

(17:2:5)

320

Random Vectors and Mean-Square Estimation

The quantity E[XX T] is deﬁned as the outer product of the vector X, which is equal to the P correlation matrix RX. The inner product RX is deﬁned by E½XT X ¼ ni¼1 E½Xi2 , which is equal to the trace of RX or E[X TX] ¼ Tr[RX]. The expected value of a scalar function g(X) of a vector random variable X can be deﬁned as follows: ð1 ð1 ð1 g(X)fX (x)dx ¼ g(X)fX (x)dx1 dxn (17:2:6) E½g(X) ¼ 1

1

1

Example 17.2.2 The correlation matrix of Example 17.2.1 can be obtained from Eq. 17.2.4 assuming that X1, X2,X3 are independent and hence the joint densities fXi Xj (xi , xj ) are given by the product of individual densities as shown below: fXi Xj (xi , xj ) ¼ fXi (xi )fXj (xj ), i = j, i, j ¼ 1,2,3 or fX1 X2 (x1 , x2 ) ¼ 6e(2x1 þ3x2 ) u(x1 )u(x2 ) fX2 X3 (x2 , x3 ) ¼ 12e(3x2 þ4x3 ) u(x2 )u(x3 ) fX1 X3 (x1 , x3 ) ¼ 8e(2x1 þ4x3 ) u(x1 )u(x3 ) Hence RX is given by 2

E½X12 6 RX ¼ 4 E½X2 X1 E½X3 X1

E½X1 X2 E½X22 E½X3 X2

2

1 6 E½X1 X3 62 7 61 ¼ E½X2 X3 5 6 66 41 E½X32 8 3

3 1 1 6 8 7 7 2 1 7 7 9 12 7 1 1 5 12 8

The trace of the matrix Tr(Rx Þ ¼ 12 þ 29 þ 18 ¼ 61 72. We note that the correlation matrix is symmetric. Covariance Matrix

The covariance matrix CX associated with a real random value of the outer product of the vector (X 2 mX) and is given 2 2 sX1 sX1 X2 sX1 Xj 6 6 sX2 X1 s2X2 sX2 Xj 6 6 . .. .. 6 .. . . T CX ¼ E½X mX ½X mX ¼ 6 6s 6 Xi X1 sXi X2 sXi Xj 6 .. .. 6 .. 4 . . . sXn X1 sXn X2 sXn Xj

vector X is the expected by 3 sX 1 X n 7 sX 2 X n 7 7 .. 7 . 7 7 (17:2:7) sXi Xn 7 7 7 .. 7 . 5 s2Xn

where sXi Xj is the covariance between the random variables Xi and Xj given by sXi Xj ¼ E(Xi mXi )(Xj mXj )

(17:2:8)

Equation (17.2.7) can be expanded to express the covariance matrix CX in terms of the correlation matrix RX: CX ¼ E XXT E X mTX mX E XT þ mX mTX ¼ E XXT mX mTX ¼ RX mX mTX (17:2:9)

17.2

Moments of Random Vectors

321

Example 17.2.3 We will ﬁnd the covariance matrix for Examples 17.2.1 and 17.2.2. The correlation matrix RX has already been found. The matrix mX mTX is given by 2 3 1 2 3 E(X1 ) 6 2 7 6 7 7 6 617 1 1 1 6 7 T mX mX ¼ 6 E(X2 ) 7 E(X1 ) E(X2 ) E(X3 ) ¼ 6 7 4 5 637 2 3 4 415 E(X3 ) 4 2 3 1 1 1 64 6 8 7 6 7 1 7 61 1 ¼6 7 6 6 9 12 7 41 1 1 5 8 12 16 From Eq. (17.2.9) the covariance matrix can be given by 2 3 2 1 1 1 1 1 62 6 6 7 8 7 64 6 61 2 1 7 61 1 6 T CX ¼ RX mX mX ¼ 6 76 6 6 9 12 7 6 6 9 41 1 1 5 41 1 8 12 8 8 12

3 2 1 1 6 7 8 7 64 1 7 6 7 ¼ 60 12 7 6 1 5 4 0 16

0 1 9 0

0

3

7 7 7 0 7 7 1 5 16

Since the cross-terms are zero, the random variables X1,X2, and X3 are pairwise uncorrelated. Joint Moments of Two Random Vectors

Given a random n-vector X and m-vector Y, with joint density fXY(x,y), the expectation of a scalar function g(X,Y) of the random vectors X,Y is deﬁned by ð1 ð1 g(x,y)fXY (x,y)dx dy E½g(X,Y) ¼ 1

1

1

1

ð1 ð1

g(x,y)fXY (x,y)dx1 dxn dy1 dym

¼

(17:2:10)

The cross-correlation matrix RXY represents the set of all joint moments of the components of real random vectors X and Y of the form ð1 xi yj fXi Yj (xi ,yj )dxi dyj , i ¼ 1, . . . , n, j ¼ 1, . . . ,m (17:2:11) E½Xi Yj ¼ 1

and is deﬁned by the outer product E[XY T] as follows: 2 3 E½X1 Y1 E½X1 Y2 E½X1 Ym 6 7 6 E½X2 Y1 E½X2 Y2 E½X2 Ym 7 6 7 E½XYT ¼ RXY ¼ 6 7 .. .. .. 6 7 . . . 4 5 E½Xn Y1 E½Xn Y2 E½Xn Ym

(17:2:12)

It is not a square matrix unless mP¼ n. The inner product E[X TY] is not deﬁned unless m ¼ n, in which case E½XT Y ¼ ni¼1 E½Xi Yi .

322

Random Vectors and Mean-Square Estimation

Cross-Covariance Matrix

The cross-covariance matrix CXY associated with random n-vector X and a random m-vector Y is expected value of the outer product (X 2 mX) and (Y 2 mY) and is given by 2

CXY

sX 1 Y1 6 sX 2 Y1 6 6 . 6 .. 6 T ¼ E½X mX ½Y mY ¼ 6 6 sXi Y1 6 6 .. 4 . s X n Y1

sX 1 Y2 sX 2 Y2 .. . sX i Y2 .. . sX n Y2

sX1 Yj sX2 Yj .. . s X i Yj .. . sXn Yj

3 sX 1 Ym sX 2 Ym 7 7 .. 7 . 7 7 (17:2:13) sXi Ym 7 7 7 .. 7 . 5 sX n Ym

Again CXY is not a square matrix unless m ¼ n. If m ¼ n then Tr½CXY ¼ Sni¼1 E½Xi Yi . Similar to Eq. (17.2.9), Eq. (17.2.13) can be expanded to express the cross-covariance matrix CXY in terms of the cross-correlation matrix RXY: CXY ¼ E XYT E X mTY mX E YT þ mX mTY ¼ E XYT mX mTY ¼ RXY mX mTY

(17:2:14)

Example 17.2.4 We will now ﬁnd the cross-covariance matrix of the two-dimensional vectors X and Y given the joint density

fXY (x,y) ¼

1 (4x1 y1 þ 2x1 y2 þ 5x2 y1 þ 5x2 y2 ), 90

0 , x1 1, 0 , x2 2 1 , y1 2, 1 , y2 3

We will ﬁrst ﬁnd the four densities, fX1 (x1 ), fX2 (x2 ), fY1 (y1 ), fY2 (y2 ) as follows: ð2 ð2 ð3 1

1 4x1 þ 7 (4x1 y1 þ 2x1 y2 þ 5x2 y1 þ 5x2 y2 )dy2 dy1 dx2 ¼ 9 1 90

1

1 7x2 þ 2 (4x1 y1 þ 2x1 y2 þ 5x2 y1 þ 5x2 y2 )dy2 dy1 dx1 ¼ 90 18 1

0

1 14y1 þ 24 (4x1 y1 þ 2x1 y2 þ 5x2 y1 þ 5x2 y2 )dy2 dx2 dx1 ¼ 90 45 1

0

1 4y1 þ 7 (4x1 y1 þ 2x1 y2 þ 5x2 y1 þ 5x2 y2 )dy1 dx2 dx1 ¼ 90 30 1

fX1 (x1 ) ¼ 0

ð1 ð2 ð3 fX2 (x2 ) ¼ 0

ð1 ð2 ð3 fY1 (y1 ) ¼ 0

ð1 ð2 ð2 fY2 (y2 ) ¼ 0

The mean vectors mX,mY and the outer product mXmTY can be obtained from the densities as follows: 2

3 29 6 54 7 7 mX ¼ 6 4 5 5, 27

2

3 134 6 45 7 7 mY ¼ 6 4 119 5, 180

2

3 1943 3451 6 1215 9720 7 7 mX mTY ¼ 6 4 134 119 5 243 972

17.3

Vector Gaussian Random Variables

323

To ﬁnd the cross-correlation matrix RXY, we need the joint densities fX1 Y1 (x1 ,y1 ), fX1 Y2 (x1 ,y2 ), fX2 Y1 (x2 ,y1 ), fX2 Y2 (x2 ,y2 ). They can be determined from fXY(x,y) and are shown below: 8x1 y1 þ 8x1 þ 10y1 þ 20 45 4x1 y2 þ 12x1 þ 10y2 þ 15 fX1 Y2 (x1 ,y2 ) ¼ 90 5x2 y1 þ 10x2 þ 2y1 þ 2 fX2 Y1 (x2 ,y1 ) ¼ 45 10x2 y2 þ 15x2 þ 2y2 þ 6 fX2 Y2 (x2 ,y2 ) ¼ 180 fX1 Y1 (x1 ,y1 ) ¼

We can now ﬁnd the various joint expectations as follows: 332 405 778 E½X2 Y1 ¼ 405 E½X1 Y1 ¼

454 405 , 1066 E½X2 Y2 ¼ 405

2

E½X1 Y2 ¼

hence

RXY

3 454 405 7 7 1066 5 405

332 6 405 ¼6 4 778 405

From Eq. (17.2.14) we can evaluate CXY ¼ RXY mX mTY : 3 2 332 454 1943 6 405 405 7 6 1215 7 6 ¼6 4 778 1066 5 4 134 243 405 405 1:516 0:766 ¼ 1:370 2:510 2

CXY

3 2 163,963 3451 6 108,135 9720 7 7¼6 119 5 4 1664 972 1215

3 1489 1944 7 7 12,197 5 4860

Deﬁnitions 1. Two random vectors X and Y are independent if fXY(x,y) ¼ fX(x) fY(y). 2. Two random vectors X and Y are uncorrelated if RXY ¼ mXmTY, or CXY ¼ 0. 3. Two random vectors X and Y are orthogonal if RXY ¼ 0. The deﬁnitions and properties of correlation and covariance matrices for two real random vectors X and Y are shown in Table 17.2.1.

B 17.3

VECTOR GAUSSIAN RANDOM VARIABLES

Gaussian random variables play an important role in signal processing and communication problems. For a real n-vector random variable, the multivariate Gaussian density takes the form fX (X) ¼

1 e(1=2) (x mx )T C1 X (x mx ) (2p)n=2 j CX j1=2

(17:3:1)

324

Random Vectors and Mean-Square Estimation

TABLE 17.2.1 Property

Correlation functions

Covariance functions

Autocorrelation and Autocovariance RX ¼ E[XX T] RX ¼ RTX Positive deﬁnite Greater than 0 Orthogonal Greater than 0 RX ¼ CX þ mXmTX

Deﬁnition Symmetry Deﬁniteness Eigenvalues Eigenvectors Principal diagonals Relationship

CX ¼ E[(X 2 mX) (X 2 mX)T] CX ¼ CTX Positive deﬁnite Greater than 0 Orthogonal Greater than 0 CX ¼ RX 2 mXmTX

Cross-Correlation and Cross-Covariance RXY ¼ E[XY T] RXY ¼ RTYX Nondeﬁnite RXY ¼ CXY þ mXmTY RXY ¼ mXmTY RXY ¼ 0 ¼ RYX RXþY ¼ RX þ RXY þ RYX þ RY RXþY ¼ RX þ RY if X, Y are orthogonal

Deﬁnition Symmetry Deﬁniteness Relationship Uncorrelated Orthogonal Sum of X and Y

CXY ¼ E[(X 2 mX) (Y 2 mY)T] CXY ¼ CTYX Nondeﬁnite CXY ¼ RXY 2 mXmTY CXY ¼ 0 ¼ CYX CXþY ¼ CX þ CXY þ CYX þ CY CXþY ¼ CX þ CY if X, Y are uncorrelated

The joint density of two random variables X and Y is given by a bivariate Gaussian as in Eq. (10.5.1) ( "

1 1 x mx 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp fXY (x,y) ¼ sx 2(1 r2 ) 2psx sy 1 r2

#) (x mx )(y my ) y my 2 þ 2r sy s x sy

(10:5:1)

where r is the correlation coefﬁcient between random variables X and Y given by r ¼ sxy =(sx sy ). This can also be expressed in the matrix form of Eq. (17.3.1) with n ¼ 2 ( 1 1 fXY (x,y) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp ½x mx 2 2p jCX j

s2 y my X rsX sY

rsX sY s2Y

1

x mx y my

)!

(17:3:2) where the covariance matrix CX, its inverse CX 21, and the mean vector m are given by " CX ¼

s2X rsX sY

rsX sY s2Y

2

# ,

3 1 r 6 s2X (1 r2 ) sX sY (1 r2 ) 7 6 7, C1 X ¼4 5 r 1 2 2 2 sX sY (1 r ) sY (1 r )

mX m¼ mY

(17:3:3)

17.3

Vector Gaussian Random Variables

325

2 In a similar fashion, when n ¼ 1, CX ¼ s2X, C21 X ¼ 1/sX, r ¼ 0, and the density for a single-dimensional random variable X from Eq. (17.3.1) is 2 1 fX (x) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e(1=2)½(xmX )=sX 2psX

(17:3:4)

a result that has already been stated in Eq. (6.4.1). If random n-vector X and m-vector Y are Gaussian-distributed, then the joint density fXY(x,y) can be obtained by forming an (n þ m) vector

X Z¼ Y

(17:3:5)

with the corresponding mean vector and the covariance matrix given by mZ ¼ E

X

¼

mX

mY CX X mX (X mX )T (Y mY )T ¼ CZ ¼ E Y mY CYX Y

CXY

(17:3:6)

CY

Since CZ is block symmetric, we note that CTYX ¼ CXY. The joint density fZ(z) ¼ fXY(x,y) can be obtained from Eq. (17.3.1) as fXY (x,y) ¼

1 (2p)

ðnþmÞ=2

j CZ

j1=2

e(1=2)(zmZ )

T

C1 Z (zmZ )

(17:3:7)

where the determinant jCZj is obtained from Eq. (16.4.3) jCZ j ¼ jCX j j CY CYX C1 X CXY j

(17:3:8)

and the inverse C1 Z , from Eq. (16.4.6) CXY 1 CX CYX CY 2 1 1 1 CX þ C1 X CXY DCY CYX CX 4 ¼ 1 D1 CY CYX CX

C1 Z ¼

1 C1 X CXY DCY

D1 CY

3 5

(17:3:9)

where the inverse of the Schur component D1 CY of CY is obtained from Eq. (16.4.7): 1 1 D1 CY ¼ (CY CYX CX CXY )

It can be seen from Eq. (17.3.7) that when the densities fX(x) and fY(y) are jointly Gaussian, the density fXY(x,y) is also jointly Gaussian.

326

Random Vectors and Mean-Square Estimation

The conditional density fY j X(y j x) is obtained from fY j X(y j x) ¼ fXY(x,y)/ fX(x). Performing the indicated division and after some involved matrix manipulation, we obtain fY j X (y j x) ¼

m=2

(2p)

T 1 1 e(1=2)(ymY j X ) CY j X (ymY j X ) 1=2 j CY j X j

(17:3:10)

1 where mY j X ¼ mY þ CYX C1 X (x mX ) and CY j X ¼ CY CYX CX CXY . We note that the conditional density is also Gaussian.

Example 17.3.1 Two-dimensional zero mean Gaussian random vectors X and Y have covariance matrices equal to

1 CX ¼ 0

0 9

4 CY ¼ 0

and

0 16

The cross-covariance matrix between X and Y is given by CXY ¼

CX1 Y1

CX1 Y2

CX2 Y1

CX2 Y2

2 6 ¼4

1 18 5

3 6 5 7 ¼ CT YX 12 5 5

We have to ﬁnd the joint density fXY(x,y) and the conditional density fY j X(y j x) using Eqs. (17.3.7) and (17.3.10).

Joint Density

From the data given, we can form the matrix CZ ¼

CX CYX

CXY CY

with

CX ¼

1 0 , 0 9

CY ¼

4 0

0 16

and the cross-covariance matrices CXY and CYX, given by 2

1

6 CXY ¼ 6 4 18 5

3 6 5 7 7 12 5 5

2 and

1

6 CYX ¼ CTXY ¼ 6 46 5

3 18 5 7 7 12 5 5

The pﬃﬃﬃﬃﬃﬃﬃﬃﬃmean vectors mX and mY are zero vectors. From Eq. (17.3.8), jCZj and jCZ j are jCZ j ¼

95904 ¼ 153:446 625

and

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 36 pﬃﬃﬃﬃﬃ jCZ j ¼ 74 ¼ 12:387 25

17.3

Vector Gaussian Random Variables

327

The inverse matrix C1 Z is obtained from Eq. (17.3.9): " C1 Z

¼

1 1 1 C1 X þ CX CXY DCY CYX CX

1 C1 X CXY DCY

1 D1 CY CYX CX

D1 CY

2

#

3 250 50 215 35 6 7 111 222 148 7 6 111 6 7 6 50 275 40 25 7 6 7 6 111 7 999 111 333 6 7 ¼6 7 725 75 7 6 215 40 6 7 6 222 111 888 592 7 6 7 4 35 25 75 325 5 148 333 592 3552 2 3 2:252 0:45 0:968 0:236 6 7 6 7 6 7 6 0:45 7 0:275 0:36 0:075 6 7 6 7 ¼6 7 6 7 6 0:968 0:36 0:816 0:127 7 6 7 6 7 4 5 0:236

0:075

0:127

0:091

With this information, the joint density fXY(x,y) is given by

fXY (x,y) ¼

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (2p) 153:446 8 2 0 2:252 > > > B 1< 6 B 6 0:45 expB ½x j y6 @ 2> 4 0:968 > > : 0:236 2

0:45 0:275 0:36 0:075

91 > > =C 7 > 0:075 7 x C C 7 A 0:127 5 y > > > ; 0:091

0:968 0:236 0:36 0:816 0:127

3

or

fXY (x,y) ¼

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (2p) 153:446 2 3 0:408y21 þ (0:127y2 þ 0:968x1 þ 0:36x2 )y1 6 7 6 7 exp6 0:045y22 þ (0:236x1 þ 0:075x2 )y2 1:126x21 7 4 5 0:45x1 x2 0:1375x22 2

328

Random Vectors and Mean-Square Estimation

Conditional Density. We will now calculate the conditional density fY j X(y j x) from Eq. (17.3.10). The conditional expectation vector mY j X is given by 0 T 1 mY j X ¼ mY þ CXY CX (x mX ) ¼ 0 2 2 3 3 18 2 2x2 3 1 x1 þ 1 0 6 6 x1 5 7 5 7 7 74 þ6 ¼6 15 4 6x 4 6 12 5 x 4x2 5 0 2 1 þ 9 5 15 5 5 The conditional covariance matrix CY j X is obtained as follows: 2 3 2 3 18 2 1 0 61 5 7 6 1 4 0 56 74 CY j X ¼ CY CTXY C1 6 1 X CXY ¼ 4 6 12 5 4 18 0 16 0 9 5 5 5 2 3 39 54 6 25 25 7 7 ¼6 4 54 348 5 25

3 6 5 7 7 12 5 5

25

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 10656 ¼ 17:05 and jCY j X j ¼ 4:129 and The determinant jCY j X j ¼ 625 2 2 3 3 725 75 39 54 1 6 25 6 888 592 7 0:816 0:127 25 7 6 6 7 7 ¼ ¼ C1 ¼ YjX 4 54 348 5 4 75 0:127 0:091 325 5 592 3552 25 25 Substituting these values in Eq. (17.3.10), we obtain 8 0 > 5 2p 17:05 :

0:816 0:127

y1

6x1 4x2 5 15

2 3 91 y1 x1 2x2 > = 0:127 6 C 5 7 A 4 5 6x1 4x2 > 0:091 ; y1 5 15

Or expanding this equation results in the conditional density fY j X(y j x) as follows: fY j X (y j x) ¼

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2p 17:05 " # 0:408y21 þ (0:127y2 þ 0:968x1 þ 0:36x2 )y1 0:045y22 exp þ(0:236x1 þ 0:075x2 )y2 0:626x21 0:45x1 x2 0:082x22

Linear Transformations

In Section 13.5 we discussed the general case of the transformation Z ¼ g(X), where Z and X are random n-vectors and g is a function n-vector. We will now discuss a linear

17.3

Vector Gaussian Random Variables

329

transformation of the form 2

3 2 y1 a11 6 y2 7 6 a21 6 7 6 Y¼6 . 7¼6 . 4 .. 5 4 .. an1 yn

a12 a22 .. . an2

32 3 a1n x1 6 x2 7 a2n 7 76 7 .. 76 .. 7 ¼ AX . 54 . 5 xn ann

(17:3:11)

where X is a Gaussian random n-vector and A is a square nonsingular n-matrix faijg. The density function for X is given in Eq. (17.3.1): fX (x) ¼

T 1 1 e(1=2)(xmX ) CX (xmx ) (2p)n=2 jCX j1=2

(17:3:1)

Using the transformation techniques developed in Chapter 13, the solution for X is given by X ¼ A21Y. The Jacobian determinant kJk ¼ jAj. The density function fY(y) is given by fY (y) ¼

(2p)

n=2

(2p)

n=2

(2p)

n=2

¼ ¼

1 1 T 1 1 eð1=2Þ(A ymx ) CX (A ymx ) 1=2 jAjjCX j

1 ð1=2Þ(yAmX )T (AT CX1 A1 )(yAmX ) e T 1=2 jAjjCX jjA j 1 ð1=2Þ(yAmX )T (ACX AT )1 (yAmX ) e T 1=2 jAjjCX jjA j

(17:3:12)

Taking the expectation and outer product of Eq. (17.3.11), we obtain mY ¼ AmX E½YYT ¼ CY ¼ E½AXXT AT ¼ AE½XXT AT ¼ ACX AT

(17:3:13)

Substituting Eqs. (17.3.13) into Eq. (17.3.12), we can write fY(y) as fY (y) ¼

(2p)

n=2

1 ð1=2Þ(ymY )T C1 Y (ymY ) 1=2 e jCY j

ð17:3:14Þ

Example 17.3.2 Starting with the same example (Example 13.5.1), we will solve for fY(y) using matrix methods. A zero mean Gaussian random 3-vector X has the density function fX(x) fX (x) ¼

1 T 1 e(1=2)x CX x 3=2 (2p)

where 2

1 CX ¼ 4 0 0

0 1 0

3 0 0 5, 1

2

jCX j ¼ 1,

C1 X

1 0 ¼ 40 1 0 0

3 0 05 1

330

Random Vectors and Mean-Square Estimation

The transformation Y ¼ AX is given by 2 3 2 2 y1 Y ¼ 4 y2 5 ¼ 4 5 y3 3

2 3 2

32 3 1 x1 3 54 x2 5 ¼ AX x3 2

Taking expectation on both sides of the preceding mX ¼ 0. The covariance matrix CY is given by 2 32 32 2 2 1 1 0 0 2 T CY ¼ ACX A ¼ 4 5 3 3 54 0 1 0 54 2 3 2 2 0 0 1 1

equation we have mY ¼ 0, since 3 2 5 3 9 19 3 2 5 ¼ 4 19 43 3 2 12 27

3 12 27 5 17

The inverse of CY is 2

C1 Y

2 1 ¼4 1 9 13 15

3 3 15 5 26

and the determinant of CY ¼ 1. Hence fY(y) is obtained from Eq. (17.3.14) 8 2 2 < 1 1 T4 exp y fY (y) ¼ 1 : 2 (2p)3=2 1 13

1 9 15

3 9 3 = 15 5y ; 26

or fY (y) ¼

1 e(1=2) (2p)3=2

2y21 þ9y22 þ26y23 þ2y1 y2 30y2 y3 6y3 y1

which is exactly the same as obtained in Example 13.5.1.

B 17.4 DIAGONALIZATION OF COVARIANCE MATRICES Diagonalization Principles

In Section 16.2 we discussed diagonalization of a nonsingular square matrix by another transformation matrix of eigenvectors called the modal matrix. Since covariance matrices are symmetric and positive deﬁnite, the modal matrix is orthogonal. This property makes diagonalization of covariance matrices somewhat simpler. Diagonalization is needed to make a problem simpler, and correlated Gaussian random variables become independent. Given any positive deﬁnite covariance n-matrix C with distinct eigenvalues, we ﬁnd the eigenvalue n-matrix L by solving the characteristic equation jlI 2 Cj ¼ 0 as outlined in Section 16.2. 2 3 l1 0 0 6 0 l2 0 7 6 7 (17:4:1) L¼6 . .. 7 .. 4 .. . 5 . 0 0 ln

17.4

Diagonalization of Covariance Matrices

331

We ﬁnd the eigenvectors ffi, i ¼ 1, . . . , ng corresponding to each of the eigenvalues fli, i ¼ 1, . . . , ng and form the modal matrix M of eigenvectors: 2 3 f11 f21 fn1 7 6 6 f12 f22 fn2 7 M ¼ f1 f2 fn ¼ 6 . (17:4:2) 7 . . .. .. 5 4 ..

f1n

f2n

Since C is symmetric, M will be orthogonal: 1, i ¼ j and fTi fj ¼ 0, i = j

fnn

MT ¼ M1

(17:4:3)

With these deﬁnitions, the following eigenequation can be formed: CM ¼ ML

or

M1 CM ¼ L

and

MT CM ¼ L

(17:4:4)

The problem now is to ﬁnd a transformation T between two random n-vectors X and Y such that the covariance matrix CX is diagonalized to the covariance matrix CY. We take the expected value and the covariance of Y in the transformation (17:4:5) (17:4:6)

Y ¼ TX mY ¼ TmX Thus E½(Y mY )(Y mY )T ¼ E½T(X mX )(X mX )T TT ,

or CY ¼ TCX TT (17:4:7)

Or, identifying terms in Eqs. (17.4.4) and (17.4.7), we obtain CY L, CX C,

T MT

(17:4:8)

T

with the required transformation matrix T M . Diagonalizing to an Identity Matrix

We can carry the diagonalization process one step further and ﬁnd a transformation matrix. matrix D such that CY is an identityp ﬃﬃﬃﬃT Since pﬃﬃﬃﬃ L is a diagonal matrix, we can take the identity n-matrix. Incorpositive square root and write L ¼ L In L ,where pﬃﬃﬃﬃ pﬃﬃﬃﬃ TIn isan L , we obtain porating this result in Eq. (17.4.4) and noting that L ¼ pﬃﬃﬃﬃ pﬃﬃﬃﬃ pﬃﬃﬃﬃ1 pﬃﬃﬃﬃ1 L In L (17:4:9) MT CM ¼ or L MT CM L ¼ In Thus the required transformation matrix D in the transformation D TCD ¼ In is pﬃﬃﬃﬃ1 (17:4:10) D¼M L The transformation T is orthogonal, whereas D is not. Example 17.4.1 given by

The covariance matrix CX of a zero mean Gaussian random 3-vector is 2

3

6 CX ¼ 4 1 0

1 0

3

7 2 15 1 3

332

Random Vectors and Mean-Square Estimation

We have to determine 1. The joint density fX(x) 2. The correlation coefﬁcient matrix r, given by 2 3 1 rX1 X2 rX1 X3 6 7 1 rX2 X3 7 r¼6 4 rX 2 X 1 5 rX3 X1 rX3 X2 1 3. The modal matrix M and show that it is orthogonal 4. The transformation matrix T in Y ¼ TX that will make CY diagonal 5. The transformation matrix D that will make CX an identity matrix 6. The joint density fY(y) 1. With jCXj ¼ 12 and 2

C1 X

5 1 6 12 4 6 6 6 1 3 ¼6 6 4 4 6 4 1 1 12 4

3 1 12 7 7 7 1 7 7 4 7 7 5 5 12

in the equation fX (x) ¼

T 1 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ e(1=2)X CX X jCX j (2p)

3=2

the joint density is given by fX (x) ¼

1 5x21 x3 x2 x2 x3 3x22 5x23 p ﬃﬃﬃﬃﬃ þ x þ þ exp 1 6 2 2 4 12 2 12 (2p)3=2 12 1

2. From the given covariance matrix CX, we have s2X1 ¼ 3,

s2X2 ¼ 2, s2X3 ¼ 3 and

sX1 X2 ¼ 1, sX1 X2 ¼ 0,

Hence the correlation matrix r is given by 2 3 1 1 pﬃﬃﬃ 0 6 7 6 6 7 6 1 1 7 7 p ﬃﬃ ﬃ p ﬃﬃ ﬃ 1 r¼6 6 6 67 6 7 4 5 1 0 pﬃﬃﬃ 1 6 3. From the determinant of the matrix 2 l3 jlI CX j ¼ 4 1 0

1 l2 1

3 0 1 5 ¼ 0 l3

sX2 X3 ¼ 1

17.4

Diagonalization of Covariance Matrices

333

we obtain the characteristic equation, l3 2 8l2 þ 19l 2 12 ¼ (l 2 1)(l 2 3) (l 2 4) ¼ 0, and the eigenvalues are l1 ¼ 1, l2 ¼ 3, l4 ¼ 4. The eigenvectors corresponding to these eigenvalues are obtained by the methods described in Example 16.2.5 and are shown below with the modal matrix M: 2

1 3 pﬃﬃﬃ 6 67 6 7 6 2 7 6 pﬃﬃﬃ 7 f1 ¼ 6 7, 6 67 6 7 4 1 5 pﬃﬃﬃ 6 2 1 1 pﬃﬃﬃ pﬃﬃﬃ 6 6 2 6 6 2 6 M ¼ 6 pﬃﬃﬃ 0 6 6 6 4 1 1 pﬃﬃﬃ pﬃﬃﬃ 6 2

2

3 1 pﬃﬃﬃ 6 27 6 7 6 7 f2 ¼ 6 0 7 , 6 7 4 1 5 pﬃﬃﬃ 2

2

1 3 pﬃﬃﬃ 6 37 6 7 6 1 7 6 pﬃﬃﬃ 7 f3 ¼ 6 7, 6 37 6 7 4 1 5 pﬃﬃﬃ 3

1 3 pﬃﬃﬃ 37 7 1 7 pﬃﬃﬃ 7 7 37 7 1 5 pﬃﬃﬃ 3

By direct multiplication we can see that fT1 f2 ¼ 0, fT1 f3 ¼ 0, fT2 f3 ¼ 0, and fT1 f1 ¼ fT2 f2 ¼ fT3 f3 ¼ 1, thus showing orthogonality. Further, jMj ¼ 1. 4. From Eq. (14.7.8) the transformation matrix T is 2 1 pﬃﬃﬃ 6 6 6 6 1 T ¼ MT ¼ 6 6 pﬃﬃ2ﬃ 6 4 1 pﬃﬃﬃ 3

2 1 3 pﬃﬃﬃ pﬃﬃﬃ 6 67 7 1 7 0 pﬃﬃﬃ 7 27 7 1 1 5 pﬃﬃﬃ pﬃﬃﬃ 3 3

2

and

1 TCX TT ¼ CY ¼ 4 0 0

0 3 0

3 0 05 4

5. From Eq. (17.4.10) the transformation matrix D that takes CX to an identity matrix I is given by

pﬃﬃﬃﬃ1 D¼M L

2 1 1 1 32 pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 1 0 6 6 37 2 1 6 76 6 2 6 0 pﬃﬃﬃ 1 7 6 7 6 ¼ 6 pﬃﬃﬃ 0 pﬃﬃﬃ 76 3 3 74 6 6 4 1 1 1 5 0 0 pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 3 6 2 2 3 1 1 1 p ﬃﬃ ﬃ p ﬃﬃ ﬃ p ﬃﬃ ﬃ 6 6 6 2 37 6 pﬃﬃﬃ 7 6 7 1 7 2 6 ¼ 6 pﬃﬃﬃ 0 pﬃﬃﬃ 7 6 2 37 3 6 7 4 1 1 1 5 pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 6 6 2 3

0

3

7 0 7 7 7 1 5 pﬃﬃﬃ 4

334

Random Vectors and Mean-Square Estimation

6. Since CY is a diagonal matrix, the random variables Y1, Y2, Y3 are independent, Gaussian, and zero mean. Hence the joint density is fY (y) ¼

1 (2p)

3=2

pﬃﬃﬃﬃﬃ e(1=2)½(y1 =1)þ(y2 =3)þ(y3 =4) 12 2

2

2

B 17.5 SIMULTANEOUS DIAGONALIZATION OF COVARIANCE MATRICES We are given two positive deﬁnite covariance matrices A and B with distinct eigenvalues. In pattern recognition problems it will be very useful to diagonalize both matrices by means of the same transformation matrix T. The transforming matrix T will take A to an identity matrix while at the same time diagonalizing B. We shall now analyze the procedure that will accomplish this. The two covariance matrices A and B with distinct eigenvalues are 1 1 0 0 b11 b12 b1n a11 a12 a1n B a21 a22 a2n C B b21 b22 b2n C C C B B B¼B . (17:5:1) A¼B . C . . .. .. C .. .. A @ .. @ .. . . A an1 an2 ann bn1 bn2 bnn We have seen in Section 16.2 how the modal matrix of eigenvectors F ¼ ff1,f2, . . . , fng of A will transform A to a diagonal matrix of its eigenvalues fl1,l2,l3, . . . , lng as shown: 0 1 l1 0 0 B 0 l2 0 C B C (17:5:2) FT AF ¼ L ¼ B . .. C .. @ .. . A . 0 0 ln From Eq. (17.4.9) we can write pﬃﬃﬃﬃ1 pﬃﬃﬃﬃ pﬃﬃﬃﬃ pﬃﬃﬃﬃ1 or L FT AF L ¼ In L In L FT AF ¼ pﬃﬃﬃﬃ1 By deﬁning U ¼ F L , we can write Eq. (17.5.3) as follows: pﬃﬃﬃﬃ1 pﬃﬃﬃﬃ1 L FT AF L ¼ UT AU ¼ In

(17:5:3)

(17:5:4)

This process is known as “whitening” the covariance matrix A. Since B is a covariance matrix, we can show that C ¼ U TBU is also a symmetric matrix as follows: pﬃﬃﬃﬃ pﬃﬃﬃﬃ1 T pﬃﬃﬃﬃ1 pﬃﬃﬃﬃ1 1 T T T T L F BF L ¼ L F B F L C ¼ ¼

pﬃﬃﬃﬃ1 pﬃﬃﬃﬃ1 L FT BF L ¼ C

(17:5:5)

The symmetric matrix C ¼ U TBU can therefore be diagonalized by its orthogonal modal matrix M, that is, M TCM ¼ M TU TBUM ¼ D ¼ diagfd1,d2, . . . , dng. The

This section is adapted, with permission, from Chapter 31 of the book Linear Systems Properties—a Quick Reference, by Venkatarama Krishnan, published by CRC Press, 1988.

17.5

Simultaneous Diagonalization of Covariance Matrices

335

transformation UM has diagonalized B, and now we will show that UM applied to A retains the whitening transformation. Using Eq. (17.5.4), we can write pﬃﬃﬃﬃ1 pﬃﬃﬃﬃ1 MT UT AUM ¼ MT L FT AF L M ¼ MT In M ¼ MT M (17:5:6)

¼ In T

since M is an orthogonal matrix M M ¼ In. Thus, the required transformation matrix T that takes A to an identity matrix and B to a diagonal matrix is given by pﬃﬃﬃﬃ1 (17:5:7) T ¼ F L M ¼ UM With this transformation, we have TT AT ¼ In , TT BT ¼ D

(17:5:8)

For computational purposes the procedure described above can be simpliﬁed as follows. From Eq. (17.5.8) we can write AT ¼ TT

and

BT ¼ TT D

(17:5:9)

Hence BT ¼ ATD, and we have the generalized eigenequation A1 BT ¼ TD

ð17:5:10Þ

where the matrix T serves as the modal matrix for the product matrix A21B with D as its diagonal eigenvalue matrix. From Eq. (17.5.10) we can now formulate a procedure for ﬁnding the transformation matrix T that diagonalizes simultaneously the covariance matrices A and B. Summary of Procedure for Simultaneous Diagonalization 1. Calculate the n distinct eigenvalues fd1, d2, . . . , dng of the matrix R ¼ A21B from det (lI –R) ¼ 0. 2. Calculate the eigenvectors ti of R from the eigenequation (lI –R)ti ¼ 0. 3. Form the modal matrix N ¼ ft1, t2, . . . , tng. 4. Form the matrix N TAN ¼ W (normalization with respect to A). 5. Since with positive values, step 4 can be rewritten, as pﬃﬃﬃﬃﬃ W is diagonal pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ ( W)1 NT AN( W)1 ¼ I and the desired transformation is T ¼ N( W)1 . Example 17.5.1

Two covariance matrices A and B are 0 1 0 2 1 1 4 A ¼ @ 1 3 1 A, B ¼ @2 1 1 4 2

given by 1 2 2 5 2A 2 6

There is no need to ﬁnd the eigenvalues of A, and for comparison, the eigenvalues of B are 2 3 2:37 0 0 3:52 0 5 LB ¼ 4 0 0 0 9:11

336

Random Vectors and Mean-Square Estimation

1. The matrix R ¼ A – 1B is given by 2

2 R ¼ A1 B ¼ 4 0 0

3 0:176 0:235 1:588 0:118 5 0:059 1:412

2. The characteristic equation of R is (l 2 2)(l2 –3l þ 2.235) ¼ 0, and the eigenvalues are 2

3 0 0 1:621 0 5 0 1:379

2 D ¼ 40 0

3. The modal matrix N of eigenvectors of R is 3 0:525 0:188 0:82 0:481 5 0:23 0:856

2

1 N ¼ 40 0 4. The matrix W ¼ N TAN is given by 2

2 0 W ¼ 4 0 2:053 0 0

3 0 0 5 2:733

and

2 1:414 pﬃﬃﬃﬃﬃ W¼4 0 0

3 0 0 1:433 0 5 0 1:653

pﬃﬃﬃﬃﬃ 5. The desired transfromation T ¼ N W 21 is given by 2 0:707 0:366 pﬃﬃﬃﬃﬃ1 0:572 T¼N W ¼4 0 0 0:161

3 0:114 0:291 5 0:518

Comments 1. The matrix T is not orthogonal since 2

0:707

0

0

3

6 7 0:572 0:616 5 = T1 TT ¼ 4 0:366 0:114 0:291 0:518 2 3 1:414 0:707 0:707 6 7 ¼4 0 1:51 0:848 5 and jTj ¼ 0:243 0 0:161 0:518

17.6

Linear Estimation of Vector Variables

2. We can check the diagonalization process: 2 3 2 1 0 0 2 6 7 6 T T T AT ¼ 4 0 1 0 5, T BT ¼ 4 0 0 0

1

0 1:621

0 0

0

1:379 3

0 2

2:37 6 whereas LB ¼ 4 0 0

0 3:52 0

337

3 7 5

0 7 0 5 9:11

3. The diagonal elements in the transformation of B are the eigenvalues of R and not the eigenvalues LB of B.

B 17.6

LINEAR ESTIMATION OF VECTOR VARIABLES

We will take up more detailed estimation problems in the next chapter. We are interested in estimating the vector X given the equation Y ¼ H X þ N m1 mn n1 m1

(17:6:1)

where Y is an m 1 observation vector, H is an m n system vector, X is an n 1 vector to be estimated, and N is an m 1 noise vector with zero mean and covariance matrix, CN ¼ E[NN T] ¼ s2I. We need a linear estimator of the form ^ ¼ AY X

(17:6:2)

where A is an n m matrix to be determined. We have to choose a scalar criterion ˆ , which can be function J, which is some measure of the estimation error 1 ¼ Y 2 HX ˆ minimized with respect to the vector X. It is reasonable to choose a sum-squared error function given by T ^ ^ ¼ Y HX ^ Y HX (17:6:3) J(X) This is known as the linear least-squares estimation. Using Eq. (16.3.21), we can minimize Eq. (17.6.3) and write T @ ^ ¼ @ ^ ^ J(X) Y HX Y HX ^ ^ @X @X T @ T ^ T HT Y YT HX ^ þX ^ T HT HX ^ Y YX ¼ ^ @X h i ^ ¼0 ¼ 0 ¼ 2HT Y þ 2HT HX ð17:6:4Þ or ^ ¼ HT H 1 HT Y X Thus, the desired n m matrix A in Eq. (17.6.2) is given by 1 A ¼ HT H H T

(17:6:5)

338

Random Vectors and Mean-Square Estimation

The quantity (H TH)21H T is called the pseudoinverse. We have essentially derived the ˆ deterministically without involving the noise covariance CN ¼ s2I. In this estimator X ˆ is not random. We ask the question as to the case the estimation error 1 ¼ Y 2 HX ˆ . If m . n, then we have an overdetermined meaning of this deterministic estimator X system where the number of observations are more than the unknowns and the n n ˆ is the one matrix H TH is invertible. In this case among the many possible solutions, X that gives a minimum mean-square solution. If m ¼ n, then H is a square matrix that may be invertible, in which case we have (H TH)21H T ¼ H 21H 2TH T ¼ H 21 and X ¼ H 21Y giving an exact solution. If m , n, then we have an underdetermined system where the number of observations is less than the number of unknowns and the ˆ can be given only in n n matrix H TH will be singular. The m-vector estimator X terms of the n 2 m unknown quantities of the vector X. ˆ random that estimates the The presence of the noise vector N makes the estimator X deterministic parameter vector X, and the criterion function J is J¼E

T ^ ^ Y HX Y HX

(17:6:6)

ˆ ¼ (H TH)21H TY is in the minimum mean-square error sense. and the estimator X ˆ ¼ (H TH)21H TY in Eq. (17.6.3), we can ﬁnd the least-square By substituting X value as T Jmin ¼ Y HðHT HÞ1 HT Y Y HðHT HÞ1 HT Y ¼ YT I HðHT HÞ1 HT Y

(17:6:7)

since I HðHT HÞ1 HT I HðHT HÞ1 HT ¼ I HðHT HÞ1 HT or the matrix ðI HðHT HÞ1 HT Þ is idempotent. Example 17.6.1

An observation system, Y ¼ HX þ N, is given by 3 2 1 3 4 5 5 ¼ 42 1 10 2

Y

H

2 3 3 1 n1 x 4 5 1 þ 4 n2 5 x2 n3 2 X N

We have to estimate in the least-squares sense the 2-vector X. We also will ﬁnd the noise sequence and the least squared error. From Eq. (17.6.4), we have 2 3 1 1 6 11 1 2 1 6 7 HT H ¼ 42 45 ¼ 11 6 1 4 2 1 2 2 3 2 3 21 11 2 1 2 6 T 1 T 6 5 1 2 1 5 7 5 5 7 7 7 H H H ¼6 ¼6 4 4 11 5 1 4 2 2 1 5 6 1 5 5 5 5

17.6

Linear Estimation of Vector Variables

ˆ is given by Hence the estimator X

^ ¼ HT H X

1

6 HT Y ¼ 11

11 6

1 1

2 4

2 3 3 1 4 5 2 5 ¼ 2 1 10

The error vector N is given by 2

3 2 3 2 3 3 3 0 ^ ¼ 4 5 5 4 8 5 ¼ 4 3 5 N ¼ Y HX 10 4 6 and the least-squared error Jmin is obtained from Eq. (17.6.7) as Jmin ¼ YT I H(HT H)1 HT Y ¼ 45

339

CHAPTER

18

Estimation Theory B 18.1

CRITERIA OF ESTIMATORS

Estimation is a process where we draw inferences about a set, random or otherwise, from knowledge of a suitably selected ﬁnite observation set. This inference is based on the minimization of some criterion function. The collection of elements in the entire set is called the population. The suitably selected observation set is called the sample. Any function of the elements of the sample is called an estimator or a statistic. Generally, estimators are random variables, but they can be constants under some circumstances. The criterion used for estimation in the previous chapter is the least-squares error. There are several other criteria for judging the proximity to the true value and the reasonableness of estimators. We shall enumerate some of these criteria assuming that u is a variable, random or otherwise, to be estimated by u^ from a set of ﬁnite observations: ^ of an estimator (u) ^ is deﬁned by 1. Bias. The bias B(u) ^ ¼ E½u^ u B(u)

(18:1:1)

^ ¼ 0 and the pdf of the estimator is cenAn estimator is said to be unbiased if B(u) tered around u. Generally we would like to select an unbiased estimator. In some cases a biased estimator may be preferable to an unbiased one. 2. Variance. The variance of an estimator is deﬁned by ^ 2 ^ ¼ s2^ ¼ E{u^ E½u} var(u) u

(18:1:2)

Generally we would choose an estimator with minimum variance. However, this may not be compatible with a minimum bias. 3. Mean-Square Error. The mean-square error is deﬁned by ^ ¼ E½u^ u2 MSE (u)

(18:1:3)

^ ¼ E{u^ E½u ^ þ E½u ^ u}2 MSE(u) n o ^ 2 þ ðE½(u ^ u)2 þ (u^ E½u)(E½ ^ ^ u) ¼ E (u^ E½u) u n o ^ u^ uÞ if u is a constant ¼ s2u^ þ E ðE½u^ uÞ2 þ ðu^ E½uÞðE½ n o ^ ¼ s2u^ þ B2 þ E ðu^ E½uÞB ¼ s2u^ þ B2 if u^ is a constant (18:1:4)

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

340

18.1

Criteria of Estimators

341

If the estimator is unbiased, then the minimum mean square is the same as minimum variance. However, in the presence of a bias we will be able to obtain a mean-square error smaller than the variance. 4. Consistency. Let u^ n be an estimator obtained from n observations. Then u^ n is a consistent estimator if lim P½(u^ n u) . 1 ! 0

n!1

for every 1 . 0

(18:1:5)

or u^ n converges in probability to u as n ! 1: We would prefer to have a consistent estimator; otherwise we will have a bias for inﬁnite samples that may not be desirable. 5. Maximum Likelihood. In Example 4.6.2 we estimated the population N of tigers in a wildlife sanctuary by maximizing the probability of occurrence of N using the maximum-likelihood criterion. We will expand on this concept further. The maximum-likelihood estimate of the parameter u from a probability density function f (x,u) of a random variable X is obtained by maximizing the likelihood function formed from n independent observations of the population fXi ¼ xi, i ¼ 1, . . . , ng. The likelihood function L(X,u) is the product of the density functions f(xi,u), i ¼ 1, . . . , n: L(X,u) ¼ f (x1 ,u) f (xn ,u)

(18:1:6)

Under certain regularity conditions the value of u^ that maximizes Eq. (18.1.6) is called the maximum-likelihood (ML) estimator. The monotonicity of the logarithm allows us to maximize the log likelihood function lnL(X,u). Thus the ML estimator is obtained from n @ @ @ X ln½L(X,u) ¼ ln ½ f (x1 ,u) . . . f (xn ,u) ¼ ln f (xi , u) @u @u @u i¼1

¼

n X i¼1

1 @f (xi ,u) f (xi ,u) @u

(18:1:7)

This method works well for large samples, but for small samples it may be biased. ^ and if it 6. Efﬁciency. The basic problem in estimation is to ﬁnd the best estimator u, exists, it will be unique. However, in some problems ﬁnding the best estimator may not be simple. In such cases we are satisﬁed if we can ﬁnd an estimator with the greatest lower bound. This bound is called the Rao –Cramer bound of an estimator, given by 1 ^ ( var(u) 2 ) @ ln f (X,u) E @u

(18:1:8)

An unbiased estimate that satisﬁes the Rao – Cramer bound is called an efﬁcient estimator.

342

Estimation Theory

B 18.2

ESTIMATION OF RANDOM VARIABLES

It is not possible for any estimator to satisfy all the criteria mentioned above. We will discuss unbiased minimum variance estimators under three broad categories: 1. Estimation of a random variable by a constant 2. Estimation of a random variable by a function of another random variable 3. Estimation of a constant by a random variable called parameter estimation We will discuss the ﬁrst two in this section and the third one in the next section.

Estimation of a Random Variable by a Constant

This is the simplest of the estimation problems. A random variable X is to be estimated by a constant a such that the mean-square error J ¼ E[X 2 a]2 is minimized. J ¼ E½X a2 ¼ E½X 2 2Xa þ a2 dJ d ¼ ½E½X 2 2mX a þ a2 ¼ 2mX þ 2a ¼ 0 dx dx or a ¼ mX :

(18:2:1)

Thus, the constant that minimizes the mean-square error is the expected value E[X ]. The minimum mean-square error Jmin ¼ E[X – mX]2 ¼ s2X is the variance. Equation (18.2.1) can be interpreted in a different way. Since ð1 (x a)2 fX (x)dx (18:2:2) E½X a2 ¼ 1

the value of a that minimizes the mean-square error is ð1 xfX (x)dx ¼ mX

(18:2:3)

1

Estimation of a Random Variable Y by a Function of Another Random Variable g(X)

Here we address the problem of estimating a random variable Y by a function of another random variable g(X ) using the mean-square error criterion J ¼ E[Y – g(X )]2: ð1 ð1 ½y g(x)2 fXY (x,y)dy dx (18:2:4) J ¼ E½Y g(X)2 ¼ 1 1

Inserting fXY (x,y) ¼ fY j X ( y j x)fX (x) in Eq. (18.2.4), we obtain ð 1 ð 1 2 J¼ ½ y g(x) fY j X ( y j x)dy fX (x)dx 1

(18:2:5)

1

Minimizing Eq. (18.2.5) amounts to minimizing the inner integral ð1 ½ y g(x)2 fY j X ( y j x)dy 1

(18:2:6)

18.2

Estimation of Random Variables

343

From Eqs. (18.2.2) and (18.2.3), the value of g(x) that minimizes Eq. (18.2.6) is ð1 g(x) ¼ yfY j X (y j x)dy 1

or g(X) ¼ Y^ ¼ E½Y j X

(18:2:7)

Thus, the estimator Yˆ is the conditional expectation of Y given X. This estimator is unbiased because ^ ¼ E½Y j X ¼ E½Y E½Y It is also minimum variance, and the minimum mean-square error is the conditional variance: s2Y j X ¼ E{Y E½Y j X}2 In general, the conditional expectation E[Y j X ] is a complicated nonlinear function of X that may not be expressible in an explicit form. We conﬁne ourselves to a linear function of the form E½Y j x ¼ g(x) ¼ Y^ ¼ a0 þ a1 x þ þ an xn ¼

n X

ai xi

(18:2:8)

i¼0

This form is known as the multiple linear regression, and the coefﬁcients {ai} are called linear regression coefﬁcients. It is linear because no powers of ai are involved. If E[Y j x] is of the form E½Y j x ¼ g(x) ¼ Y^ ¼ ax þ b

(18:2:9)

then it is called simple linear regression. The equation for the regression line is Y^ ¼ aX þ b

(18:2:10)

The estimator Y^ is unbiased because ^ ¼ E½aX þ b ¼ amX þ mY amX ¼ E½Y E½Y It is also minimum variance. Evaluation of Regression Coefﬁcients

We will now evaluate the linear regression coefﬁcients a and b for the case where Y is regressed on X, using the minimum mean-square error J ¼ E½Y aX b2

(18:2:11)

as the criterion function. We will ﬁrst solve for b by differentiating J with respect to b: @J ¼ 2E½Y aX b ¼ 0 @b

344

Estimation Theory

Hence b ¼ mY amX

(18:2:12)

We can now solve for a by substituting Eq. (18.2.12) into Eq.(18.2.11) and differentiating J with respect to a: J ¼ E½Y mY a(X mX )2 @J ¼ 2E{½Y mY a(X mX )(X mX )} ¼ 0 @a ¼ E½(X mX )(Y mY ) a(X mX )2 ¼ 0 ¼ sXY as2X ¼ 0 Hence a¼

sXY s2X

and substituting

sXY ¼ rsX sY ,

a¼r

sY sX

(18:2:13)

The equations obtained by setting @J=@a ¼ 0 and @J=@b ¼ 0 are called normal equations. With these values of a and b the mean-square error can be found as follows: J ¼ E½Y mY a(X mX )2 ¼ s2Y 2asXY þ a2 s2X ¼ s2Y 2 ¼ s2Y

s2XY ¼ s2Y (1 r2 ) s2X

s2XY þ s2XY s2X (18:2:14)

The simple linear regression plot shown in Fig.18.2.1 indicates the slope a and the offset b. The line of regression divides the cluster of points into two halves such that the mean square between the points and the line of regression is minimum. In an analogous manner, we can also regress X on Y and ﬁnd the corresponding regression coefﬁcients a and b in the expression X^ ¼ aY þ b

FIGURE 18.2.1

(18:2:15)

18.2

Estimation of Random Variables

345

All the parameters such as the means, variances, and correlation coefﬁcients will be the same. In much the same way as in Eqs. (18.2.12) and (18.2.13), the regression coefﬁcients a and b are given by a¼

sXY s2Y

and

b ¼ mX amY

(18:2:16)

Substituting sXY ¼ rsX sY in Eq. (18.2.16), a can also be given as a¼r

sX sY

(18:2:17)

Multiplying the two slopes a and a, we obtain aa ¼ r

sY sX r ¼ r2 sX sY

(18:2:18)

Since the true means and variances are not known, Eq. (18.2.18) is a more accurate method for computing the correlation coefﬁcient r in a real-life situation rather than using either Eq. (18.2.13) or (18.2.17). The sign of r can be obtained only from the physics of the problem. The regression coefﬁcients for a multiple linear regression problem Y^ ¼ a0 þ a1 X1 þ þ an Xn can be obtained in principle by forming " J¼E Y

n X

(18:2:19)

#2 ai Xi

i¼0

and solving the (n þ 1) normal equations obtained from @J=@a0 ¼ 0, @J=@a1 ¼ 0, . . . , @J=@an ¼ 0: A simpler method will be discussed in the next section. Example 18.2.1 The relationship between random variables X and Y is shown in Table 18.2.1. The parameters of X and Y are given by mX ¼ 0:905, s2X ¼ 0:198, mY ¼ 1:939, s2Y ¼ 0:308, sXY ¼ 0:080:

TABLE 18.2.1 No.

X

Y

No.

X

Y

1 2 3 4 5 6 7 8 9 10

0.05 0.29 0.74 0.55 1.07 0.47 1.06 0.7 0.54 0.65

1.82 2.05 1.28 1.19 2.06 1.96 2.46 1.76 1.13 2.35

11 12 13 14 15 16 17 18 19 20

1.54 0.72 0.66 1.23 1.35 0.97 1.3 0.96 1.73 1.52

1.49 2.15 0.88 1.49 2.18 2.74 2.1 2.69 2.18 2.82

346

Estimation Theory

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The correlation coefﬁcient rXY ¼ sXY =sX sY ¼ 0:08= (0:198 0:308) ¼ 0:324: We will ﬁnd the regression coefﬁcients a, b, a, b: a¼

sXY 0:08 ¼ ¼ 0:404, b ¼ my amX ¼ 1:939 0:404 0:905 ¼ 1:574 2 0:198 sX

a¼

sXY 0:08 ¼ ¼ 0:26, b ¼ mX amY ¼ 0:905 0:26 1:939 ¼ 0:40 0:308 s2Y

The regression lines y^ ¼ ax þ b and x^ ¼ ay þ b are shown in Figs. 18.2.2 and 18.2.3 respectively. The minimum mean-square error (MMSE) for Y on X regression ¼ 0.308 (1 2 0.3242) ¼ 0.276, and the MMSE for X on Y regression ¼ 0.194 (1 2 0.3242) ¼ 0.174. pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The square root of the product aa ¼ 0:404 0:26 ¼ 0:324 ¼ rXY is the correlation coefﬁcient obtained earlier. In general, the means and variances of random variables are not known a priori and we can only estimate them. We will revisit this example after discussing parameter estimation. Orthogonality Principle. We will motivate the orthogonality principle by taking the following expectations E½Y aX bb

(18:2:20a)

E½(Y aX b)aX

(18:2:20b)

with a ¼ sXY =s2X and b ¼ mY amX . Equation (18.2.20a) becomes E½Y aX bb ¼ ½mY amX ðmY amX )b ¼ 0: For Eq. (18.2.20b) we have E½(Y aX b)aX ¼ {E½XY aE½X 2 bmX }a ¼ {E½XY aE½X 2 (mY amX )mX }a ¼ {sXY as2X }a ¼ {sXY sXY }a ¼ 0 Combining both equations, we can write Y aX b aX þ b ¼0 E error 1 estimate Y^

FIGURE 18.2.2

18.2

Estimation of Random Variables

347

FIGURE 18.2.3

Equation (18.2.21) is the orthogonality principle, which states that under MMSE criterion the estimate Y^ is orthogonal to the error 1. In other words ^ ¼0 E½1Y

(18:2:21)

This is a powerful result that is used extensively in estimation problems. We can give a geometric interpretation to the orthogonality principle by means of Fig. 18.2.4. The two observations are b and a, represented by vectors x 0 and x 1 in a two-dimensional subspace. The unknown quantity y is not in the subspace. However, the estimate y^ , which is a linear combination of x 0, and x 1 is in the subspace. The error vector 1 is given by 1 ¼ y y^ is also shown in Fig. 18.2.4. From the geometry of the diagram, it can be seen that 1 is a minimum only when it is orthogonal to the subspace spanned by x 02x 1. Hence 1 is orthogonal to the estimate y^ . We will use the orthogonality principle to ﬁnd the regression coefﬁcients for a multiple linear regression problem of Eq. (18.2.19) expressed in matrix form Y^ ¼ a0 þ a1 X1 þ þ an Xn ¼ aT X T

(18:2:19)

where a is the (n þ 1) column vector [a0, a1, . . . , an] and X is the (n þ 1) random variable vector [1, X1, X2, . . . , Xn]T. Using the orthogonality principle that the error

FIGURE 18.2.4

348

Estimation Theory

1 ¼ Y aT X is orthogonal to the subspace spanned by the vector X, we have E½(Y aT X)XT ¼ 0

(18:2:22)

From Eq. (17.2.5) we can deﬁne E½XXT ¼ RX and E½XY ¼ RXY , and Eq. (18.2.22) can be rewritten as RX a ¼ E½XY

(18:2:23)

a ¼ R1 X E½XY

We can also ﬁnd the minimum mean-square error using the orthogonality principle: J ¼ E½(Y aT X)(Y aT X) ¼ E½1(Y aT X)

(18:2:24)

Using orthogonality, we obtain Jmin ¼ E½1Y ¼ E½(Y aT X)Y ¼ E½(Y 2 aT E½XY

(18:2:25)

a result arrived at with comparative ease. Example 18.2.2 We will now solve the simple linear regression problem by using Eq. (18.2.23). We are to estimate Y in Y ¼ b þ aX þ 1. The estimator Y^ ¼ b þ aX. The vector a ¼ [b,a]T, and the vector X ¼ [1,X ]T. The correlation matrix RX is given by 1 1 mX RX ¼ E ½1 X ¼ X mX m2 1 mY E½XY ¼ E Y ¼ E(XY) X Hence the regression coefﬁcients from Eq. (18.2.23) are m2 1 mX 1 mY b 1 ¼ 2 a¼ ¼ sX mX mX m2 E(XY) a 1 m2 mY mX E(XY) ¼ 2 sX sXY 1 mY (m2 m2X ) mX (E½XY mX mY ) ¼ 2 sX sXY

mX 1

mY

E(XY)

or mY mX a b sXY ¼ a s2X which is the same result as obtained before but without differentiation. We can also ﬁnd the minimum mean-square error using the orthogonality principle:

12 ¼ E½(Y aX b)(Y aX b) ¼ E½1(Y aX b)

18.2

Estimation of Random Variables

349

Using orthogonality, we obtain

12min ¼ E½1Y ¼ E½(Y aX b)Y ¼ E½Y 2 aE½XY mY ½mY amX ¼ E½Y 2 m2y a{E½XY mx mY } ¼ s2Y asXY ¼ s2Y (1 r2 ) a result arrived at with comparative ease. Example 18.2.3 A random variable Y ¼ aX 2 þ bX þ c þ 1 is to be estimated by a linear regression line Y^ ¼ aX 2 þ bX þ c. We want to ﬁnd the coefﬁcients a,b,c. The column vector a ¼ [a,b,c]T, and the column vector X ¼ [X 2,X,1]T. The correlation matrix RX is 82 3 9 2 3 m4 m3 m2 < X2 = (18:2:26) RX ¼ E 4 X 5½ X 2 X 1 ¼ 4 m3 m2 m 5 : ; 1 m2 m 1 and 82 3 9 2 3 E½X 2 Y < X2 = E½XY ¼ E 4 X 5Y ¼ 4 E½XY 5 : ; 1 E½Y

(18:2:27)

Hence the regression coefﬁcients are 2 3 2 a m4 4 b 5 ¼ 4 m3 c m2

31 2 3 m2 E½X 2 Y m 5 4 E½XY 5 1 E½Y

m3 m2 m

(18:2:28)

Using Eq. (18.2.25), we can derive the minimum mean-square error as follows: 2

E½X 2 Y

3

6 7 12min ¼ E½Y 2 ½a,b,c4 E½XY 5 E½Y ¼ E½Y 2 {aE½X 2 Y þ bE½XY þ cE½Y}

(18:2:29)

Example 18.2.4 We will continue the previous example with observed data for random variables X and Y shown in Table 18.2.2. It is desired to estimate Y from X. The correlation matrix RX is given as 2

E½X 4 E½X 3 4 RX ¼ E½X 3 E½X 2 E½X 2 E½X

3 2 4:53 E½X 2 E½X 5 ¼ 4 2:78 1:79 1

2:78 1:79 1:23

3 1:79 1:23 5 1

In addition, the cross-moments of X with Y are E[X 2Y ] ¼ 3.79, E[X Y ] ¼ 2.67, with E[Y ] ¼ 2.21 and E[Y 2] ¼ 4.92.

350

Estimation Theory

TABLE 18.2.2 No.

X

Y

No.

X

Y

1 2 3 4 5 6 7 8 9 10 11

0.91 0.22 0.23 1.92 1.87 0.55 0.94 1.4 1.06 1.38 1.91

2.6 2.21 2.2 2.03 2.07 2.38 2.48 2.37 2.44 2.34 1.9

12 13 14 15 16 17 18 19 20 21

1.47 1.37 0.63 1.39 0.79 1.53 1.51 1.35 1.86 1.64

2.25 2.28 2.26 2.23 2.27 2.11 2.1 2.17 1.77 1.95

FIGURE 18.2.5

The regression coefﬁcients are obtained from Eq. (18.2.28) as follows: 3 2 3 2 3 2 31 2 3:79 0:46 a 4:53 2:78 1:79 4 b 5 ¼ 4 2:78 1:79 1:23 5 4 2:67 5 ¼ 4 0:79 5 2:21 2:06 c 1:79 1:23 1 The line of regression is y ¼ 0:46x2 þ 0:79x þ 2:06. The original data y and the line of regression are shown in Fig. 18.2.5. The minimum mean-square error is obtained from Eq. (18.2.29) and is given by 12min ¼ E½Y 2 aE½X 2 Y þ bE½XY þ cE½Y ¼ 0:0109

B 18.3 ESTIMATION OF PARAMETERS (POINT ESTIMATION) A population is represented by the random variable X characterized by a parameter u, which is an unknown constant, and whose distribution function FX(x,u) is of a known form. We estimate this parameter by observing n realizations {x1, . . . , xn} of the

18.3

Estimation of Parameters (Point Estimation)

351

random variable X. Associated with each of these realizations is a set of random variables {X1, . . . , Xn} having the same characteristics as the population. We form a function ^ If X is the n-vector equal to [X1, . . . , Xn]T, then u^ = g(X1, . . . , Xn) for the estimator u. g(X) is known as the point estimator for u. As mentioned earlier, any function g(X) is ^ we cannot conclude that also called a statistic for u. Having determined the estimator u, it is exactly equal to the parameter u. However, we may be able to conclude that u is in the proximity of u^ within the interval u1 ¼ g1(X) and u2 ¼ g2(X). The interval estimator for u is the random interval, fQ1, Q2g, such that P{Q1 , u , Q2 } ¼ g

(18:3:1)

where g is called the conﬁdence coefﬁcient and the interval (u1,u2) is the g –conﬁdence interval. The coefﬁcient a ¼ (1 2 g) is called the signiﬁcance level. We will discuss interval estimators in the next section.

Estimation of Mean

The point estimator m^ X , called the sample mean for the mean mX of the random variable X, can be given as the weighted sum of the independent observation random variables fX1, . . . , Xng. Thus m^ X ¼

n X

a i Xi ¼ aT X

(18:3:2)

i¼1

where the weight vector a has to be determined according to the criteria of unbiasedness and minimum variance. The unbiasedness criterion applied to Eq. (18.3.2) yields " # n n n X X X ai Xi ¼ ai E½Xi ¼ mX ai ¼ m X (18:3:3) E½m^ X ¼ E i¼1

i¼1

i¼1

Hence we have the ﬁrst condition: n X

ai ¼ 1

(18:3:4)

i¼1

The variance of the estimator m^ X of Eq. (18.3.2) under the assumption that the Xi values are independent is given by var(m^ X ) ¼ s2m^ X ¼

n X

a2i u2X

(18:3:5)

i¼1

P The problem is to minimize var(m^ X ) subject to the constraint ni¼1 ai ¼ 1: We make use of Lagrange multipliers to ﬁnd a solution. We treat this problem as an unconstrained minimization by adjoining the constraint equation as shown below: ! n n X X 2 2 a i sX þ l ai 1 (18:3:6) J¼ i¼1

i¼1

P where l is the Lagrange multiplier that has to be determined. Since l ni¼1 ai 1 ¼ 0, we have not changed the minimization criterion. We treat Eq. (18.3.6) as unconstrained

352

Estimation Theory

and differentiate with respect to ai: " !# n n X @J @ X 2 2 ¼ a s þl ai 1 @ai @ai i¼i i X i¼1 ¼ 2ai s2X þ l ¼ 0

and

ai ¼

l , 2s2X

i ¼ 1, . . . , n

(18:3:7)

We have determined faig in terms of the unknown Lagrange multiplier l. Substituting for ai from Eq. (18.3.7) into the constraint Eq. (18.3.4), we obtain n X l ¼1 2 2s X i¼1

or l ¼

2s2X n

(18:3:8)

Substituting Eq. (18.3.8) back into Eq. (18.3.7), we obtain the weighting coefﬁcients ai as 1 ai ¼ , i ¼ 1, . . . , n n

(18:3:9)

and the minimum variance unbiased estimator for mX is given by m^ X ¼

n 1X Xi n i¼1

(18:3:10)

We can compute the variance of m^X from Eq. (18.3.5): var(m^ X ) ¼ s2m^ X ¼

n X s2 i¼1

X n2

¼

s2X n

(18:3:11)

In Eq. (18.3.11), as n ! 1, variance of m^ X tends to 0. We have already shown in Eq. (14.1.2) that if the variance is zero, then m^ X ! mX with probability 1. Hence we can conclude that m^ X is a consistent estimator since it tends to mX as n ! 1. The density of m^ X , which is centered at mX , is shown in Fig. 18.3.1 for different values of n.

FIGURE 18.3.1

18.3

Estimation of Parameters (Point Estimation)

353

Minimum Mean-Square Estimator for the Mean

Following the lines of Eq. (18.3.3), we will ﬁnd a minimum mean-square estimator (MMSE) m X such that mX ¼

n X

ai Xi

and

E½mX ¼

n X

i¼1

ai E½Xi ¼ kmX

(18:3:12)

i¼1

and the corresponding bias BmX in the estimator mP X is BmX ¼ (k 1)mX . We have to ﬁnd n 2 2 2 g such that the variance s ¼ the coefﬁcients fa i i¼1 ai sX is minimized subject to the m Pn X constraint i¼1 ai ¼ k. Using Lagrange multipliers as in the previous case, the coefﬁcients are found to be k ai ¼ , n

i ¼ 1, . . . , n

(18:3:13)

and the corresponding minimum variance is equal to s2mX ¼

k2 s2X n

(18:3:14a)

and the biased estimator in terms of the unbiased estimator is given by k mX ¼ m^ X n

(18:3:14b)

We will ﬁnd k that minimizes the mean-square error criterion given by MSE ¼ 12 ¼ E½mX mX 2 ¼ E½(mX E½mX ) þ (E½mX mX )2 ¼ s2mX þ B2mX ,

E½(mX E½mX ) ¼ 0

(18:3:15)

Substituting Eqs. (18.3.3) and (18.3.4) in Eq. (18.3.15), we have

12 ¼ s2mX þ B2mX ¼

k2 s2X þ (k 1)2 m2X n

(18:3:16)

Differentiating Eq. (18.3.16) with respect to k, we obtain @ 2 s2 ½1 ¼ 2k X þ 2(k 1)m2X ¼ 0 n @k or kmin ¼

m2X s2X þ m2X n

and

mX ¼

m2X s2X þ m2X n

m^ X

(18:3:17)

The variance of the estimator mX is obtained by substituting for k in Eq. (18.3.14a) and 2 s2mX ¼ kmin

s2X m4X s2X ¼ 2 n n s2X þ m2X n

(18:3:18)

354

Estimation Theory

which is less than (m^ X ) but biased with bias equal to

B mX

s2X mX ¼ (kmin 1)mX ¼ 2n sX þ m2X n

(18:3:19)

However, mX is consistent and asymptotically unbiased as n ! 1. This may not be a useful estimator because it is dependent on s2X and mX , which are both unknown quantities. At the very least, the ratio sX =mX must be known. This example illustrates that biased estimators in some cases may be better estimators than unbiased ones.

Estimation of Variance

Analogous to formulation of the sample mean, we can express the sample variance to be of the form s2X ¼

n 1X (Xi mX )2 n i¼1

(18:3:20)

and indeed s2X will be an unbiased minimum variance estimator; but mX is unknown, and hence s2X may not be a useful estimator. However, if we substitute the sample mean m^ X instead of mX in Eq. (18.3.20), then we can write the new estimator s~ 2X as follows: s~ 2X ¼

n 1X (Xi m^ X )2 n i¼1

(18:3:21)

To determine whether s2X is unbiased, we take expectations on both sides of Eq. (18.3.21) and write ( " #) n n n n X 1X 2 X 1X 2 2 E(s~ X ) ¼ E X Xi Xj þ 2 Xj Xk n i¼1 i n j¼1 n j¼1 k¼1 8 2 39 > > > n n j¼1 n j¼1 n j¼1 k¼1 :n i¼1 ; j=i

n 1X

j=k

2 n 2(n 1) 2 n(n 1) 2 2 2 mX þ 1 þ 2 (sX þ mX ) mX ¼ n i¼1 n n n n2 n 1X n1 2 n 1 2 2(n 1) 2 (n 1) 2 n1 2 mX þ mX ¼ sX ¼ sX þ mX n i¼1 n n n n n As we can see from Eq. (18.3.22), the estimator s~ 2X is biased. To obtain an unbiased estimator, we have to divide Eq. (18.3.21) by (n 2 1) instead of by n and write the unbiased minimum variance estimator s^ 2X , also called the sample variance, as s^ 2X ¼

n 1 X (Xi m^ X )2 n 1 i¼1

(18:3:23)

In forming an unbiased estimator for the variance, the degrees of freedom in the estimator determines the divisor. In Eq. (18.3.20) there are n degrees of freedom since there are n

18.3

Estimation of Parameters (Point Estimation)

355

independent values of Xi, and hence the divisor P is n. However, in Eq. (18.3.23) there are only (n 2 1) degrees of freedom since m^ X ¼ ni¼1 (Xi =n), which takes away one degree of freedom. Thus, the divisor is (n 2 1). The square root of the sample variance s^ X is called the standard error. We will now compute the variance of the sample variance "

var(s^ 2X )

#2 n 1 X 2 ¼E (Xi m^ X ) (s2X )2 n 1 i¼1 ( ) n n X X 1 ¼E (Xi m^ X )2 (Xj m^ X )2 s4X (n 1)2 i¼1 j¼1 8 2 39 = < 1 n n X n X X 4 2 25 4 s4X ¼E (X m ^ ) þ (X m ^ ) (X m ^ ) i i j X X X :(n 1)2 i¼1 ; i¼1 j¼1 i=j

n n(n 1) 4 n s4 v þ s s4X ¼ v þ X 2 4 2 X 2 4 n1 (n 1) (n 1) (n 1)

(18:3:24)

where v4 ¼ E(Xi mX )4 from Eq. (10.4.2). We can also conﬁrm that s^ 2X is a consistent estimator since var(s^ 2X ) tends to 0 as n tends to 1. In fact, all three estimators, s2X , s~ 2X , and s^ 2X , are consistent while only s2X and s^ 2X are unbiased. However, for large n all the estimators are asymptotically unbiased.

Estimation of Covariance

The unbiased estimator for the covariance sXY between two random variables X and Y can be derived with the following assumptions. Corresponding to each of the random variables, we have two sets of independent observations fX1, . . . , Xng and fY1, . . . , Yng with E[Xi] ¼ mX, E[Yi] ¼ mY, E[XiYi] ¼ sXY þ mX mY, and E[XiYj] ¼ mX mY for i = j. The estimator s^ XY for the covariance can be expressed as s^ XY ¼

n 1X (Xi m^ X )(Yi m^ Y ) K i¼1

(18:3:25)

where the divisor K is found from the following steps to satisfy the unbiasedness condition E(s^ XY ) ¼ sXY : " # n X X (Xi m^ X )(Yi m^ Y ) ¼ E½Xi Yi Xi m^ Y Yi m^ X þ m^ X m^ Y E i¼1

i

X 1X 1X 1 XX X i Yj E X i Yi X i ¼ Yj Yi Xj þ 2 n n n i " X X i Y i X X i Y j X i Yi X Yi X j 1 X þ 2 E X i Yi ¼ Xi Yi n n n n n i j=i j=1 # 1 XX Xi Yj þ 2 n j=i i

356

Estimation Theory

X

1 2(n 1) n(n 1) mX mY þ ¼ (sXY þ mX mY ) 1 mX mY n n n2 i Xn 1 n1 2(n 1) n1 sXY þ mX mY mX mY þ mX mY ¼ n n n n i ¼ (n 1)sXY

(18:3:26)

From Eq. (18.3.26) K is found to be (n 2 1), and the unbiased minimum variance estimator for the covariance is s^ XY ¼

n 1 X (Xi m^ X )(Yi m^ Y ) n 1 i¼1

(18:3:27)

a result very similar to that of the estimator s^ 2X of the variance. We will now compute the variance of the sample covariance "

#2 n 1 X (Xi m^ X )(Yi m^ Y ) (sXY )2 var(s^ XY ) ¼ E n 1 i¼1 ( ) n n X

X

1 ¼E (Xi m^ X )(Yi m^ Y ) (Xj m^ X )(Yj m^ Y ) s2XY (n 1)2 i¼1 j¼1 8 2 n 39 P 2 > > > > ½(X m^ X )(Yi m^ Y ) > = < 1 6 i¼1 i 7> 6 7 s2XY ¼E n n 6 7 P P 24 > > 5 (n 1) > > þ ½(X m ^ )(Y m ^ )½(X m ^ )(Y m ^ ) i i j j > > X Y X Y ; : i¼1 j¼1 j=i

n n(n 1) 2 n s2 v XY þ s s2XY ¼ vXY þ XY 2 2 2 XY 2 2 n1 (n 1) (n 1) (n 1)

(18:3:28)

2 where we have deﬁned vXY as the second cross-central 2 ¼ E ½(Xi mX )(Yi mY ) moment between X and Y. Since var(s^ XY ) tends to 0 as n tends to 1, we conclude that (s^ XY ) is also a consistent unbiased minimum variance estimator. In Eqs. (18.2.13) and (18.2.17) the means and variances were explicitly stated. In general, these quantities can only be estimated. We will revisit the linear regression problem later. Example 18.3.1 We will ﬁnd the estimated means and variances of the random variables X and Y from the data given in Example 18.2.1. From Eq. (18.3.10) the unbiased, minimum variance estimators for the mean of X and Y are m^ X ¼

1 18:1 ½0:05 þ 0:29 þ þ 1:52 ¼ ¼ 0:905 20 20

m^ Y ¼

1 38:78 ½1:82 þ 2:05 þ þ 2:82 ¼ ¼ 1:939 20 20

18.3

Estimation of Parameters (Point Estimation)

357

Similarly, from Eq. (18.3.9) the estimated variances of X and Y are 1 ½(0:05 0:905)2 þ (0:29 0:905)2 þ þ (1:52 0:905)2 19 3:767 ¼ 0:198 ¼ 19 1 s^ 2Y ¼ ½(1:82 1:939)2 þ (2:05 1:939)2 þ þ (2:82 1:939)2 19 5:857 ¼ 0:308 ¼ 19

s^ 2X ¼

The estimated covariance between X and Y is 1 ½(0:05 0:905)(1:82 1:939) þ þ (1:52 0:905)(2:82 1:939) 19 1:521 ¼ 0:080 ¼ 19

s^ X Y ¼

The estimated correlation coefﬁcient is r^ XY ¼

s^ XY 0:08 ¼ 0:324 ¼ s^ X s^ Y 0:247

These parameters are the same numbers that we used in Example 18.2.1. However, the regression coefﬁcients a and b will be estimates and will be governed by a probability distribution. Example 18.3.2 We will now ﬁnd the estimated means and variances of the random variables X and Y from the data given in Example 18.2.3. From Eq. (18.3.10) the unbiased, minimum variance estimators for the mean of X and Y are m^ X ¼

1 25:93 ½0:91 þ 0:22 þ þ 1:64 ¼ ¼ 1:235 21 21

m^ Y ¼

1 46:41 ½2:6 þ 2:21 þ þ 1:95 ¼ ¼ 2:21 21 21

Similarly, from Eq. (18.3.9) the estimated variances of X and Y are 1 ½(0:91 1:235)2 þ (0:22 1:235)2 þ þ (1:64 1:235)2 20 5:50 ¼ 0:275 ¼ 20

s^ 2X ¼

1 ½(2:6 2:21)2 þ (2:21 2:21)2 þ þ (1:95 2:21)2 20 0:795 ¼ 0:0398 ¼ 20

s^ 2Y ¼

358

Estimation Theory

We will now ﬁnd estimated values for the higher-order moments of X. They are given by ^ 4 ¼ ^ 4X ¼ E½X m

1 95:1 ½0:914 þ 0:224 þ þ 1:644 ¼ ¼ 4:53 21 21

^ 3 ¼ ^ 3X ¼ E½X m

1 58:43 ½0:913 þ 0:223 þ þ 1:643 ¼ ¼ 2:78 21 21

^ 2 ¼ ^ 2X ¼ E½X m

1 37:52 ½0:912 þ 0:222 þ þ 1:642 ¼ ¼ 1:79 21 21

The estimated second moment of Y is given by ^ 2 ¼ ^ 2X ¼ E½Y m

1 103:36 ½2:62 þ 2:212 þ þ 1:952 ¼ ¼ 4:92 21 21

The estimated cross-moments of X with Y are 1 56:1 ^ E½XY ¼ ½0:91 2:6 þ 0:22 2:21 þ þ 1:64 1:95 ¼ ¼ 2:67 21 21 ^ 2 Y ¼ 1 ½0:912 2:6 þ 0:222 2:21 þ þ 1:642 1:95 ¼ 79:6 ¼ 3:79 E½X 21 21 The estimated correlation matrix RX is, 2

^ 4 E½X ^ X ¼ 4 E½X ^ 3 R ^E½X 2

3 2 ^ 3 E½X ^ 2 4:53 E½X 5 ¼ 4 2:78 ^E½X 2 E½X ^ ^ 1:79 E½X 1

2:78 1:79 1:23

3 1:79 1:23 5 1

These values are the same values given in Example 18.2.3. Again, the regression coefﬁcients a, b, and c will be only estimates. Estimated Regression Coefﬁcients from Data

As stated in previous examples, the evaluated regression coefﬁcients will be estimates of the true values since the means and variances are only estimates. We will now discuss the methodology of ﬁnding the estimated regression coefﬁcients and determining their variances from the given data. The linear regression problem for each yi can be stated as yi ¼ a1 xi1 þ þ am xim þ wi ¼ xTi a þ wi ,

i ¼ 1, . . . , n

with n . m. Or more succinctly y ¼ XT a þ w

(18:3:29)

where the n-vector y is the realization of the random variable Y with E(y) ¼ X Ta where a is an unknown m-vector parameter to be estimated, and xi is the observation m-vector given by xi ¼ [xi1, . . . , xim]T. In linear regression, the coefﬁcient a1 represents the offset and hence xi1¼1 for all i. The n-vector w is a realization of the measurement noise random variable W. The elements wi of w are assumed to be uncorrelated with zero mean and variance E[w2i ] ¼ s2W for all i. X is an m n (n . m) matrix of

18.3

Estimation of Parameters (Point Estimation)

359

observations given by ½x 2 1 x11 6x 6 12 6 . 6 . . X¼ 6 6 6 x1j 6 6 . 6 . 4 . x1m

x2 x21 x22 .. . x2j .. .

xi xi1 xi2 .. . xij .. .

xn 3 2 xn1 1 6 7 xn2 7 6 x12 6 .. 7 7 6 . . 7 6 .. 7¼6 6 xnj 7 7 6 x1j 7 .. 7 6 6 . . 5 4 ..

x2m xim xmn

1 x22 .. . x2j .. .

3 1 1 xi2 xn2 7 7 .. .. 7 7 . . 7 7 (18:3:30) xij xnj 7 7 .. 7 .. 7 . 5 .

x1m x2m xim xnm

Given this information, we have to estimate the regression coefﬁcient vector a using the least-squares error. Thus, the criterion function J(a) is given by J(a) ¼ (y XTa )T (y XTa ) ¼ yT y aT Xy yT XT a þ aT XXT a

(18:3:31)

From Eq. (16.3.4) the derivative with respect to the vector a of Eq. (18.3.31) is @ @ J(a) ¼ ½yT y aT Xy yT XT a þ aT XXT a @a @a ¼ 2Xy þ 2XXTa ¼ 0

(18:3:32)

From Eq. (18.3.32) the estimated regression coefﬁcients a^ are given by a^ ¼ (XXT )1 Xy where XX T expanded in terms of the P 2 n xi2 n 6P P 6 xi2 x2i2 6 n n 6 6 . .. 6 . 6 . T P. XX ¼ 6 P 6 xij xij xi2 6 n n 6 6 . .. 6 .. 4P P . xim xim xi2 n

n

(18:3:33)

observation points fxijg is given by P P 3 xij xim n n 7 P P xi2 xij xi2 xim 7 7 n n 7 7 .. .. 7 7 . . P 2 P 7 xij xij xim 7 7 n n 7 7 .. .. 7 . . P P 2 5 xim xij xim n

(18:3:34)

n

The equation for the regression line is given by y^ ¼ XT a^

(18:3:35)

The estimator a^ is an unbiased estimator of a since E(^a) ¼ (XXT )1 XE(y) ¼ (XXT )1 XXT a ¼ a

Covariance of the Estimated Regression Coefﬁcients a^

Before ﬁnding the covariance of a, we will express E½^a a in a more convenient form given by E½^a a ¼ E½(XXT )1 Xy a ¼ E½(XXT )1 X(XT a þ w) a ¼ E½(XXT )1 Xw

(18:3:36)

360

Estimation Theory

The covariance matrix Ca^ of the estimated regression coefﬁcients can now be given by Ca^ ¼ cov(^a) ¼ cov½(^a a)(^a a)T ¼ cov{(XXT )1 XwwT XT (XXT )T } ¼ (XXT )1 X cov(wwT )XT (XXT )T ¼ (XXT )1 Xs2w IXT (XXT )T ¼ (XXT )1 s2w

(18:3:37)

where cov(wwT ) ¼ s2w Im . The variances of the regression coefﬁcients are given by the diagonal terms of the covariance matrix Ca^ . Since we do not know the variance s2w of the measurement noise, we use its estimate s^ 2w , given by s^ 2w ¼

(y y^ )T (y y^ ) (y XT a^ )(y XT a^ ) ¼ nm nm

(18:3:38)

In this equation, the divisor (n 2 m) reﬂects the reduction in the number of degrees of freedom n by the number of regression coefﬁcients m. This yields an unbiased estimator for s2w.

Variance of the Estimated Regression line y^ i

We will now ﬁnd the variance of the estimated mean y^ i , given by y^ i ¼ a^ 1 xi1 þ þ a^ m xim ¼ a^ T xi ,

i ¼ 1, . . . , n

(18:3:39)

where y^ i is an unbiased estimator of yi since E½^yi ¼ y. The variance of y^ i can be given by E½ y^ i yi 2 ¼ E½(^yi yi )T (^yi yi ) ¼ xTi E ½(^a a)(^a a)T xi ¼ xTi cov(^a)xi

(18:3:40)

Substituting for cov(^a) from Eq. (18.3.37), we have var( y^ i ) ¼ s2y^ i ¼ xTi (XXT )1 xi s^ 2w , i ¼ 1, . . . , n

(18:3:41)

and since we do not know the variance, we substitute the estimated variance s^ 2w and obtain the estimated variance of y^ i as s^ 2y^ i ¼ xTi (XXT )1 xi s^ 2w ,

i ¼ 1, . . . , n

(18:3:42)

which is a function of i, and is chi-square-distributed with (n 2 m) degrees of freedom.

Simple Linear Regression

The governing equation for a simple linear regression is yi ¼ a1 þ a2 xi þ wi ,

i ¼ 1, . . . , n

(18:3:43)

with the usual deﬁnitions. From the data, we will derive expressions for the estimated regression coefﬁcients for a1 and a2. The normal equations can be obtained from Eqs. (18.3.32) and (18.3.34) as follows: X 3 2 2 X 3 n xi yi 6X 7 a^ 1 6X 7 n n X ¼4 (18:3:44) 4 5 5 2 ^ a 2 xi xi xi yi n

n

n

18.3

Estimation of Parameters (Point Estimation)

361

The determinant of the rectangular matrix of Eq. (18.3.44) is X

!2 xi X X n 2 X ¼n xi xi x2i n n

n X xi n

n

¼n

X

X

x2i m2 m^ 2x ¼ n

n

(xi m^ x )2

(18:3:45)

n

P where m^ x ¼ (1=n) n xi . Solving Eq. (18.3.44) and substituting in Eq. (18.3.45), the estimated coefﬁcients a^ 1 and a^ 2 are

a^ 1 a^ 2

2 X

x2i 1 6 nX ¼ X 4 n (xi m^ x )2 xi n

2X

n

X

xi

n

n

X

32 X 76 X n 54

yi

xi yi

3 7 5

n

X

X

3 yi xi xi yi 1 6 n Xn 7 n n X X ¼ X 4 5 n (xi m^ x )2 n xi yi xi yi x2i

n

n

n

(18:3:46)

n

In Eq. (18.3.46) n

X

xi yi

X

n

xi

X

n

yi ¼ n

X

n

xi yi n2 m^ x m^ y ¼ n

n

X

(xi m^ x )(yi m^ y )

n

Hence X n

a^ 2 ¼

(xi m^ x )(yi m^ y ) X (xi m^ x )2

(18:3:47)

n

Also, in Eq. (18.3.46) X

x2i

n

X n

yi

X n

xi

X

xi yi ¼ nm^ y

X

n

n

x2i

X n

" xi

X

# (xi m^ x )(yi m^ y ) þ nm^ x m^ y

n

"

# X

X x2i m^ 2X nm^ x (xi m^ x )(yi m^ y ) ¼ nm^ y n

n

Hence X a^ 1 ¼ m^ y

n

(xi m^ x )(yi m^ y ) X m^ x ¼ m^ y a^ 2 m^ x (x2i m^ 2X )

(18:3:48)

n

The estimated regression line equation is y^ i ¼ a^ 1 þ a^ 2 xi , i ¼ 1, . . . , n

(18:3:49)

362

Estimation Theory

We can now compute the variances of a^ 1 and a^ 2 from Eqs. (18.3.33) X 3 2 X 2 xi xi 1 a^ 1 6 7 2 n n var ¼ X 4 X 5s^ w a^ 2 (xi m^ x )2 n xi n n

(18:3:50)

n

where we have substituted the estimated variance s^ 2w from Eq. (18.3.38) for the true variance s2w . Simplifying Eq. (18.3.50), we obtain the variances of a^ 1 , and a^ 2 as var(^a1 ) ¼ s^ 2a^ 1 ¼ X n

s^ 2w (xi m^ x )2

2X

6 var(^a2 ) ¼ s^ 2a^ 2 ¼ s^ 2w 4

n

3 2 3 (xi m^ x )2 þ n(m^ 2x ) 2 m^ x 7 61 7 X 5 ¼ s^ 2w 4 þ X 2 25 n (xi m^ x ) (xi m^ x ) n n

(18:3:51)

n

Finally, we can compute the estimated variance of the estimated regression line y^ i . From Eq. (18.3.42), we have X 3 2 X 2 xj xj

6 n 1 7 1 2 n 2 X X 1 xi 4 s^ , i ¼ 1, . . . , n (18:3:52) s^ y^ i ¼ 5 2 xi w (xj m^ x ) n xj n n

n

Equation (18.3.52) can be simpliﬁed as follows P 3 2 P 2 xj xj X X 1 n 5 ½ 1 xi 4 nP x2j 2xi xj þ nx2i ¼ xj n xi n n n

¼

X

2 xj m^ x þn(xi m^ x )2

(18:3:53)

n

where we have subtracted and added m^ 2x . Substituting Eq. (18.3.53) in Eq. (18.3.52), we obtain " # X 1 2 2 2 (xj m^ x ) þ n(xi m^ x ) s^ 2w s^ y^ i ¼ P n n (xj m^ x )2 n 1 (xi m^ x )2 þP ¼ (18:3:54) s^ 2w , i ¼ 1, . . . , n 2 n (x m ^ ) j x n Example 18.3.3 From the data in Example 18.2.1 we will determine the estimated regression coefﬁcients and their variances using the techniques developed above. The linear equation used to solve for the regression coefﬁcients is y i ¼ a1 þ a 2 x i þ w i ,

i ¼ 1, . . . , 20

In Eq. (18.3.29) the 20-vector y is given by 1:82, 2:05, 1:28, 1:19, 2:06, 1:96, 2:46, 1:76, 1:13, 2:35, T y¼ 1:49, 2:15, 0:88, 1:49, 2:18, 2:74, 2:10, 2:69, 2:18, 2:82

18.3

Estimation of Parameters (Point Estimation)

363

and the 20-vector x is given by, 0:05, 0:29, 0:74, 0:55, 1:07, 0:47, 1:06, 0:7, 0:54, 0:65, T x¼ 1:54, 0:72, 0:66, 1:23, 1:35, 0:97, 1:3, 0:96, 1:73, 1:52 a is a 2-vector of regression coefﬁcients given by a ¼ ½a1 ,a2 T and w is an unknown measurement 20-vector noise whose variance can only be estimated. We can now form the 2 20 matrix X, where the ﬁrst row is always 1 since a1 multiplies 1 for all i. Hence the matrix X is 82 39 1 1 1 1 1 1 1 1 1 1 > > > = 7 X¼ 6 4 5 > > 1 1 1 1 1 1 1 1 1 1 > > : ; 1:54 0:72 0:66 1:23 1:35 0:97 1:3 0:96 1:73 1:52 The quantity XXT and (XXT )1 are given by 20 18:1 0:267 XXT ¼ (XXT )1 ¼ 18:1 20:147 0:24

0:24 0:265

The estimated regression coefﬁcients a^ are obtained from Eq. (18.3.33): 0:267 0:24 38:78 1:574 a^ ˆa ¼ 1 ¼ (XXT )1 (Xy) ¼ ¼ a^ 2 0:24 0:265 36:617 0:404 with Xy ¼

38:78 36:617

where a^ 1 and a^ 2 exactly match the quantities b and a in Example 18.2.1. The equation of the regression line y^ i ¼ a^ 1 þ a^ 2 xi ¼ 1:574 þ 0:404xi ,

i ¼ 1, . . . , 20

has already been shown in Fig. 18.2.2. We will now compute the estimated variance of the noise term from Eq. (18.3.38): s^ 2W ¼

(y XT aˆ )T (y XT aˆ ) 4:74 ¼ 0:263 ¼ 20 2 nm

This compares within the limits of data analysis with MMSE ¼ 0.276, obtained in Example 18.2.1. From Eq. (18.3.37), the covariance matrix of a^ is 0:267 0:24 0:0704 0:0633 Caˆ ¼ cov(ˆa) ¼ (XXT )1 s^ 2W ¼ 0:263 ¼ 0:24 0:266 0:0633 0:0699 Hence var(^a1 ) ¼ 0:0704 and var(^a2 ) ¼ 0:0699. We can now obtain the variance ofPy^ i using Eq. (18.3.54), where n ¼ 20, the average value m^ x ¼ 0:905, and the summation n (xj m^ x )2 ¼ 3:7665. Substituting these values, we obtain 2 3 2 (xi m^ x ) 7 1 (xi 0:905)2 61 þ s^ 2y^ i ¼ s^ 2w 4 þ P ¼ 0:263 , i ¼ 1, . . . , 20 5 20 20 3:7665 (xj m^ x )2 20

364

Estimation Theory

B 18.4 INTERVAL ESTIMATION (CONFIDENCE INTERVALS) In the previous section we discussed point estimators u^ for a parameter u. However, a point estimator may not be meaningful without knowledge of the probability of how far away it is from the true value. Hence, we would also like to know an interval about the estimator u^ and the probability that the true value u lies in that interval. Such an interval for u is the random interval, fQ1, Q2g, such that P{Q1 , u , Q2 } ¼ g

(18:4:1)

where g is the conﬁdence coefﬁcient. The realized interval (u1, u2) is the g conﬁdence ^ In essence, interval. The terms u1 and u2 could possibly be functions of the estimator u. we can replace the empirical point estimator with an interval estimator that gives more information about the population parameter. The coefﬁcient a ¼ (1 2 g) is called the signiﬁcance level. In a two-sided conﬁdence interval as in Eq. (18.4.1), it is usual to split the signiﬁcance level a on both sides of the density. We will apply the conﬁdence intervals to various parameters such as the mean and variance. We have already come across the basic ideas of conﬁdence intervals in Section 7.9 and Examples 14.8.1 and 14.8.4. Example 18.4.1 In a presidential election, a news agency polled 1000 probable voters, and 470 of them said that they would vote Democratic while 460 of them said that they would vote Republican. The agency reported that the Democrat winning was 47% and the Republican winning was 46%, with a margin of error of +4%. What this means is that the true probability of the Democrat winning lies in the interval (51%, 43%) and that of the Republican lies in the interval (50%, 42%): P{0:43 , P(Democrat winning) , 0:51} ¼ x P{0:42 , P(Republican winning) , 0:50} ¼ x The conclusion arrived at is that the results are within the margin of error and hence the election is a dead heat. What is never mentioned is x, the probability of these results occurring! Conﬁdence Interval for the Unknown Mean of Population (Known Variance). We P have already established in Eq. (18.3.10) that m^ X ¼ (1=n) ni¼1 Xi is an unbiased minimum variance point estimator for the mean mX. If n is large enough, we can assume that m^ X is Gaussian-distributed with variance s2X =n. We will ﬁnd the conﬁdence interval under the assumption that s2X is known. The random variable Z given by Z¼

m^ X mX pﬃﬃﬃ sX = n

(18:4:2)

is a standard Gaussian, and for a conﬁdence coefﬁcient g ¼ 1 2 a we can write P{Za=2 , Z z1a=2 } ¼ 1 a ¼ g or m^ m P za=2 , X pﬃﬃﬃX z1a=2 ¼ g sX = n

(18:4:3)

where we have split the signiﬁcance level a equally on both sides of the density function. The values za=2 ¼ P(Z a=2) and z1a=2 ¼ P(Z 1 a=2) for a conﬁdence level of g are extracted from the Gaussian table (Table 18.4.1) for

18.4

Interval Estimation (Conﬁdence Intervals)

365

TABLE 18.4.1 Gaussian Tables g%

P(Z z)

z

P(Z z)

z

99 98 95 90 85 80 75 70 65 60 55 50

0.005 0.01 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25

22.57583 22.32635 21.95996 21.64485 21.43953 21.28155 21.15035 21.03643 20.93459 20.84162 20.75542 20.67449

0.995 0.99 0.975 0.95 0.925 0.9 0.875 0.85 0.825 0.8 0.775 0.75

2.57583 2.32635 1.95996 1.64485 1.43953 1.28155 1.15035 1.03643 0.93459 0.84162 0.75542 0.67449

probabilities of a=2 and (1 a=2) respectively. Since the Gaussian curve is symmetric, za=2 ¼ z1a=2 . The z values for 95% conﬁdence are z0.025 ¼ – 1.96 and z0.975 ¼ 1.96. The corresponding z values for 99% conﬁdence are z0.005 ¼ – 2.576 and z0.995 ¼ 2.576 and for 90% conﬁdence z0.05 ¼ – 1.645 and z0.95 ¼ 1.645. From Eq. (18.4.3) we have m^ X mX pﬃﬃﬃ z1a=2 ¼ 1 a P za=2 , sX = n from which it follows that za=2 sX z1a=2 sX P pﬃﬃﬃ , m^ X mX pﬃﬃﬃ ¼1a n n and ﬁnally

z1a=2 sX za=2 sX ¼1a P m^ X pﬃﬃﬃ , mX m^ X pﬃﬃﬃ n n

(18:4:4)

Equation (18.4.4)pis ﬃﬃﬃ to be interpreted pﬃﬃﬃas the probability that the random interval (m^ X z1a=2 sX = n, m^ X za=2 sX = n) will contain the true population mean mX is 100(1 2 a)%. The probability density of m^ X centered about the true mean mX and the conﬁdence intervals are shown in Fig. 18.4.1. The 99% conﬁdence interval is given by 2:576sX 2:576sX ¼ 0:99 (18:4:5) P m^ X pﬃﬃﬃ , mX m^ X þ pﬃﬃﬃ n n In an analogous manner, the 95% conﬁdence interval is given by 1:96sX 1:96sX ¼ 0:95 P m^ X pﬃﬃﬃ , mX m^ X þ pﬃﬃﬃ n n

(18:4:6)

366

Estimation Theory

FIGURE 18.4.1

and the 90% conﬁdence interval is given by 1:645sX 1:645sX ¼ 0:90 P m^ X pﬃﬃﬃ , mX m^ X þ pﬃﬃﬃ n n

(18:4:7)

It can be seen from Eqs. (18.4.5)– (18.4.7) that as the conﬁdence level g decreases and conversely as the signiﬁcance level a increases, the bounds become tighter. If we want to have 100% conﬁdence or 0% signiﬁcance, then the conﬁdence interval is 1! The equations also show that as the number of samples n increases, the bounds also become tighter. The conﬁdence level is usually ﬁxed a priori at 95% or 90% before sampling. Conﬁdence Interval for the Unknown Mean of Population (Unknown Variance). In the previous section we assumed that the variance is known. Since the true variance is known in very few cases, we use the estimated variance s^ 2X . In that case we have to ﬁnd the distribution of the random variable T, given by T¼

m^ X mX pﬃﬃﬃ s^ X = n

(18:4:8)

which is a Student-t distribution given by Eq. (7.9.2) with n degrees of freedom. Equation (18.4.8) can rewritten as T¼

pﬃﬃﬃ (m^ X mX )=sX = n {½(n 1)(s^ X =sX )2 =(n 1)}1=2

(18:4:9)

pﬃﬃﬃ where the term (m^ X mX )=sX = n is a standard Gaussian, and we will show that the term n (n 1)s^ 2X X Xi m^ X 2 ¼ sX s2X i¼1 is chi-square-distributed with (n – 1) degrees of freedom. Adding and subtracting mX to (Xi m^ X ) in the numerator of the righthand side of the equation

18.4

Interval Estimation (Conﬁdence Intervals)

367

above, we have n n X Xi m^ X 2 X Xi mX (m^ X mX ) 2 ¼ sX sX i¼1 i¼1 ( ) n X Xi mX 2 m^ X mX 2 2(Xi mX )(m^ X mX ) ¼ þ sX sX s2X i¼1 ( ) n X Xi mX 2 m^ mX 2 since ¼ n X sX sX i¼1 n X

(Xi mX ) ¼ n(m^ X mX )

(18:4:10)

i¼1

P The ﬁrst term on the righthand side of Eq. (18.4.10), ni¼1 ½(Xi mX )=spXﬃﬃﬃ2 is chi-square-distributed with n degrees ofpfreedom. Since (m^ X mX )=(sX = n) is ﬃﬃﬃ Gaussian with mean 0 and variance sX = n, the second term m^ mX 2 m^ X mX 2 pﬃﬃﬃ ¼ n X sX sX = n P is chi-square-distributed with one degree of freedom. Hence ni¼1 ½(Xi m^ X )=sX 2 is chi-square-distributed with (n 2 1) degrees of freedom. Further, it can be shown that s^ 2X and m^ X are independent random variables. Hence, from Example 13.4.5, T in Eq. (18.4.8) is indeed Student-t-distributed with (n 2 1) degrees of freedom. We can now obtain the 100(1 – a)% conﬁdence intervals for the population mean mX using the Student-t as follows. Analogous to Eq. (18.4.4), we can write (m^ X mX ) pﬃﬃﬃ t1a=2 ¼ 1 a ¼ g P{ta=2 , T t1a=2 } ¼ P ta=2 , s^ X = n from which it follows that pﬃﬃﬃ pﬃﬃﬃ P{ta=2 s^ X = n , (m^ X mX ) t1a=2 s^ X = n} ¼ 1 a ¼ g and ﬁnally pﬃﬃﬃ pﬃﬃﬃ P{m^ X t1a=2 s^ X = n , mX m^ X ta=2 s^ X = n} ¼ 1 a ¼ g

(18:4:11)

where ta=2 ¼ P(T a=2) and t1a=2 ¼ P(T 1 a=2) and ta=2 ¼ t1a=2 because of the symmetry of the Student-t density. From the Student-t table (Table 18.4.2), t1a=2 is extracted for n – 1 degrees of freedom and probability (1 2 a/2). Equation p (18.4.11) is to be read ﬃﬃﬃ pﬃﬃﬃ as the probability that the random interval (m^ X t1a=2 s^ X = n, m^ X ta=2 s^ X = n) includes the true population mean mX is equal to (1 2 a). These conﬁdence limits are shown in Fig. 18.4.2 on a Student-t density with (n – 1) degrees of freedom.

Example 18.4.2 The tensile strength of six samples of a material is given by 226, 223, 225, 227, 224, and 222 lb. The estimated mean m^ X ¼ 224:5 lb. The standard error, s^ X ¼ 1:871 lb. Assuming a Gaussian distribution for the samples, the t0.95 value for (6 – 1) ¼ 5 degrees of freedom from Table 18.4.2 is given by t0.95 ¼ 2.01505. Hence the

368

Estimation Theory

TABLE 18.4.2 Student-t Tables with t-Values Shown v

FT (t) ¼ 0:95

FT (t) ¼ 0:975

FT (t) ¼ 0:995

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

6.31375 2.91999 2.35336 2.13185 2.01505 1.94318 1.89458 1.85955 1.83311 1.81246 1.79588 1.78229 1.77093 1.76131 1.75305 1.74588 1.73961 1.73406 1.72913 1.72472

12.70620 4.30265 3.18245 2.77645 2.57058 2.44691 2.36462 2.30600 2.26216 2.22814 2.20099 2.17881 2.16037 2.14479 2.13145 2.11991 2.10982 2.10092 2.09302 2.08596

63.65674 9.92484 5.84091 4.60409 4.03214 3.70743 3.49948 3.35539 3.24984 3.16927 3.10581 3.05454 3.01228 2.97684 2.94671 2.92078 2.89823 2.87844 2.86093 2.84534

lower conﬁdence limit tL is pﬃﬃﬃ 1:871 tL ¼ m^ X t0:95 s^ X = n ¼ 224:5 2:01505 ¼ 223:872 6 and the upper conﬁdence limit tU with t0.05 ¼ – t0.95 is pﬃﬃﬃ 1:871 tU ¼ m^ X t0:05 s^ X = n ¼ 224:5 þ 2:01505 ¼ 225:128 6 Hence we are 90% conﬁdent that the true mean lies in the interval (223.872, 225.128) lb. On the other hand, if we are given that the true standard deviation sX ¼ 1.871 lb, then we

FIGURE 18.4.2

18.4

Interval Estimation (Conﬁdence Intervals)

369

use the Gaussian distribution for conﬁdence intervals given by Eq. (18.4.6), and the lower and upper conﬁdence intervals are pﬃﬃﬃ 1:871 ¼ 223:987 zL ¼ m^ X z0:95 sX = n ¼ 224:5 1:645 6 pﬃﬃﬃ 1:871 ¼ 225:013 zU ¼ m^ X z0:05 sX = n ¼ 224:5 þ 1:645 6 The conﬁdence interval (223.987, 225.013) for the Gaussian is a little tighter than (223.872, 225.128) as obtained for the Student-t, showing that the Student-t density is ﬂatter than the Gaussian as shown in Fig. 7.9.1. Example 18.4.3 In a probability examination for a class of 16 students, the following scores were recorded: X ¼ ½ 94

70 57

77

45

43

70 67

Estimated mean m^ X ¼

96

89

62

98 56

68

75

71

16 1 X Xi ¼ 71:125 16 i¼1

16 1 X (Xi 71:125)2 ¼ 283:183 15 i¼1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Standard error s^ X ¼ 283:183 ¼ 16:828

Estimated variance s^ 2X ¼

We have to ﬁnd the 90% conﬁdence limits for estimated mean m^ X ¼ 71:125. From the Student-t tables (Table 18.4.2) the t value for 15 degrees of freedom and 95% probability is 1.753. Hence the lower conﬁdence limit is given by pﬃﬃﬃ 16:828 ¼ 63:75 tL ¼ m^ X t0:95 s^ X = n ¼ 71:125 1:753 16 and the upper conﬁdence limit is pﬃﬃﬃ 16:828 tL ¼ m^ X t0:05 s^ X = n ¼ 71:125 þ 1:753 ¼ 78:5 16 Thus, we are 90% conﬁdent that the true mean will be in the interval (63.75, 78.5) or within +7.375 of the estimated mean 71.125. Conﬁdence Interval for the Unknown Variance (Unknown Mean). We have already seen in Eq. (18.4.10) that the random variable (n 1)s^ 2X =s2X is chi-squaredistributed with (n 2 1) degrees of freedom. This density is not symmetric, and the interval estimate will not be centered at s2, unlike the Gaussian and Studentt densities. To determine the upper and lower conﬁdence limits, we can write (n 1)s^ 2X 2 2 x1a=2,(n1) ¼ 1 a ¼ g P xa=2,(n1) , s2X from which it follows that ( ) x2a=2,(n1) x21a=2;(n1) 1 ¼g , P (n 1)s^ 2X s2X (n 1)s^ 2X

370

Estimation Theory

TABLE 18.4.3 Chi-Square Tables with u ¼ x2 Values Shown g ¼ 0.9 v 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

g ¼ 0.95

g ¼ 0.99

FU (u) ¼ 0:05 FU (u) ¼ 0:95 FU (u) ¼ 0:025 FU (u) ¼ 0:975 FU (u) ¼ 0:005 FU (u) ¼ 0:995 0.00393 0.10259 0.35185 0.71072 1.14548 1.63538 2.16735 2.73264 3.32511 3.94030 4.57481 5.22603 5.89186 6.57063 7.26094 7.96165 8.67176 9.39046 10.11701 10.85081

3.84146 5.99146 7.81473 9.48773 11.0705 12.59159 14.06714 15.50731 16.91898 18.30704 19.67506 21.02601 22.36199 23.68475 24.99576 26.29620 27.58709 28.86928 30.14351 31.41042

and ﬁnally

0.00098 0.05064 0.21580 0.48442 0.83121 1.23734 1.68987 2.17973 2.70039 3.24697 3.81575 4.40379 5.00875 5.62873 6.26214 6.90766 7.56419 8.23075 8.90652 9.59078

(

5.02389 7.37776 9.34840 11.14329 12.83250 14.44938 16.01276 17.53455 19.02277 20.48306 21.91996 23.33660 24.73555 26.11891 27.48836 28.84532 30.19099 31.52636 32.85231 34.16959

(n 1)s^ 2X (n 1)s^ 2X , s2X 2 P 2 x1a=2,(n1) xa=2,(n1)

0.00004 0.01003 0.07172 0.20699 0.41174 0.67573 0.98926 1.34441 1.73493 2.15586 2.60322 3.07382 3.56503 4.07467 4.60092 5.14221 5.69722 6.26480 6.84397 7.43384

7.87944 10.59663 12.83816 14.86026 16.74960 18.54758 20.27774 21.95495 23.58935 25.18818 26.75685 28.29952 29.81947 31.31935 32.80132 34.26719 35.71847 37.15645 38.58226 39.99685

) ¼g

(18:4:12)

where x21a=2,(n1) and x2a=2,(n1) are constants to be determined from the chi-square table (Table 18.4.3) for (n – 1) degrees of freedom at probability levels of 1 a=2 and a=2 respectively. Equation (18.4.12) is to be read as the probability that the random interval ! (n 1)s^ 2X (n 1)s^ 2X , x21a=2,(n1) x2a=2,(n1) includes the population variance s2X is equal to g ¼ (1 –a). These conﬁdence limits are shown in Fig. 18.4.3 on a chi-square density with (n – 1) degrees of freedom. Example 18.4.4 We will ﬁnd the 90% conﬁdence interval for the sample variance s^ 2X ¼ 283:183 calculated in Example 18.4.3. From the chi-square table (Table 18.4.3), the x2 value for probability 0.05 and 15 degrees of freedom is 7.261, and the corresponding value for probability of 0.95 and 15 degrees of freedom is 24.996. Hence, from Eq. (18.4.12), the upper conﬁdence limit is given by x2U ¼

(n 1)s^ 2X 15 283:183 ¼ 585:008 ¼ 7:261 x2a=2,(n1)

18.4

Interval Estimation (Conﬁdence Intervals)

371

FIGURE 18.4.3

and the lower conﬁdence limit is given by x2L ¼

(n 1)s^ 2X 15 283:183 ¼ ¼ 169:937 24:996 x21a=2,(n1)

The corresponding 90% conﬁdence limits for the standard error s^ X ¼ 16:828 are pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ( 169:937 ¼ 13:036, 585:008 ¼ 24:187). This result is to be interpreted as the probability of the standard deviation sX of the students’ scores will be in the interval (13.036, 24.187) is 90%. Conﬁdence Interval for the Coefﬁcients of Simple Linear Regression. Since the residual variance s^ 2w is chi-square-distributed with (n 2 2) degrees of freedom the estimated variances s^ 2a^ 1 and s^ 2a^ 2 for the regression coefﬁcients a^ 1 and a^ 2 of the regression line y^ i ¼ a^ 1 þ a^ 2 xi , i ¼ 1, . . . ; n are also chi-square-distributed with (n 2 2) degrees of freedom as given in Eqs. (18.3.51). The terms a^ 1 and a^ 1 are unbiased a1 2, and hence the quantities (^ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃestimators of a1 and paﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a1 )=(s^ a^ 1 = n 2) and (^a2 a2 )=(s^ a^ 2 = n 2) are Student-t-distributed with (n 2 2) degrees of freedom. From Eqs. (18.4.11), we can form equations for 100(1 2 a)% conﬁdence intervals (^a1 a1 ) t1a=2,(n2) ¼ 1 a ¼ g P ta=2,(n2) , s^ a^ 1 (18:4:13) (^a2 a2 ) P ta=2,(n2) , t1a=2,(n2) ¼ 1 a ¼ g s^ a^ 2 where t1a=2,(n2) is obtained from the t table (Table 18.4.2) for (n 2 2) degrees of freedom and probability (1 2 a/2). Equation (18.4.13) can be stated more formally as P{^a1 t1a=2,(n2) s^ a^ 1 , a1 a^ 1 ta=2,(n2) s^ a^ 1 } ¼ 1 a P{^a2 t1a=2,(n2) s^ a^ 2 , a1 a^ 2 ta=2,(n2) s^ a^ 2 } ¼ 1 a

(18:4:14)

and hence Confidence interval for a1 ¼ (^a1 t1a=2,(n2) s^ a^ 1 , a^ 1 ta=2,(n2) s^ a^ 1 ) (18:4:15) Confidence interval for a2 ¼ (^a2 t1a=2,(n2) s^ a^ 2 , a^ 2 ta=2,(n2) s^ a^ 2 )

372

Estimation Theory

Example 18.4.5 We will now ﬁnd 90% conﬁdence intervals for the estimated regression coefﬁcients of Example 18.3.3. The estimated regression coefﬁcients are given by a^ 1 ¼ 1:574 and a^ 2 ¼ 0:404, with estimated variances s^ 2a^ 1 ¼ 0:0704 s^ 2a^ 2 ¼ 0:0699. The standard errors for a^ 1 and a^ 2 are s^ a^ 1 ¼ 0:265 and s^ a^ 2 ¼ 0:264. From the t table (Table 18.4.2), for 18 degrees of freedom and probability of 0.95, t0.95,18 ¼ 1.7341. From Eq. (18.4.15) we can compute the 90% conﬁdence interval for a^ 1 as La^ 1 1:115 ¼ 1:574 1:734 0:265 ¼ Ua^ 1 2:034 and the 90% conﬁdence interval for a^ 2 is La^ 2 1:115 ¼ 0:404 1:734 0:264 ¼ Ua^ 2 0:054 Conﬁdence Region for Simple Linear Regression Line. line is given as follows: y^ i ¼ a^ 1 þ a^ 2 xi ,

The estimated regression

i ¼ 1, . . . , n

(18:3:49)

yi with estimated variance Since y^ i is a Gaussian-distributed unbiased estimator of pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s^ 2y^ i given by Eq. (18.3.54), the quantity (^yi yi )=(s^ y^ i = n 2) is Student-t-distributed with (n – 2) degrees of freedom. Hence, we can form equations for 100(1 – a)% conﬁdence intervals as y^ i yi t1a=2,(n2) ¼ 1 a ¼ g, i ¼ 1, . . . , n (18:4:16) P ta=2,(n2) , s^ y^ i where t1a=2,(n2) is obtained from the t table (Table 18.4.2) for (n – 2) degrees of freedom and probability (1 – a/2). Equation (18.4.16) can be rewritten as P{^yi t1a=2,(n2) s^ y^ i , yi y^ i ta=2,(n2) s^ y^ i } ¼ 1 a,

i ¼ 1, . . . , n (18:4:17)

and the lower bound of the 100(1 – a)% conﬁdence region for yi is given by Ly^ i ¼ y^ i t1a=2,(n2) s^ y^ i

(18:4:18)

with the upper bound of the region given by Uy^ i ¼ y^ i ta=2,(n2) s^ y^ i

(18:4:19)

Example 18.4.6 In Example 18.4.5 we will ﬁnd the 90% conﬁdence region for the regression line given by y^ i ¼ 1:574 þ 0:404xi ,

i ¼ 1, . . . , 20

From Example 18.3.3 the variance of the regression line is given by 1 (xi 0:905)2 , i ¼ 1, . . . , 20 þ s^ 2y^ i ¼ 0:263 3:7665 20 This value of s^ 2y^ i can be substituted in Eq. (18.4.18) to obtain the lower bound of the 90% conﬁdence region 1=2 1 (xi 0:905)2 , i ¼ 1, . . . , 2 þ Ly^ i (xi ) ¼ 1:574 þ 0:404xi 1:7341 0:263 3:7665 20

18.5

Hypothesis Testing (Binary)

373

FIGURE 18.4.4

and the upper bound of the region can be obtained from Eq. (18.4.19): 1=2 1 (xi 0:905)2 Uy^ i (xi ) ¼ 1:574 þ 0:404xi 1:7341 0:263 , i ¼ 1, . . . , 20 þ 3:7665 20 The conﬁdence region is shown in Fig. 18.4.4 along with the regression line y^ i and the data points.

B 18.5

HYPOTHESIS TESTING (BINARY)

We have discussed some estimation problems in the previous sections. However, many probability models require us to make decisions about populations on the basis of limited knowledge. For example, a physician has to make a decision based on knowledge of her patient whether to give a particular drug. Usually this decision is binary in nature, that is, to give or not to give. Thus, we have a decision problem as opposed to an estimation problem. Such decision problems come under the category of hypothesis testing. We postulate a carefully formed assertion based on available information. This assertion is called the null hypothesis, H0. If we are interested only in accepting it or rejecting it at a signiﬁcance level a, then it is called signiﬁcance testing. On the other hand, we may be interested in postulating an alternate hypothesis H1 and ﬁnding methods of accepting H0 or H1. This is called hypothesis testing. For example, we may be interested in testing the null hypothesis H0 that the mean time between failures u (MTBF) of a memory chip is u0 hours. The alternate hypothesis may be u ¼ u1 or u, u0. We are not particularly interested in the alternate hypothesis u . u0 because it does not matter whether the chip overperforms. On the other hand, in the communication receiver we can hypothesize that the noise variance is u0. We are now interested in the alternate hypothesis u . u0 because we do not want the variance to be too high, while we are not interested in u , u0 because it does not affect the performance for low variances. The experiment on which we test a null hypothesis is based on an n random vector X, fXi, i ¼ 1, . . . , ng from a population as in the estimation problem. The n random vector will span an n-dimensional Euclidean space R n. We divide this n-space of observations into two mutually exclusive regions, R(H0) and R(H1). If the observed vector x, fxi, i ¼ 1, . . . , ng, called the test statistic, lies in the space R(H1), we reject the null

374

Estimation Theory

TABLE 18.5.1 True Situation H0 H0 H0 H0

true, H1 false true, H1 false false, H1 true false, H1 true

Accept or Reject

Decision Type

Accept H0 Reject H0 Accept H0 Reject H0

Correct decision—no error Type I error—a Type II error—b Correct decision—no error

hypothesis H0. On the other hand, if it falls in the region R(H0), we may not reject H0. The region R(H0) is called the acceptance region, and R(H1) is called the critical region or rejection region. Whether we accept or reject the null hypothesis, there is always the possibility of errors. The error created when H0 is rejected when it is true is called type I error and is denoted by a, which is also known as the signiﬁcance level. Other names for this error are false alarm and producer’s risk. The error created when H0 is accepted when it is false is called type II error and is denoted by b. Other terms for this error are missed alarm and consumer’s risk. The four possibilities, two of which are correct decisions and the other two, erroneous decisions, are shown in Table 18.5.1. A simple hypothesis is one in which the parameters are speciﬁed exactly. If they are not, it is called a composite hypothesis. A null hypothesis is almost always simple. For example, H0 ¼ u0 is simple whereas H1 . u0 is composite. In testing hypotheses we usually control the signiﬁcance level a a priori and try to minimize the type II error b. Example 18.5.1 (Signiﬁcance Testing) A pharmaceutical company wants to test the effectiveness of the new antihypertensive drug that they are developing. A group of n ¼ 25 volunteers are selected with a mean systolic pressure of m25 ¼ 150 mmHg with a standard deviation of s25 ¼ 25 mmHg. The company wants to see whether the new drug can bring the systolic pressure below 140 mmHg. We have to design a test to determine whether the drug is effective at a signiﬁcance level of a ¼ 0.01. We will assume that the systolic pressure is a Gaussian-distributed random variable X with mean mX ¼ 150 and standard deviation sX ¼ 25. Hence the sample mean m^ 25 (X) of systolic pressures afterptaking the drug is Gaussian-distributed with mean 150 and standard ﬃﬃﬃﬃﬃ deviation sm^ X ¼ 25= 25 ¼ 5 as shown in Fig. 18.5.1. We have to form the null

FIGURE 18.5.1

18.5

Hypothesis Testing (Binary)

375

hypothesis H0 carefully. If we choose the null hypothesis that the sample mean m^ 25 (X) after taking the drug is the same as the mean value of 150, then the drug has minimal effect on the systolic pressure. If we now choose a region R such that P{m^ 25 (X) [ R} ¼ P{m^ 25 (X) ra } ¼ 0:01 and the null hypothesis is rejected at a signiﬁcance level of 1%, then we can conclude that the drug is effective in controlling systolic blood pressure. From Eq. (18.4.2) the test statistic is Z¼

m^ X (X) mX m^ X (X) mX pﬃﬃﬃ ¼ sm^ X sX = n

and we can write a one-sided equation: m^ X mX m^ X 150 pﬃﬃﬃ za ¼ P za ¼ a ¼ 0:01 P 5 sX = n Referring to Table 18.4.1 the value of za corresponding to a probability of 0.01 is 22.326. Hence m^ X 150 ¼ 2:326 5

or m^ X ¼ 150 5 2:326 ¼ 138:37

so that ra ¼ 138.37 as shown in Fig. 18.5.1. Thus, we can reject the hypothesis H0 that the drug is ineffective in controlling the systolic blood pressure if the sample mean of the pressure measurements drops below 139. On the other hand, if we increase the signiﬁcance level to 2.5%, then z0.025 ¼ 21.96 and m^ X 150 ¼ 1:96 5

or

m^ X ¼ 150 5 1:96 ¼ 140:2

and we cannot reject the hypothesis at the higher signiﬁcance level of 2.5% and conclude that the drug is ineffective at that level. Thus, the signiﬁcance level has to be ﬁxed before the test begins. We shall now consider several cases of testing hypotheses. The form of the density function of a random variable X is a known function fX (x,u) that depends on a parameter u. Case 1: H0 ¼ u0 and H1 ¼ u1. Assuming a Gaussian random variable, consider a simple hypothesis H0 ¼ u0 against a simple alternate H1 ¼ u1. The decision to accept or reject H0 is dependent on the test statistic x obtained from the population. The probability densities fX(x,u0) and fX(x,u1) and the error regions a and b are shown in Fig. 18.5.2. If x , c, the null hypothesis H0 will be accepted, and if x . c, H0 will be rejected. The critical region R(H1) is x . c. The type I error a depends on the hypothesis under test H0, and the type II error b depends on both H0 and the alternate hypothesis H1. We can also see from Fig. 18.5.2 that we cannot minimize both types of error simultaneously since decreasing a increases b and vice versa. Case 2: H0 ¼ u0 and H1 ¼ u . u0 (One-Sided Test). In this case we have a simple hypothesis H0 ¼ u0 and a composite alternate H1 ¼ u . u0. The acceptance region for H0 is x , c. Here u ranges from u0 to 1 and the error probability b

376

Estimation Theory

FIGURE 18.5.2

will be a function of u. The Gaussian probability densities along with the errors a and b(u) are shown in Fig. 18.5.3. The acceptance and rejection regions are also shown in ﬁgure. The critical region will still be x . c, but the type II error function b(u) will depend on the closeness of u to u0. The function b(u) is called the operating characteristic (OC) (Fig.18.5.4) of the test. The OC shows the probability of a wrong decision. It can be visualized by sliding the density function fX(x,u) under H1 to the left starting from u0 and integrating it from 1 to c, or ðc ðc 2 1 pﬃﬃﬃﬃﬃﬃ e(1=2)½(xu)=s dx fX (x,u)dx ¼ (18:5:1) b(u) ¼ 2p:s 1 1 The complementary function P(u) ¼ 1 b(u) is called the power of the test. It shows the probability of a correct decision. The operating characteristic (OC) is shown in Fig. 18.5.4. Case 3: H0 ¼ u0 and H1 = u0 (Two-Sided Test). Here, we have a simple hypothesis H0 ¼ u0 and a composite alternate H1 ¼ {u = u0 g. The acceptance region is c1 , x , c2. The signiﬁcance level a is split between the two tails of the

FIGURE 18.5.3

18.5

Hypothesis Testing (Binary)

377

FIGURE 18.5.4

FIGURE 18.5.5

FIGURE 18.5.6

density under H0 so that P{m(x) ^ , za=2 } ¼ c1 and P{m(x) ^ , z1a=2 } ¼ c2 . Here u ranges from c1 to c2, and the error probability b will be a function of u. The Gaussian probability densities along with the errors a and b(u) are shown in Fig. 18.5.5.

378

Estimation Theory

The critical region is x , c1 and x . c2, but the type II error function b(u) (OC) extends from 1 to 1 (Fig. 18.5.6) and the value will depend on the closeness of u to u0. It can again be visualized by sliding the density function fX(x, u) under H1 to the left, starting from c1 and integrating it from c1 to c2: ð c2 ð c2 2 1 pﬃﬃﬃﬃﬃﬃ e(1=2)½(xu)=s dx fX (x,u)dx ¼ (18:5:2) b(u) ¼ 2p:s c1 c1 The acceptance region is also shown in Fig. 18.5.5 and the two-sided OC curve, in Fig. 18.5.6. Procedure for Hypothesis Testing

The following procedures are established for credible testing of hypotheses: 1. Null hypothesis H0 must be carefully formulated. 2. Deﬁne the experiment to be performed. 3. Choose a proper test statistic. 4. Establish the signiﬁcance level a. 5. Choose a value for type II error b. 6. With a and b speciﬁed, determine a suitable sample size n. We will now apply hypothesis testing to the mean and variance of a population. We will have n realizations {x1 , . . . , xn ) of a Gaussian random variable P X whose mean m we want to hypothesize. The sample mean m^ is, as usual, m^ ¼ (1=n) n xi , and the variance of the sample mean s^ 2 ¼ s2 =n. The density of m^ will be given by pﬃﬃ 1 (1=2)½(xm)=(s= n)2 (18:5:3) fm^ (x) ¼ pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ e 2p s= n These quantities will be used to form the test statistic.

Mean m—known Variance s

We will now test the hypothesis that the mean m is H0 ¼ m0 against the alternates: (1) H1 ¼ m , m0 , (2) H1 ¼ m . m0 , and (3) H1 ¼ m = m0 . The test statistic we will use is the random variable Z¼

m^ m0 pﬃﬃﬃ s= n

(18:5:4)

with a signiﬁcance level of a. Under the hypothesis H0, the random variable Z is Gaussian with zero mean and unit variance. Note that za for which P(Z za ) ¼ a will be negative for a , 0:5 and z1a for which P(Z z1a ) ¼ 1 a will be positive, and if the pdf is symmetric, then za ¼ z1a . 1. H1 ¼ m , m0 : Accept H0 if Z . za. Or, from Eq. (18.5.4) Z¼

m^ m0 pﬃﬃﬃ . za s= n

or

pﬃﬃﬃ m^ . m0 þ za s= n

(18:5:5)

pﬃﬃﬃ The acceptance region R(H pﬃﬃﬃ 0 ) ¼ {m^ . m0 þ za s= n} and thepﬃﬃﬃrejection region R(H1 ) ¼ {m^ , m0 þ za s= n}. The critical point c ¼ m0 þ za s= n} The OC curve

18.5

Hypothesis Testing (Binary)

is obtained from Eq. (18.5.1): ð1 ð1 pﬃﬃ 1 (1=2)½(xm)=(s= n)2 pﬃﬃﬃﬃﬃﬃ b(m) ¼ fm^ (x, m)dx ¼ dx pﬃﬃﬃ e 2p s= n c c 1 1 m0 m pﬃﬃﬃ þ za ¼ 1 erf pﬃﬃﬃ 2 2 s= n

379

(18:5:6)

where ðx erf(x) ¼

2 2 pﬃﬃﬃﬃ ej dj p 0

(18:5:7)

2. H1 ¼ m . m0 : In this case we accept H0 if Z . z12a. Or Z¼

m^ m0 pﬃﬃﬃ , z1a s= n

or

pﬃﬃﬃ m^ . m0 þ z1a s= n

(18:5:8)

pﬃﬃﬃ ^ , m0 þ z1a s= n} and the rejection region The acceptance region R(Hp 0 )ﬃﬃﬃ¼ {m pﬃﬃﬃ R(H1 ) ¼ {m^ . m0 þ z1a s= n}. The critical point c ¼ m0 þ z1a s= n The OC curve is obtained from Eq. (18.5.1): ðc ðc pﬃﬃ 1 (1=2)½(xm)=(s= n)2 pﬃﬃﬃﬃﬃﬃ fm^ (x, m)dx ¼ dx b(m) ¼ pﬃﬃﬃ e 2p s= n 1 1 ¼

1 1 m0 m pﬃﬃﬃ þ z1a 1 þ erf pﬃﬃﬃ 2 2 s= n

(18:5:9)

where erf(x) is given by Eq. (18.5.7) 3. H1 ¼ m = m0 : In the previous two cases we had one-sided hypothesis testing, whereas here we will have two-sided hypothesis testing. As seen previously, the signiﬁcance level a is split evenly between the two tails of the density under H0 so that the probability is a/2 on each tail. We now accept H0 if za=2 , Z , z1a=2 , or za=2 ,

m^ m0 pﬃﬃﬃ , z1a=2 s= n

or

pﬃﬃﬃ pﬃﬃﬃ m0 þ za=2 s= n , m^ , m0 þ z1a=2 s= n (18:5:10)

pﬃﬃﬃ pﬃﬃﬃ The acceptance region R(H0 ) ¼ {m0 þ za=2 s= n} and the pﬃﬃﬃ n , m^ , m0 þ z1a=2pﬃﬃs= ﬃ ,ﬃﬃﬃm0 þ za=2 s= n m^ . m0 p þﬃﬃzﬃ1a=2 s= n}. The critical rejection region R(H1 ) ¼ {m^ p points are c1 ¼ m0 þ za=2 s= n and c2 ¼ m0 þ z1a=2 s= n. The OC curve is obtained from Eq. (18.5.2): ð c2 ð c2 pﬃﬃ 1 (1=2)½(xm)=(s= n)2 pﬃﬃﬃﬃﬃﬃ fm^ (x,m)dx ¼ dx b(m) ¼ pﬃﬃﬃ e 2p s= n c1 c1 1 1 m0 m 1 m0 m p ﬃﬃ ﬃ p ﬃﬃ ﬃ p ﬃﬃ ﬃ p ﬃﬃ ﬃ þ z1a=2 erf þ za=2 ¼ erf (18:5:11) 2 2 s= n 2 s= n where erf(x) is as given by Eq. (18.5.7). Mean m—Unknown Variance s2

The procedure with an unknown variance is similar to the one with the known variance discussed earlier. We replace the unknown variance with the estimated variance

380

Estimation Theory

s^ 2X and use the Student-t distribution for analysis. The test statistic is m^ m0 pﬃﬃﬃ T¼ s= ^ n

(18:5:12)

and this is Student-t-distributed with (n 2 1) degrees of freedom. The critical regions for the three cases discussed earlier are 1. H1 ¼ m , m0

t(n1),a s^ pﬃﬃﬃ R(H0 ) ¼ m^ . m0 þ n t(n1),a s^ pﬃﬃﬃ Rejection region: R(H1 ) ¼ m^ , m0 þ n

Acceptance region:

2. H1 ¼ m . m0

(18:5:13)

Acceptance region: Rejection region:

t(n1),ð1aÞ s^ pﬃﬃﬃ R(H0 ) ¼ m^ , m0 þ n t(n1),ð1aÞ s^ p ﬃﬃ ﬃ R(H1 ) ¼ m^ . m0 þ n

(18:5:14)

3. H1 ¼ m = m0

t(n1),a=2 s^ t(n1),(1a=2) s^ pﬃﬃﬃ pﬃﬃﬃ , m^ , m0 þ Acceptance region: R(H0 ) ¼ m0 þ n n (18:5:15) ) t(n1),(a=2) s^ t(n1),(1a=2) s^ pﬃﬃﬃ pﬃﬃﬃ ; m^ . m0 þ Rejection region: R(H1 ) ¼ m^ , m0 þ n n (

(18:5:16) To determine the OC curves in these three cases we have to integrate a noncentral t density, and this is beyond the scope of our discussion. However, for large n, the t distribution becomes Gaussian and the OC curves can be approximated by Eqs. (18.5.6), (18.5.9), and (18.5.11). Example 18.5.2 The following diastolic blood pressure measurements were recorded for a patient on 10 different occasions. The pressure measurements are assumed to be Gaussian-distributed: BP measurements

78

85

88

85

78

86

79

82

80

81

We want to ﬁnd whether these measurements support the hypothesis that the pressure is less than 80 at the signiﬁcance levels of (1) 1%, (2) 2.5%, and (3) 5%. This problem is a little different from what we discussed earlier because both hypotheses P 1 (78 þ þ 81) ¼ 82:2 and the estiare composite. The estimated mean value m^ ¼ 10 mated standard deviation s^ ¼ 3:584. We choose the statistic given by Eq. (18.5.12) m^ m0 pﬃﬃﬃ T¼ s= ^ n which is Student-t-distributed. The t values for 1%, 2.5%, and 5% signiﬁcance levels for (10 2 1) ¼ 9 degrees of freedom from the Student-t table (Table 18.4.2) are 2.8214,

18.5

Hypothesis Testing (Binary)

381

2.2622, and 1.8331 respectively. We can now formulate the composite hypotheses as (1) null hypothesis H0 : m , m0 ¼ 80; and (2) alternate hypothesis H1 : m m0 ¼ 80: The acceptance regions from Eq. (18.5.14) are pﬃﬃﬃ ^ n} 1% significance: R(H0 ) ¼ {m^ , m0 þ t(n1), (1a) s= 3:5839 ¼ 80 þ 2:8214 pﬃﬃﬃﬃﬃ ¼ 83:2 10 pﬃﬃﬃﬃ 2.5% signiﬁcance: R(H0 ) ¼ m^ , 80 þ 2:2622 3:5839 ¼ 82:564 10 pﬃﬃﬃﬃ 5% signiﬁcance: R(H0 ) ¼ m^ , 80 þ 1:8331 3:5839 ¼ 82:078 10 Since the estimated mean is m^ ¼ 82:2 , 82:564 , 83:2, we conclude that the hypothesis with the diastolic pressure is 80 at signiﬁcance levels of 1% and 2.5% can be accepted. It cannot be accepted at the 5% level because m^ ¼ 82:2 . 82:078. We will not calculate the OC curve b(m) since this involves noncentral Student-t distribution. Example 18.5.3 An energy specialist recommends that a homeowner could save at least $15 a month in utilities if energy appliances were upgraded. Before the upgrade was installed, the mean monthly utility expense m0 was $255 with standard deviation s ¼ $17. After the upgrade, the mean monthly expenses (in U.S. dollars) for a 10-year period is shown below: 243 252 246 262 244 219 228 212 264 260 We have to test whether the upgrade has saved $15 a month at signiﬁcance levels of (1) 1%, (2) 2.5%, (3) 5%. We will assume that the utility expenses are Gaussian-distributed and formulate the null hypothesis H0 as mean monthly expense m0 ¼ $255 and the alternate hypothesis H1 as mean monthly expense m1 ¼ $240. If H0 is rejected and H1 is accepted at signiﬁcance levels of a ¼ 0.01,0.025,0.05, then the homeowner would have saved $15 a month. The 1 ½243 þ þ 260 ¼ estimated mean value m^ of the average monthly expense m^ ¼ 10 243 with s given as 17. The density functions of m^ for H0 and H1 are given by pﬃﬃﬃﬃ 2 1 pﬃﬃﬃﬃﬃ e(1=2)½(x255)=(17= 10) fm^ (x j H0 ) ¼ pﬃﬃﬃﬃﬃﬃ 2p 17= 10 pﬃﬃﬃﬃ 2 1 pﬃﬃﬃﬃﬃ e(1=2)½(x240)=(17= 10) fm^ (x j H1 ) ¼ pﬃﬃﬃﬃﬃﬃ 2p 17= 10 and are shown in Fig. 18.5.7. The test statistic is chosen as the normalized Gaussian m^ m0 m^ 255 pﬃﬃﬃﬃﬃ pﬃﬃﬃ ¼ Z¼ s= n 17= 10 and the simple hypotheses H0: m ¼ m0 ¼ 255 and H1: m ¼ m1 ¼ 240 at signiﬁcance levels of 1%, 2.5%, and 5% can be tested. The corresponding z values from the Gaussian table (Table 18.4.1) are z0.01 ¼ 22.326, z0.025 ¼ 21.96, and z0.05 ¼ 21.645 respectively. The acceptance regions from Eq. (18.5.6) are pﬃﬃﬃ 1% significance: R(H0 ) ¼ {m^ . m0 þ za s= n} pﬃﬃﬃﬃﬃ ¼ 255 2:326 (17= 10) ¼ 242:49

382

Estimation Theory

FIGURE 18.5.7

pﬃﬃﬃ 2:5% significance: R(H0 ) ¼ {m^ . m0 þ za s= n} pﬃﬃﬃﬃﬃ ¼ 255 1:96 (17= 10) ¼ 244:47 pﬃﬃﬃ 5% significance: R(H0 ) ¼ {m^ . m0 þ za s= n} pﬃﬃﬃﬃﬃ ¼ 255 1:645 (17= 10) ¼ 246:16 pﬃﬃﬃ The critical point c ¼ m0 þ za s= n. The 5% signiﬁcance region and the estimated mean m^ ¼ 243 are shown in Fig. 18.5.7. Since the estimated mean is m^ ¼ 243 . 242:5, we conclude that the null hypothesis m0 ¼ $255 can be accepted at the 1% level with consequent no average monthly savings of $15. On the other hand, m^ ¼ 243 , 244:47 , 246:16, and the null hypothesis is rejected at the 2.5% and 5% levels and the alternate hypothesis m1 ¼ 240 is accepted, resulting in an average monthly savings of $15. The OC curve b(m,za) for signiﬁcance level a is obtained from Eq. (18.5.6) as 1 1 m0 m 1 1 255 m pﬃﬃﬃ þ za b(m, za ) ¼ 1 erf pﬃﬃﬃ 1 erf pﬃﬃﬃ þ za ¼ 2 2 2 s= n 2 5:4259 The type II errors for a ¼ 0.01, 0.025, 0.05 are obtained by substituting m ¼ 240 and z0.01 ¼ 22.326, z0.025 ¼ 21.96, z0.05 ¼ 21.645 in the equation above. Thus, b(240, 22.326) ¼ 0.3214, b(240, 21.96) ¼ 0.2032, b(240, 21.645) ¼ 0.1261. The operating characteristic curves shown in Fig. 18.5.8 are obtained from plotting b(m,za) for 1%, 2.5%, and 5% values of za. Number of Samples n for Given a and b. In testing hypotheses the usual practice is to ﬁx the type I error a at a value of 2.5% or 5% and minimize the type II error b. The only way to minimize b is to increase the sample size n. For the case of H0: m ¼ m0 and H1 ¼ m , m0, the critical point c for Gaussian distributions is found to be pﬃﬃﬃ c ¼ m0 þ za s= n

(18:5:17)

If the allowable b is also speciﬁed, we can then obtain the required number of samples n.

18.5

Hypothesis Testing (Binary)

383

FIGURE 18.5.8

Referring to Fig. 18.5.9, we require n to be such that ð1 pﬃﬃ 1 (1=2)½(xm1 )=(s= n)2 pﬃﬃﬃﬃﬃﬃ dx b pﬃﬃﬃ e 2p s= n c pﬃﬃﬃ Substituting j ¼ (x m1 )=(s= n) in this equation, we obtain ð1 2 1 pﬃﬃﬃﬃﬃﬃ e(1=2)j dj b cm1 2p pﬃ s= n

(18:5:18)

(18:5:19)

If the integral in Eq. (18.5.19) is to be b, then the lower limit c m1 pﬃﬃﬃ . z1b s= n

(18:5:20)

where z12b is obtained from the Gaussian tables for a probability of b. Substituting for c from Eq. (18.5.17) in Eq. (18.5.20) and simplifying, we have m0 m1 pﬃﬃﬃ . z1b za s= n

FIGURE 18.5.9

384

Estimation Theory

FIGURE 18.5.10

Thus the sample size n is given by n.

s2 (z1b za )2 (m0 m1 )2

(18:5:21)

Example 18.5.4 We will use the data from Example 18.5.2 and ﬁnd the required sample size for a ¼ 0.05, and we are also given s ¼ 17, m0 ¼ 255, and m1 ¼ 240. As the number of samples n is increased, the critical point c increases and b decreases and we stop at the value of n that gives the desired value of b. In this example, for n ¼ 5,10,15 we obtain c ¼ 242.49, 246.16, 247.78 and b ¼ 0.3715, 0.1261, 0.0382 respectively. These are shown in Fig. 18.5.10. If we choose the value of b ¼ 0.1261 from the previous example, we should get n ¼ 10. From Gaussian tables z0.05 ¼ 21.645 and z12b ¼ z0.8739 ¼ 1.145. Substituting these values in Eq. (18.5.21), we obtain n . ½172 (1:145 þ 1:645)2 =(255 240)2 ¼ 9:9982 or n ¼ 10. This corresponds to the 10 samples of Example 18.5.2.

B 18.6.

BAYESIAN ESTIMATION

The previous approaches of estimating a parameter u in the probability density f (x,u) were based solely on the random samples fX1, . . . , Xng that are incorporated in the test statistic. This methodology is called the classical or the objective approach. However, in some problems we may have prior information about u that can be used to reﬁne its estimate. For example, the classical estimate of the probability u in a binomial distribution b(k;n,u), where k is the number of successes in n trials, is u^ ¼ k=n. We can then view u as the realization of a random variable Q whose density function f (u) is the prior information. The density of X will be a conditional density f (x j u). The joint density of the sample fX1, . . . , Xng and the parameter Q can be given in terms of the conditional density as f (x1 , . . . , xn ,u) ¼ f (x1 , . . . , xn j u)f (u)

(18:6:1)

18.6. Bayesian Estimation

385

Using Bayes’ theorem, we can obtain the conditional density f (u j x1 , . . . , xn ) as f (u j x1 , . . . ; xn ) ¼ Posterior

f (x1 , . . . , xn , u) f (x1 , . . . , xn , ju) ¼ f (u) f (x1 , . . . , xn ) f (x1 , . . . , xn ) Prior

where the marginal density f (x1 , . . . , xn ) is given by ð1 f (x1 , . . . , xn ,u) du f (x1 , . . . , xn ) ¼

(18:6:2)

(18:6:3)

1

The density f (u j x1 , . . . , xn ) is called the posterior density. What we have accomplished is modifying the prior density f(u) by knowledge of the joint density of the random samples fX1, . . . , Xng to the posterior density. This method of estimation is called the subjective approach to estimation or Bayesian estimation. It is used in medical imaging problems, which are discussed in Chapter 23. We will now present Bayes’ estimation of the probability p of an event where p is a realization of a random variable X with probability density function fX( p) whose range is [0,1]. Prior estimates of p can be obtained from ð1 (18:6:4) p^ ¼ pfX ( p)dp 0

To improve on the estimate of p, we conduct an experiment of tossing a die n times and observing the number of aces to be k. We can apply Bayes’ theorem from Eq. (8.3.4) and write the posterior density as fX (p j B) ¼ Ð 1 0

P(B j X ¼ p)fX ( p) P(B j X ¼ p)fX ( p)dp

where B ¼ fk aces in n trialsg. From binomial probability, we obtain n k P(B j X ¼ p) ¼ p (1 p)nk k Substituting Eq. (18.6.6) into Eq. (18.6.5), we obtain n k p (1 p)nk fX (p) k fX (p j B) ¼ Ð1 n nk k fX (p)dp 0 k p (1 p)

(18:6:5)

(18:6:6)

(18:6:7)

The updated estimate of p can be obtained by substituting fX ( p j B) from Eq. (18.6.7) for fX ( p) in Eq. (18.6.4), and the result is ð1 (18:6:8) p^ post ¼ pfX (p j B)dp 0

If we now assume that fX( p) is uniformly distributed in f0,1] instead of a general distribution in the range [0,1], Eq. (18.6.7) can be simpliﬁed. The integral ð1 m!(n k)! (18:6:9) pm (1 p)nk dp ¼ (n þ m k þ 1)! 0 can be shown to be true using mathematical induction.

386

Estimation Theory

Substituting fX( p) ¼ 1 and from Eq. (18.6.9), we can evaluate fX(B) as ð1 n! k!(n k)! 1 n fX (B) ¼ pk (1 p)nk 1dp ¼ ¼ k k!(n k)! (n þ 1)! n þ 1 0

(18:6:10)

we can express the conditional density fX( p j B) as follows: n! pk (1 p)nk 1 (n þ 1)! k k!(n k)! p (1 p)nk fX (p j B) ¼ ¼ 1 k!(n k)! (n þ 1)

(18:6:11)

The posterior estimate for p is obtained from Eq. (18.6.8) as ð (n þ 1)! 1 kþ1 p (1 p)nk dp p^ post ¼ k!(n k)! 0 and from Eq. (18.6.9) p^ post is given by p^ post ¼ Example 18.6.1

(n þ 1)! (k þ 1)!(n k)! k þ 1 ¼ k!(n k)! (n þ 2)! nþ2

(18:6:12)

We have to estimate the parameter Q in the following problem Yi ¼ Q þ Wi ,

i ¼ 1, . . . , n

where fWig are independent identically distributed zero mean Gaussian random variables with variance s2W and Q is also a Gaussian-distributed random variable with mean mQ and variance s2Q . We want to ﬁnd the Bayesian estimate m^ Q of Q. From Eq. (18.6.2), the conditional density of Q conditioned on the n-vector random variable Y can be given as f (y, u) f (y j u)f (u) f (y j u)f (u) ¼ ¼ Ð1 f (y) f (y) 1 f (y j u)f (u)du Ð1 Ð1 Since 1 f (u j y)du ¼ ½1=f (y) Ð 1 1 f (y j u)f (u)du ¼ 1, the factor f(y) is independent of u and normalizes the integral 1 f (y j u)f (u)du to 1. The joint density function f (y j Q ¼ u) is governed by the joint density of W, and hence f (u j y) ¼

f (y j u) ¼

n Y

n Y

2 1 pﬃﬃﬃﬃﬃﬃ e(1=2)½(yi u)=sW 2psW i¼1 Pn 1 (1=2) ½(yi u)=sW 2 i¼1 ¼ e n=2 n (2p) sW

f (yi j u) ¼

i¼1

with 2 1 f (u) ¼ pﬃﬃﬃﬃﬃﬃ e(1=2)(u=sQ ) 2psQ

Multiplying f (y j u) and f (u), we have " 2 #) n 1 X yi u 2 u f (y j u)f (u) ¼ f (y, u) ¼ exp þ 2 i¼1 sW sQ (2p)(nþ1)=2 snW sQ 1

(

18.6. Bayesian Estimation

The factor

n P

387

(yi u)2 can be written as,

i¼1 n X

n X

i¼1

i¼1

(yi u)2 ¼

( yi m^ Y þ m^ Y u)2 ¼

n X

(yi m^ Y )2 þ n(m^ Y u)

i¼1

since the cross product vanishes. Substitution of this result in f (y,u) yields ( " #) n 1 1 X yi m^ Y 2 f (y,u) ¼ exp sW 2 i¼1 (2p)(nþ1)=2 snW sQ 1 n 1 2 2 ( m ^ u) þ u exp 2 s2W Y s2Q

1 n 1 2 2 ¼ C exp (m^ u) þ 2 u 2 s2W Y sQ

2 2 1 n 2 2 nsQ þ sW ¼ C exp (m^ 2m^ Y u) þ u 2 s2W Y s2W s2Q

ns2Q m^ 2Y 1 ns2Q þ s2W 2 2ns2Q m^ Y u ¼ C exp u 2 þ 2 s2W s2Q nsQ þ s2W ns2Q þ s2W where C is a function of only the sample values fyig and the parameters. Completing the squares in u in the exponent and after some algebra, there results 8 2 32 9 2 > > ns m ^ > > > > > 6u 2 Q Y 2 7 > = < 7 ns þ s 16 W7 Q f (y,u) ¼ C exp 6 sW s Q 6 7 > > > 2 4 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 5 > > > > > ; : ns2Q þ s2W 2 1 nm^ 2Y exp ¼ K e(1=2)½(umu )=su 2 ns2Q þ s2W where the quantities, mu and su are deﬁned by mu ¼

ns2Q m^ Y , ns2Q þ s2W

sW sQ su ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ns2Q þ s2W

and K is another function independent of u. The marginal density f(y) can be given as ð1 pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ 2 1 pﬃﬃﬃﬃﬃﬃ e(1=2)½(umu )=su du ¼ K 2p su f (y) ¼ K 2p su 2p su 1 Substituting for f(y,u) and f(y) in the posterior density f(ujy), we obtain 2

f (u j y) ¼

2 f (y,u) K e(1=2)½(umu )=su 1 pﬃﬃﬃﬃﬃﬃ ¼ ¼ pﬃﬃﬃﬃﬃﬃ e(1=2)½(umu )=su f (y) K 2p su 2p su

388

Estimation Theory

Hence, the posterior density is also Gaussian-distributed with mean mu and variance s2u , and the conditional expectation of Q given the observed values of y using Bayesian estimation techniques is E½Q j y ¼ mu ¼

ns2Q m^ Y ns2Q þ s2W

Note that using classical estimation techniques where there is no knowledge of the parameters of Q, the conditional expectation of Q given the observed values of y is ^ ¼ m^ E½Q j y ¼ Q Y obtained by substituting s2Q ¼ 1 in the Bayesian estimate mu. Example 18.6.2 A machine of uncertain quality manufactures microchips. In the absence of any information, we shall assume a priori that the bad microchips are uniformly distributed in the interval [0,1]. We take a sample of 10 microchips and ﬁnd that 3 are bad. We want to estimate the probability of the bad microchips manufactured. 3 ¼ 0:3: We ﬁnd The prior probability of bad microchips from classical method ¼ 10 the posterior probability from Eq. (18.6.2). Here n ¼ 10 and k ¼ 3; hence the posterior probability is (3 þ 1)/(10 þ 2) ¼ 0.3333, indicating that the proportion of bad chips is worse than what we might obtain by the classical method. On the other hand, if the 2 ¼ 0:2. The Bayesian probnumber of bad chips out of 10 is only 2, then the prior will be 10 ability will yield (2 þ 1)/(10 þ 2) ¼ 0.25. In this case also, the proportion of bad chips is worse than that obtained from the classical method.

Maximum A Posteriori Probability (MAP) Estimation

We will now discuss the important problem of maximizing the posterior probability to arrive at a decision. A communication receiver with additive noise is given by xi ¼ si þ wi ,

i ¼ 1, . . . , n, or

x¼sþw

(18:6:13)

where fxig is the observation process deﬁned by the n-vector x, fsig is the signal process deﬁned by the n-vector s, and fwig is a zero mean random noise deﬁned by the n-vector w. Without loss in generality, we will assume that the signal vector is either all 0 or 1. We observe x and decide on one of the following two hypotheses: H0 : xi ¼ wi , i ¼ 1, . . . , n,

or x ¼ w:

H1 : xi ¼ si þ wi , i ¼ 1, . . . , n or

0 transmitted

x ¼ s þ w:

1 transmited

where H0 is the null hypothesis and H1 is the alternate hypothesis. The prior probabilities of these hypotheses are P(H0) and P(H1). The probability density functions of observing x conditioned on H0 and H1 are fX(x j H0) and fX (x j H1). The posterior probabilities of H0 and H1 after observing x are P(H0 j x) and P(H1 j x). The decision will be based on maximizing the posterior probabilities. We choose the following decision criterion: P(H0 j x) . P(H1 j x):

choose H0

P(H0 j x) , P(H1 j x):

choose H1

18.6. Bayesian Estimation

389

FIGURE 18.6.1

Or more succinctly, the decision criterion is H P(H1 jx) 1 _ P(H0 j x) H0

1

(18:6:14)

If fHi, i ¼ 1,2g are the two hypotheses, then, from Bayes’ rule, we obtain P(Hi j x) ¼

P(Hi , x) fX (x j Hi )P(Hi ) ¼ , fx (x) fX (x)

i ¼ 0,1

(18:6:15)

Substituting Eq. (18.6.15) in Eq. (18.6.14), the decision rule becomes fX (x j H1 )P(H1 ) H1 fX (x) _ fX (x j H0 )P(H0 ) H0 fX (x)

1

or

H fX (x j H1 )P(H1 ) 1 _ fX (x j H0 )P(H0 ) H0

1

(18:6:16)

Transposing the prior probabilities to the right of the inequality, we have L(x) ¼

H fX (x j H1 ) 1 P(H0 ) ¼g _ fX (x j H0 ) P(H1 ) H0

(18:6:17)

where L(x) is deﬁned as the likelihood function or likelihood ratio and g is the threshold value or the critical point. Since L(x) is the ratio of n-fold probability densities, we prefer to take logarithm of L(x). The inequality of Eq. (18.6.17) will be preserved since the logarithm is a monotone function. Thus the log likelihood ratio ln½L(x) is given by H1 P(H0 ) ln½L(x) _ ln ¼ ln g P(H1 ) H0

(18:6:18)

The inequality given by this equation maximizes the posterior probability. We can now represent the decision receiver as shown in Fig. 18.6.1.

Minimization of Average Probability of Error

In the previous section the posterior probability was maximized to arrive at a decision rule. We can approach this problem through minimizing the average probability of error. The probabilities of missed alarm a and false alarm b are given by a ¼ P(accept H1 j H0 true) ¼ P(H1 j H0 ) b ¼ P(accept H0 j H1 true) ¼ P(H0 j H1 ) a, b, and the acceptance and rejection regions are as shown in Fig. 18.6.2.

(18:6:19)

390

Estimation Theory

FIGURE 18.6.2

The probability of average error Pe is given by Pe ¼ P(average error) ¼ P(H1 j H0 )P(H0 ) þ P(H0 j H1 )P(H1 ) ð ð ¼ P(H0 ) fX (x j H0 )dx þ P(H1 ) fX (x j H1 )dx R1

(18:6:20)

R0

If R is the entire region of integration, then R ¼ R0 þ R1 and ð ð fX (x j H0 )dx ¼ fX (x j H1 )dx ¼ 1 R

(18:6:21)

R

Using Eq. (18.6.21), we can convert Eq. (18.6.20) into integration over the region R0 ð ð Pe ¼ P(H0 ) fX (x j H0 )dx fX (x j H0 )dx R

R0

ð

þ P(H1 )

fX (x j H1 )dx R0

ð ¼ P(H0 ) þ

½P(H1 )fX (x j H1 ) P(H0 )fX (x j H0 )dx

(18:6:22)

R0

where P(H0) is always positive, and minimizing Pe depends on the function g(x) ¼ P(H1 )fX (x j H1 ) P(H0 )fX (x j H0 ) under the integral sign. A plot of g(x) is shown in Fig. 18.6.3. From the ﬁgure, we obtain P(H1 )fX (x j H1 ) . P(H0 )fX (x j H0 ):

g(x) is positive ) x [ R1

P(H1 )fX (x j H1 ) , P(H0 )fX (x j H0 ):

g(x) is negative ) x [ R0

(18:6:23)

Hence we have the decision criterion H fX (x j H1 )P(H1 ) 1 _ fX (x j H0 )P(H0 ) H0

1

or

H fX (x j H1 ) 1 P(H0 ) _ fX (x j H0 ) P(H1 ) H0

which is exactly the same as Eq. (18.6.17) derived for the MAP criterion.

(18:6:24)

18.6. Bayesian Estimation

391

FIGURE 18.6.3 Maximum-Likelihood Estimation

We have already formulated the basic ML criterion in Eq. (18.1.7) as the maximization of the loglikelihood function ln L(x,u). We will discus this important estimation criterion more formally. X is a continuous random variable with density function fX (x;u), where u is a k-vector given by u ¼ [u1, u2, . . . , uk]T are parameters to be estimated by observing n (n . k) independent observations x1, x2, . . . , xn of the random variable X. The likelihood function L(x,u) is deﬁned by L(x,u) ¼

n Y

fX (xi ,u)

i¼1

and the loglikelihood function, by ln½L(x,u) ¼

n X

ln ½fX (xi ,u)

(18:6:25)

i¼1

Maximizing the loglikelihood function ln[L(x,u)] with respect to fui, i ¼ 1, . . . , kg and setting them equal to zero yields k equations given by @ {ln½L(x,u1 , . . . , uk )} ¼ 0, @ui

i ¼ 1, . . . , k

(18:6:26)

The estimates of the k parameters (u1,u2, . . . , uk) are obtained by solving the k equations in Eq. (18.6.26). We will brieﬂy discuss the properties of ML estimators: 1. They provide a consistent approach to parameter estimation problems. 2. They can be used to generate hypothesis tests for parameters. 3. The likelihood equations need to be speciﬁcally worked out for any given distribution and estimation problems. 4. The numerical estimation problem is usually nontrivial. Simple solutions are possible in only a few cases. 5. Maximum-likelihood estimates can be heavily biased for small samples. Example 18.6.3 In an election we want to estimate the proportion p of voters who will vote Democratic. We will assign 1 if a person votes Democratic and 0 if he

392

Estimation Theory

does not vote Democratic. The probability f (x;p) for each person is a Bernoulli distribution with f (x;p) ¼ px (1 p)1x ,

x ¼ 0,1;

0p1

The value of p has to be estimated using the ML rule based on an opinion poll of n voters chosen at random. Thus, there will be n sample values x1, x2, . . . , xn, each with probability f (xi ;p) ¼ pxi (1 p)1xi ,

xi ¼ 0,1; 0 p 1; i ¼ 1, . . . , n

The likelihood function and the loglikelihood functions are given by n Pn Pn Y L(x;p) ¼ pxi (1 p)1xi ¼ p i¼1 xi (1 p)n i¼1 xi i¼1

ln½L(x;p) ¼

n X

xi ln(p) þ n

i¼1

n X

! xi ln(1 p)

i¼1

Differentiating ln[L(x;p)] with respect to p, we have

P Pn n ni¼1 xi @ i¼1 xi ¼0 ln½L(x;p) ¼ 1p p @p We obtain the estimate p^ by solving for p in this equation: p^ ¼

n 1X xi ¼ m^ X n i¼1

where p^ is an unbiased estimator because E½^p ¼ E½m^ X ¼ p 1 þ (1 p) 0 ¼ p. Example 18.6.4 Telephone calls coming into an exchange are Poisson-distributed with parameter l and are given by fX (x;l) ¼ el

lx , x!

x ¼ 0,1, . . .

where l is to be estimated using the ML rule by observing a random sample x1, x2, . . . , xn, of n telephone calls, each with probability fX (xi ;l) ¼ el

lxi , xi !

i ¼ 1,2, . . . , n

The likelihood function and the loglikelihood functions are given by Pn n xi Y enl l i¼1 xi l l ¼ Qn e L(x;l) ¼ xi ! i¼1 xi ! i¼1 ln½L(x;l) ¼ nl þ

n X i¼1

xi ln l

n X

ln (xi !)

i¼1

Differentiating ln[L(x; l)] with respect to l, we have Pn @ xi ln½L(x; l) ¼ n þ i¼1 ¼ 0 l @l

18.6. Bayesian Estimation

393

We obtain the estimate l^ by solving for l in the preceding equation: 1 l^ ¼ n

n X

xi ¼ m^ X

i¼1

^ ¼ E½m^ X ¼ l. where l^ is an unbiased estimator because E½l Example 18.6.5 A sample of observations fxi, i ¼ 1, . . . , ng drawn from a popu2 lation governed by a form of Weibull density is given by fX (x) ¼ 2lxelx u(x) with parameter l. We want to estimate lusing the ML criterion. The likelihood function is given by ½L(x;l) ¼

n Y

fX (xi ) ¼ (2l)n

i¼1

n Y

xi el

Pn

x2 i¼1 i

,

xi . 0

i¼1

and the loglikelihood function by ln½L(x;l) ¼ n ln(2) þ n ln(l) þ

n X

ln(xi ) l

i¼1

n X

x2i ,

xi . 0

i¼1

Differentiating ln[L(x;l)] with respect to l, we obtain n @ n X ln½L(x; l) ¼ x2 ¼ 0 @l l i¼1 i

Solving for l in this equation, we obtain the estimate l^ as n l^ ¼ Pn

2 i¼1 xi

¼

1 b m2

where

m2 ¼ E½X 2

^ 2 ¼ m2 , but E½1=b We have E½X 2 ¼ m2 ¼ 1=l and E½m m2 = 1=E½b m2 ¼ 1=m2 ¼ l, and hence l^ is a biased estimator. Example 18.6.6 A random sample of n is drawn from a Gaussian-distributed population of mean mX and variance s2X. We want to estimate the mean and the variance using the ML rule. The likelihood function and the loglikelihood function are shown below: n Y

2 1 2 pﬃﬃﬃﬃﬃﬃ e(1=2sX )(xi mX ) 2p s X i¼1 Pn 2 2 1 ¼ e(1=2sX ) i¼1 (xi mX ) n=2 n (2p) sX

L(x;mX , s2X ) ¼

n n n 1 X ln½L(x;mX ,s2X ) ¼ ln(2p) ln(s2X ) 2 (xi mx )2 2 2 2sX i¼1

394

Estimation Theory

Differentiating with respect to mX and s2X , and setting them equal to zero, we have n @ 1 X ln½L(x;mX , s2X ) ¼ 2 (xi mX ) ¼ 0 @mX sX i¼1 n @ n 1 X 2 ln½L(x;m , s ) ¼ þ (xi mX )2 ¼ 0 X X @s2X 2s2X 2s4X i¼1

Solving these equations, we ﬁnd the estimators as follows: m^ X ¼

n 1X xi n i¼1

and

s^ 2X ¼

n 1X (xi mX )2 n i¼1

In these estimators, while the estimator for the mean m^ X is unbiased, the estimator for the variance s^ 2X is biased because E½s^ 2X ¼ ½(n 1)=ns2X .

ML Rule in Hypothesis Testing

In the previous section we saw that the MAP rule minimizes the average probability of wrong decisions using the prior probabilities PfHi,i ¼ 0,1g of the two hypotheses. In many cases the priors may not be exactly known. In such a case, we choose whichever conditional probability P(x j H0) or P(x j H1) is greater, thereby making the assumption that both hypotheses H0 and H1 are equiprobable. We will show later in an example that when the priors are not known precisely, the decision errors will be more than that when the priors are equiprobable. This decision rule where the priors are assumed equiprobable is called the maximum-likelihood (ML) rule for testing hypotheses and is stated as follows: P(x j H0 ) . P(x j H1 ): P(x j H0 ) , P(x j H1 ):

accept H0 accept H1

(18:6:27)

In terms of density functions Eq. (18.6.27) can be given more succinctly as H fx (x j H1 ) 1 _ fx (xjH0 ) H0

1

(18:6:28)

In many applications the ML rule is widely used, especially when the average probability of error Pe is low. Further, the missed-alarm probability is much more important than the false-alarm probability. For example, missing the diagnosis of cancer has more serious consequences than does diagnosing cancer when there is none. The MAP rule is prone to yielding higher miss probabilities than is the ML rule as we will see in the following example. Example 18.6.7 The output x of a system consists of an input s ¼ 0 or 1 and a zero mean additive noise w with variance s2W ¼ 14. The prior probability of sending s ¼ 0 is P(s ¼ 0) ¼ 3 1 4 and that of s ¼ 1 is P(s ¼ 1) ¼ 4. We have to ﬁnd the following using both the MAP and the ML rules: (1) the decision threshold g, (2) the average probability of error Pe (x), and (3) the probabilities of missed alarm and false alarm. We can formulate the hypotheses as

395

18.6. Bayesian Estimation

FIGURE 18.6.4

follows: 3 4 1 H1 : {1 transmitted} with P(H1 ) ¼ 4

H0 : {0 transmitted} with P(H0 ) ¼

The conditional probabilities of x under the two hypotheses are 2 2 fX (x j H0 ) ¼ pﬃﬃﬃﬃﬃﬃ e(1=2)½4x 2p

2 2 fX (x j H1 ) ¼ pﬃﬃﬃﬃﬃﬃ e(1=2)½4(x1) 2p

and

These probability density functions are shown in Fig. 18.6.4. The likelihood ratio L(x) is given by 2

fX (x j H1 ) e(1=2)½4(x1) 2 2 L(x) ¼ ¼ (1=2)½4x2 ¼ e2½x 2xþ1x ¼ e2(2x1) fX (x j H0 ) e MAP Criterion. The decision rule from the loglikelihood function of Eq. (18.6.18) is ln½e

2(2x1)

H1 3=4 _ ln 1=4 H0

or

H1 1 ln 3 x_ þ 2 4 H0

or

H1 x _ 0:7747 H0

Thus, the critical point for MAP is g ¼ 0.7747 as shown in Fig. 18.6.4. The probability of average error Pe(x) for any value of x from Eq. (18.6.22) is given by ðx PeMAP (x) ¼ P(H0 ) þ

½P(H1 )fX (j j H1 ) P(H0 )fX (j j H0 )dj 1

ð 3 2 x 1 2(j1)2 3 2j2 p ﬃﬃﬃﬃﬃﬃ e ¼ þ e dj 4 4 2p 1 4 pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ 1 1 ¼ þ erf½ 2(x 1) 3 erf½ 2p ; 2 8

2 erf(x) ¼ pﬃﬃﬃﬃ p

ðx 0

2

ej dj

396

Estimation Theory

The minimum error is obtained when x ¼ g ¼ 0.7747 and Pe(min) ¼ 0.127. The probability of missed alarm is obtained from the following integral ðg PMAMAP ¼

2 2 pﬃﬃﬃﬃﬃﬃ e2(x1) dx ¼ 0:3261 2p 1

and the probability of false alarm from ð1 PFAMAP ¼ g

2 2 pﬃﬃﬃﬃﬃﬃ e2x dx ¼ 0:0607 2p

These probabilities are also shown in Fig. 18.6.4. The function gMAP (x) under the integral, that governs the extremum for PeMAP (x) is given by 2 2 1 3 2 gMAP (x) ¼ pﬃﬃﬃﬃﬃﬃ e2(x1) e2x 4 2p 4 ML Criterion. We will now obtain the same quantities using the ML criterion. Here the decision criterion becomes H1 1 x+ 2 H0 and the critical point for ML is 0.5. Hence the probability of error PeML (x) is given by ð 1 1 x PeML (x) ¼ þ ½ fX (j j H1 ) fX (j j H0 ) dj 2 2 1 ð 2 2 1 1 x ½e2(j1) e2j dj ¼ þ pﬃﬃﬃﬃﬃﬃ 2 2p 1 h hpﬃﬃﬃﬃﬃii pﬃﬃﬃ 1 1 ¼ þ erf 2(x 1) erf 2x 2 4 The minimum error is obtained when x ¼ g ¼ 0.5 and Pe(min) ¼ 0.159. The probability of missed alarm is obtained from the following integral ð 0:5 2 2 pﬃﬃﬃﬃﬃﬃ e2(x1) dx ¼ 0:1587 PMAML ¼ 2p 1 and the probability of false alarm from, ð1 2 2 pﬃﬃﬃﬃﬃﬃ e2x dx ¼ 0:1587 PFAML ¼ 2p 0:5 Note that both probabilities are the same for the maximum-likelihood criterion. The function gML(x) under the integral that governs the extremum for PeML (x) is given by i 2 2 h 2 gML (x) ¼ pﬃﬃﬃﬃﬃﬃ e2(x1) e2x 2p The average probabilities of error PeMAP (x) and PeML (x) are shown in Fig. 18.6.5. Table 18.6.1 compares the two decision criteria.

18.6. Bayesian Estimation

397

FIGURE 18.6.5

TABLE 18.6.1 Category P(H0)—probability of null hypothesis P(H1)—probability of alternate hypothesis g—critical point Pe(min)—minimum probability of average error PMA—probability of missed alarm PFA—probability of false alarm

MAP Rule

ML Rule

0.75 0.25 0.7747 0.127 0.3261 0.0607

0.5 0.5 0.5 0.159 0.1587 0.1587

Comparing the two methods, we see that the minimum probability of average error is smaller for the MAP than for the ML. However, the probability of the more important missed alarm is higher for MAP than for ML. Thus, in most communication problems the ML criterion is the preferred method of estimation. The functions gMAP (x) and gML (x) for both MAP and ML criteria are shown in Fig. 18.6.6.

FIGURE 18.6.6

398

Estimation Theory

In Fig. 18.6.6 the function gML(x) corresponding to the ML rule has an odd symmetry, and hence it integrates to zero. The critical point is 12, as seen before. Example 18.6.8 (Discrete Probability) An extrusion company has contracted to supply plastic shopping bags to supermarkets. Ordinarily, the probability of tearing in this group A0 of bags is 0.01. Because of a production malfunction, a second group A1 of bags have a higher probability of 0.05 of tearing. The prior probabilities of choosing A0 or A1 are 0.7 and 0.3 respectively. To ﬁnd out which group of bags came from A1, the company tests a sample of n bags sequentially until it comes across a ﬁrst failure. From this information it must decide whether the bags came from group A0 or A1. Using both the MAP and the ML criteria, we have to ﬁnd (1) the number n of bags required to be tested, (2) the probability of average error Pe(n), and (3) probabilities of missed and false alarms PMA and PFA. We will ﬁrst approach the problem in generality and later substitute numbers to obtain results. Let q0 be the tearing probability in group A0 and q1 be the tearing probability in group A1. The two hypotheses can be deﬁned by H0 : H1 :

bag came from group A0 bag came from group A1

(18:6:29)

The conditional probabilities of the waiting times upto the ﬁrst tearing are geometrically distributed and are given by P(n j H0 ) ¼ q0 (1 q0 )n1 ;

P(n j H1 ) ¼ q1 (1 q1 )n1

(18:6:30)

MAP Criterion. According to the MAP criterion of Eq. (18.6.14), the decision rule is P(H1 j n) H 1 _1 P(H0 j n) H 0

or

P(n j H1 )P(H1 ) H 1 _1 P(n j H0 )P(H0 ) H 0

or

P(n j H1 )P(H1 ) H1 P(n) _1 P(n j H0 )P(H0 ) H 0 P(n)

or (18:6:31)

P(n j H1 ) H 1 P(H0 ) _ P(n j H0 ) H 0 P(H1 )

Substituting the conditional probabilities from Eq. (18.6.30) into Eq. (18.6.31), we have q1 1 q1 n1 H 1 P(H0 ) _ q 0 1 q0 H 0 P(H1 )

(18:6:32)

Taking logarithms on both sides of Eq. (18.6.32) and simplifying, we obtain q0 P(H0 ) H 1 ln q1 P(H1 ) (n 1) _ 1 q1 H0 ln 1 q0

or

H1 n _ nMAP H0

q0 P(H0 ) ln q1 P(H1 ) ¼1þ 1 q1 ln 1 q0

(18:6:33)

18.6. Bayesian Estimation

399

where nMAP is the critical value of n. Thus we have the decision criterion for n: If n . nMAP : accept hypothesis H1 If n , nMAP : accept hypothesis H0

(18:6:34)

This MAP decision criterion gives us the minimum number n that are needed to be tested sequentially. Probabilities of Missed and False Alarms and Average Error. We will now calculate the probabilities of missed and false alarms as functions of n. The probability of missed alarm PMA(n) is given by PMA (n) ¼ P(N . n j H1 ) ¼

1 X

q1 (1 q1 )k1

k¼nþ1

¼1

n X

q1 (1 q1 )k1 ¼ 1 q1

k¼1

1 (1 q1 )n ¼ (1 q1 )n 1 (1 q1 )

(18:6:35)

where we have used the result for the ﬁnite summation of a geometric series. Similarly, the probability of false alarm PFA (n) is given by PFA (n) ¼ P(N n j H0 ) ¼

n X

q0 (1 q0 )k1

k¼1 n

¼ q0

1 (1 q0 ) ¼ 1 (1 q0 )n 1 (1 q0 )

(18:6:36)

Finally, the probability of average error PeMAP(n) is given by PeMAP (n) ¼ (1 q1 )n P(H1 ) þ ½1 (1 q0 )n P(H0 )

(18:6:37)

We obtain the various probabilities by substituting n ¼ nMAP in Eqs. (18.6.35) – (18.6.37). ML Criterion. For the ML criterion, the hypotheses H0 and H1 are equiprobable, and substituting P(H0) ¼ P(H1) in Eq. (18.6.36), we obtain q0 ln H1 q n _ nML ¼ 1 þ 1 (18:6:38) 1 q1 H0 ln 1 q0 with the decision criterion similar to that of Eq. (18.6.37). Equations for the probabilities of missed and false alarms for the ML criterion as functions of n are exactly the same as those of the MAP criterion except that we substitute n ¼ nML. The probability of average error for the ML criterion can be obtained from Eq. (18.6.37) by substituting P(H0) ¼ P(H1) ¼ 12, and the resulting equation is PeML (n) ¼

1 1 þ (1 q1 )n (1 q0 )n 2 2

(18:6:39)

Numerical Solutions. The conditional probability functions P(n j H0) ¼ 0.01 (1 2 0.01)n and P(n j H1) ¼ 0.05 (1 2 0.05)n are shown in Fig. 18.6.7.

400

Estimation Theory

FIGURE 18.6.7

Substituting for the MAP criterion, q0 ¼ 0.01, q1 ¼ 0.05, P(H0) ¼ 0.7, and P(H1) ¼ 0.3 in Eq. (18.6.39), we obtain the critical value nMAP: 0:01 0:7 ln 0:05 0:3 ¼ 19:4793 nMAP ¼ 1 þ 1 0:05 ln 1 0:01 Or, using the MAP criterion, the minimum number of bags to test sequentially is n ¼ 20. The missed-alarm probability is obtained from Eq. (18.6.38): PMA (n) ¼ (1 q1 )n ¼ 0:95n , and the a error is 0.9520 ¼ 0.3585. The false-alarm probability is obtained from Eq. (18.6.39): PFA (n) ¼ 1 (1 q0 )n ¼ 1 0:99n , and the b error is 1 2 0.9920 ¼ 0.1821. The probability of average error PeMAP(n) is obtained from Eq. (18.6.37): PeMAP (n) ¼ 0:95n 0:3 þ (1 0:99n ) 0:7 and PeMAP(min) ¼ PeMAP(20) ¼ 0.235. For the ML criterion, we substitute q0 ¼ 0.01, q1 ¼ 0.05, P(H0) ¼ 0.5, and P(H1) ¼ 0.5 in Eq. (18.6.38) and obtain the critical value nML: 0:01 ln 0:05 ¼ 40:0233 nML ¼ 1 þ 1 0:05 ln 1 0:01 Or, using the ML criterion, the minimum number of bags to test sequentially is n ¼ 41. The missed-alarm probability is obtained from Eq. (18.6.38): PMA (n) ¼ 1 (1 q1 )n ¼ 0:95n , and the a error is 0.9541 ¼ 0.1221. The false-alarm probability is obtained from Eq. (18.6.39): PFA (n) ¼ 1 (1 q0 )n ¼ 1 0:99n , and the b error is 1 2 0.9941 ¼ 0.3377. The probability of average error PeMAP(n) is obtained from Eq. (18.6.39): PeML (n) ¼ 12 þ 12 ½0:95n þ 0:99n and PeML(min) ¼ PeML(41) ¼ 0.2299. Probabilities PeMAP (n) and PeML (n) are shown in Fig. 18.6.8 along with the minimum number of bags needed for both the MAP and ML criteria. The results indicate that the number of bags needed using the MAP is half the number needed for ML. The more important missed-alarm probability for the ML criterion

18.6. Bayesian Estimation

401

FIGURE 18.6.8

is considerably lower than that of the MAP criterion. The ML criterion is more often used in communication problems. The MAP criterion does ﬁnd applications in medical imaging problems such as single-photon emission computed tomography (SPECT).

Hypothesis Testing with Added Cost for Errors

In the previous sections we never considered the costs associated with type I and type II errors. We have already mentioned that the missed-alarm or type I error is more serious than the false-alarm or type II error. We will now form a decision rule that takes into account the costs associated with the errors. We shall postulate the true hypotheses by H0 and H1 and the deﬁne the accepted hypotheses by A0 and A1. There will be four pairs of decisions, two of which, fA0,H0g and fA1,H1g, are correct decisions and the other two of which, fA0,H1g (missed alarm) and fA1,H0g (false alarm) are incorrect decisions. We will associate a cost Cij for each pair fAi,Hj,i,j ¼ 1,2g of the four decisions. The average cost of the decisions C for the MAP criterion is C ¼ C00 P(A0 , H0 ) þ C11 P(A1 ,H1 ) þ C01 P(A0 ,H1 ) þ C10 P(A1 ,H0 )

(18:6:40)

where P(Ai,Hj) i,j ¼ 1,2 represents the joint probability of accepting Ai, given that Hj is true. We have to form a decision rule that minimizes the average cost. Expressing Eq. (18.6.40) in terms of conditional probabilities, we have C ¼ C00 P(A0 j H0 )P(H0 ) þ C01 P(A0 j H1 )P(H1 ) þ C11 P(A1 j H1 )P(H1 ) þ C10 P(A1 j H0 )P(H0 )

(18:6:41)

where C01 is the cost of the missed alarm and C10 is the cost of the false alarm. Substituting conditional density functions for the conditional probabilities, there results ð ð C MAP ¼ C00 P(H0 ) fX (x j H0 )dx þ C01 P(H1 ) fX (x j H1 )dx R0

R0

ð

ð

þ C11 P(H1 )

fX (x j H1 )dx þ C10 P(H0 ) R1

fX (x j H0 )dx R1

(18:6:42)

402

Estimation Theory

We can transform the integrals in Eq. (18.6.42) to be over the same region R0: ð ð C MAP ¼ C00 P(H0 ) fX (x j H0 )dx þ C01 P(H1 ) fX (x j H1 )dx R0

R0

ð ð fX (x j H1 )dx þ C10 P(H0 ) 1 fX (x j H0 )dx (18:6:43) þ C11 P(H1 ) 1 R0

R0

Combining like terms in Eq. (18.6.43), we obtain ð CMAP ¼ C11 P(H1 ) þ C10 P(H0 ) þ ½P(H1 )(C01 C11 )fX (x j H1 ) R0

P(H0 )(C10 C00 )fX (x j H0 )dx

(18:6:44)

At this point it is natural to assume that the cost for an incorrect decision exceeds the cost for a correct decision, or C10 . C00 and C01 . C11. With this assumption, the terms P(H1 )ðC01 C11 ÞfX fX (x j H1 ) and P(H0 )ðC10 C00 ÞfX (x j H0 ) are both positive. Hence, if the integrand I ¼ P(H1 )(C01 C11 ÞfX (x j H1 ) PðH0 Þ(C10 C00 )fX (x j H0 ) is negative, then we can conclude that x [ R0 and choose H0. Thus the decision rule for the MAP criterion is fX (x j H1 )P(H1 )(C01 C11 ) H 1 _1 fX (x j H0 )P(H0 )(C10 C00 ) H 0 or L(x) ¼

fX (x j H1 ) H 1 P(H0 )(C10 C11 ) _ fX (x j H0 ) H 0 P(H1 )(C01 C11 )

(18:6:45)

and the decision rule for the ML criterion can be obtained by substituting P(H0) ¼ P(H1) in Eq. (18.6.45): L(x) ¼

fX (x j H1 ) H 1 C10 C00 _ fX (x j H0 ) H 0 C01 C11

(18:6:46)

In Eqs. (18.6.45) and (18.6.46), C10 is the cost associated with a false alarm and C01 is for a missed alarm. We weigh the missed alarm more heavily than the false alarm with the result C01 . C10. Hence the ratio (C10 C00 =(C01 C11 ) will always be lesser than 1 if the costs associated with right decisions are the same. Example 18.6.9 We will reexamine Example 18.6.7, assigning cost functions to wrong decisions. We will assume that the costs for the right decisions are the same, C00 ¼ C11 ¼ 1, and the cost for a missed alarm to be C01 ¼ 4C00 and the cost for a false alarm, C10 ¼ 2C00. The prior probabilities are P(H0) ¼ 34 and P(H1) ¼ 14. The conditional probabilities of x from Example 18.6.7 under the two hypotheses are 2 2 fX (x j H0 ) ¼ pﬃﬃﬃﬃﬃﬃ e(1=2)½4x 2p

and

2 2 fX (x j H1 ) ¼ pﬃﬃﬃﬃﬃﬃ e(1=2)½4(x1) 2p

The likelihood ratio L(x) is L(x) ¼

fX (x j H1 ) ¼ e2(2x1) fX (x j H0 )

18.6. Bayesian Estimation

403

From Eq. (18.6.45), the MAP criterion for minimization of cost functions is H 1 P(H0 )(C10 C00 ) 3C00 (2 1) ¼ ¼1 L(x) ¼ e2(2x1) _ C00 (4 1) H 0 P(H1 )(C01 C11 Þ The critical point from the loglikelihood function can be obtained as follows: H1 2(2x 1) _ 0 H0

or

H1 1 x_ H0 2

or gMAP ¼

1 2

The average cost function C MAP (x) obtained from Eq. (18.6.44) is given by ð 1 3 2 x CMAP (x) ¼ C00 þ 2 C00 þ pﬃﬃﬃﬃﬃﬃ 4 4 2p 1 1 3 2(j1)2 2j2 (4 1)e (2 1)e dx 4 4 i pﬃﬃﬃ o 7 3 n h pﬃﬃﬃ ¼ C00 þ erf 2(x 1) erf 2x 4 8 Assuming that C00 ¼ 1, the minimum cost function C MAP(min) ¼ CMAP (0:5) ¼ 1:238. The probability of missed alarm is obtained from the following integral ð gMAP 2 2 1 1 pﬃﬃﬃﬃﬃﬃ e2(x1) dx ¼ 1 erf pﬃﬃﬃ ¼ 0:1587 PMAMAP ¼ 2 2p 2 1 and the probability of false alarm, from ð1 2 2x2 1 1 pﬃﬃﬃﬃﬃﬃ e dx ¼ 1 erf pﬃﬃﬃ ¼ 0:1587 PFAMAP ¼ 2 2p 2 gMAP Similarly, the ML criterion for minimization of cost functions can be obtained by substituting P(H0) ¼ P(H1) ¼ 12 in Eq. (18.6.44). The result is H 1 C10 C00 C00 (2 1) 1 ¼ ¼ L(x) ¼ e2(2x1) _ H 0 C01 C11 C00 (4 1) 3 The critical point from the loglikelihood function can be obtained as follows: H 1 1 H 1 1 ln (3) or gML ¼ 0:2253 2(2x 1) _ ln or _ 3 4 H0 H0 2 The average cost function C ML (x) is ðx 2 2 C00 2 (1 þ 2) þ pﬃﬃﬃﬃﬃﬃ C00 (4 1)e2(j1) (2 1)e2j dx 2 2 2p 1 h i 1 h pﬃﬃﬃ i pﬃﬃﬃ 3 ¼ C00 2 þ erf 2(x 1) erf 2x 4 4

CML (x) ¼

Assuming that C00 ¼ 1, the minimum cost function C ML(min) ¼ CML (0:2253) ¼ 1:254. The probability of missed alarm is obtained from the following integral ð gML 2 2 pﬃﬃﬃﬃﬃﬃ e2(x1) dx ¼ 0:0607 PMAML ¼ 2p 1

404

Estimation Theory

FIGURE 18.6.9

and the probability of false alarm, from ð1 PFAML ¼

2 2 pﬃﬃﬃﬃﬃﬃ e2x dx ¼ 0:3261 2p gML

The probability density functions, the critical point gML, and the error regions for the ML criterion are shown in Fig. 18.6.9. The cost functions C MAP (x) and C ML (x) along with their critical points gMAP and gML are shown in Fig. 18.6.10. The function gMAP(x) under the integral that governs the extremum for PeMAP(x) is 2 2 1 gMAP (x) ¼ pﬃﬃﬃﬃﬃﬃ e2(j1) 3e2j 2 2p

FIGURE 18.6.10

18.6. Bayesian Estimation

FIGURE 18.6.11

TABLE 18.6.2 Category P(H0)—probability of null hypothesis P(H1)—probability of alternate hypothesis C00 ¼ C11—cost of correct decisions C01—cost of missed alarm C10—cost of false alarm g—critical point C min —minimum cost of average error PMA—probability of missed alarm PFA—probability of false alarm

MAP Rule

ML Rule

0.75 0.25 1 4 2 0.5 1.238 0.1587 0.1587

0.5 0.5 1 4 2 0.2253 1.254 0.0607 0.3261

The function gML(x) under the integral, which governs the extremum for PeML(x), is 2 2 1 gML (x) ¼ pﬃﬃﬃﬃﬃﬃ 3e2(j1) e2j 2 2p These two functions gMAP(x) and gML(x) are shown in Fig. 18.6.11. Table 18.6.2 compares the two decision criteria.

405

CHAPTER

19

Random Processes

B 19.1

BASIC DEFINITIONS

In the earlier sections a random variable X was deﬁned as a function that maps every outcome ji of points in the sample space S to a number X(ji) on the real line R. A random process X(t) is a mapping that assigns a time function X(t,ji) to every outcome ji of points in the sample space S. Alternate names for random processes are stochastic processes and time series. More formally, a random process is a time function assigned for every outcome j [ S according to some rule X(t,j), t [ T, j [ S, where T is an index set of time. As in the case of a random variable, we suppress j and deﬁne a random process by X(t). If the index set T is countably inﬁnite, the random process is called a discrete-time process and is denoted by Xn. Referring to Fig. 19.1.1, a random process has the following interpretations: 1. X(j,t1) is random variable for a ﬁxed time t1. 2. X(ji,t) is a sample realization for any point ji in the sample space S. 3. X(ji,t2) is a number. 4. X(j,t) is a collection or ensemble of realizations and is called a random process. An important point to emphasize is that a random process is a ﬁnite or an inﬁnite ensemble of time functions and is not a single time function. Example 19.1.1 A fair coin is tossed. If heads come up, a sine wave x1(t) ¼ sin(5pt) is sent. If tails come up, then a ramp x2(t) ¼ t is sent. The resulting random process X(t) is an ensemble of two realizations, a sine wave and a ramp, and is shown in Fig. 19.1.2. The sample space S is discrete. Example 19.1.2 In this example a sine wave is in the form X(t) ¼ A sin(vt þ F), where F is a random variable uniformly distributed in the interval (0, 2p). Here the sample space is continuous, and the sequence of sine functions is shown in Fig. 19.1.3.

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

406

19.1

FIGURE 19.1.1

FIGURE 19.1.2

FIGURE 19.1.3

Basic Deﬁnitions

407

408

Random Processes

Distribution and Density Functions. Since a random process is a random variable for any ﬁxed time t, we can deﬁne a probability distribution and density functions as FX (x; t) ¼ P(j, t: X(j; t) x) for a fixed t

(19:1:1)

and fX (x; t) ¼

@ FX (x þ Dx; t) FX (x; t) FX (x; t) ¼ lim Dx!0 @x Dx ¼ lim P(x , X(t) x þ Dx) Dx!0

(19:1:2)

These are also called ﬁrst-order distribution and density functions, and in general, they are functions of time. Means and Variances. Analogous to random variables, we can deﬁne the mean of a random process as ð1 xfX (x; t)dx (19:1:3) mX (t) ¼ E ½X(t) ¼ 1

and the variance as s2X (t) ¼ E ½X(t) mX (t)2 ¼ E ½X 2 (t) m2X (t) ð1 ¼ ½x mX (t)2 fX (x; t)dx

(19:1:4)

1

where E½X 2 (t) ¼

ð1

x2 fX (x; t)dx

(19:1:5)

1

Since the density is a function of time, the means and variances of random processes are also functions of time.

Example 19.1.3 We shall now ﬁnd the distribution and density functions along with the 7 : mean and variance for the random process of Example 19.1.1 for times t ¼ 0, 12 , 10 t ¼ 0,

x1 (0) ¼ 0,

x2 (0) ¼ 0

At t ¼ 0 the mapping diagram from the sample space to the real line is shown in Fig. 19.1.4a along with the corresponding distribution and density functions. The mean value is given by mX (0) ¼ 0; 12 þ 0 12 ¼ 0. The variance is given by 2 sX (0) ¼ (0 0)2 12 þ (0 0)2 12 ¼ 0: 1 1 1 1 t ¼ , x1 ¼ 1, x2 ¼ 2 2 2 2 At t ¼ 12 the mapping diagram from the sample space to the real line is shown in Fig. 19.1.4b along with the corresponding distribution and density functions. The mean value is given by mX 12 ¼ 12 12 þ 1 12 ¼ 34 ¼ 0:75:

19.1

Basic Deﬁnitions

409

FIGURE 19.1.4

The variance is given by s2X t¼

1 2

7 , 10

2 2 1 34 12 þ 1 34 12 ¼ 16 ¼ 0:0625: 7 7 7 x1 ¼ 1, x2 ¼ 10 10 10

¼

1 2

7 the mapping diagram from the sample space to the real line is shown in At t ¼ 10 Fig. 19.1.4c along with the corresponding distribution and density functions.

7 3 ¼ 0:15 ¼ 7 1 1 1 ¼ 20 The mean value is given by mX 10 7 10 322 1 2 2 7 3 21 289 The variance is given by sX 10 ¼ 10 þ 20 2 þ 1 þ 20 2 ¼ 400 ¼ 0:7225: Example 19.1.4 A random process, given by X(t) ¼ A sin(vt), is shown in Fig. 19.1.5, where A is a random variable uniformly distributed in the interval (0,1].

410

Random Processes

FIGURE 19.1.5

The density and distribution functions of A are fA (a) ¼

2

1, 0 , a 1 0, otherwise

0, FA (a) ¼ 4 a, 1,

a0 0,a1 a.1

We have to ﬁnd the distribution function FX (x; t). For any given t, x ¼ asin(vt) is an equation to a straight line with slope sin(vt), and hence we can use the results of Examples 12.2.1 and 12.2.2 to solve for FX (x; t). The cases of sin(vt) . 0 and sin(vt) , 0 are shown in Fig. 19.1.6. Case 1: sin(vt) . 0. There are no points of intersection on the a axis for x 0, and hence FX(x; t) ¼ 0. For 0 , x sin(vt) we solve x ¼ a sin(vt) and obtain a ¼ x=½sin(vt). The region Ia for which a sin(vt) x is given by Ia ¼ {0 , a

FIGURE 19.1.6

19.1

Basic Deﬁnitions

411

x=½sin(vt)} (Fig. 19.1.6). Thus x x x FX (x; t) ¼ FA FA (0) ¼ FA ¼ sin(vt) sin(vt) sin(vt) Finally for x . sin(vt), the region Ia for which asin(vt) x, is given by Ia ¼ f0 , a 1g and FX (x; t) ¼ 1. Thus, for sin(vt) . 0, we have 2 0; x0 6 x , 0 , x sin(vt) FX (x; t) ¼ 4 sin(vt) 1, x . sin(vt) Case 2: sin(vt) , 0. The region Ia for which x . 0 is given by Ia ¼ f0 , a 1g, and hence FX (x; t) ¼ 1. For 2jsin(vt)j , x 0, we solve x ¼ –ajsin(vt)j and obtain a ¼ ½x=( j sin(vt)j). The region Ia for which –a jsin(vt)j x is given by x ,a1 Ia ¼ j sin(vt)j (Fig. 19.1.6). Thus, FX (x; t) ¼ FA (1) FA

x x ¼1 j sin(vt)j j sin(vt)j

Finally, for x – jsin(vt)j, the region Ia for which –a jsin(vt)j x is given by Ia ¼ f1 , a 1g and FX (x; t) ¼ 0. Thus, for sin(vt) , 0, we have 2 0, x j sin(vt)j x 61 , j sin(vt)j , x 0 FX (x; t) ¼ 4 j sin(vt)j 1, x.0 Case 3: sin(vt) ¼ 0. The region Ia for which x . 0 is given by Ia ¼ f0 , a 1g and FX (x; t) ¼ 1. For x 0, Ia ¼ 1 and FX (x; t) ¼ 0. Thus, for sin(vt) ¼ 0, we have 0, x 0 FX (x; t) ¼ 1, x . 0 Example 19.1.5 We shall now ﬁnd the distribution and density functions along with the mean and variance for the random process of Example 19.1.2 and see how this process differs from the previous ones. We are given that X(t) ¼ A sin(vt þ F), where A is a constant and fF(f) ¼ 1/2p in the interval (0, 2p) and we have to ﬁnd fX (x; t). We will solve this problem by (1) ﬁnding the distribution FX (x; t) and differentiating it, and (2) by direct determination of the density function. 1. Determination of Distribution Function FX (x; t). The distribution function for F is given by FF (f) ¼ (f=2p), 0 , f 2p. The two solutions for x ¼ A sin(vt þ f) are obtained from the two equations: sin(vt þ f1 ) ¼

x A

and

sin(p vt f2 ) ¼

x A

412

Random Processes

Hence the solutions are given by x vt f1 ¼ sin1 A

and

f2 ¼ p sin1

x A

and are shown in Fig. 19.1.7. For x –A, there are no points of intersection and hence FX (x; t) ¼ 0. For 2A , x A, the set of points along the f axis such that A sin(vt þ f) x is (0, f1] < (f2, 2p]. Hence FX (x; t) is given by FX (x; t) ¼ FF (f1 ) FF (0) þ FF (2p) FF (f2 ) h x io 1 n 1 x sin vt 0 þ 2p p sin1 vt ¼ 2p A A n o 1 x 1 x 1 ¼ 2 sin1 þ p ¼ sin1 þ , A , x A 2p A p A 2 Finally, for x . A, the entire curve A sin(vt þ f) is below x, and FX(x;t) ¼ 1. 2. Determination of Density Function fX (x; t) (a) The two solutions to x ¼ A sin(vt þ f) have been found earlier. (b) The absolute derivatives j@x=@fjjf1 and j@x=@fjjf2 are given by

h i

@x

¼ A cos(vt þ f1 ) ¼ A cos vt þ sin1 x vt

@f

A f1 p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h x i A2 x2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ A2 x 2 ¼A ¼ A cos sin1 A A and

h i

@x

¼ A cos(vt þ f2 ) ¼

A cos vt þ p sin1 x vt

@f

A f2 p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

h x i

A2 x2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

¼ A cos p sin1 ¼ A2 x 2

¼A A A

FIGURE 19.1.7

19.1

Basic Deﬁnitions

413

(c) With the two solutions for x, the density function fX (x; t) is given by Eq. (12.3.6): 1 1 1 1 þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ , A , x A fX (x; t) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 A x 2p 2p p A2 x 2 Integration of fX (x; t) gives the distribution function FX (x; t) 2 3 0, x A x 1 61 7 FX (x; t) ¼ 4 sin1 þ , A , x A 5 p A 2 1, x.A and these two solutions are exactly the same as before. For this random process, we ﬁnd that the density and the distribution functions are both independent of time. For A ¼ 1, they become 1 fX (x; t) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ , 1 , x 1 p 1 x2 2 3 0, x 1 6 sin1 (x) 7 1 FX (x; t) ¼ 6 þ , 1 , x 1 7 4 5 p 2 1, x.1 and these functions are shown in Fig. 19.1.8. Since fX (x) has even symmetry, the mean value is 0 and the variance is obtained from ð1 x2 1 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dx ¼ sX ¼ 2 2 1 p 1 x pﬃﬃﬃ The value of s ¼ +1= 2 is also shown in Fig. 19.1.8. Later we will discuss random processes whose density and distribution functions are independent of time.

FIGURE 19.1.8

414

Random Processes

Higher-Order Distribution Functions

If t1 and t2 are different times, then X1 ¼ X(t1) and X2 ¼ X(t2) are two different random variables as shown in Fig. 19.1.9. A second-order distribution function FX (x1 , x2 ; t1 , t2 ) for X1 and X2 can be deﬁned as FX (x1 , x2 ; t1 , t2 ) ¼ P½X(t1 ) x1 , X(t2 ) x2

(19:1:6)

and the second-order density function fX (x1 , x2 ; t1 , t2 ) as fX (x1 , x2 ; t1 , t2 ) ¼

@2 FX (x1 , x2 ; t1 , t2 ) @x1 @x2

(19:1:7)

Similarly, if t1, . . . , tn are different times, then an nth-order distribution function is deﬁned as FX (x1 , . . . , xn ; t1 , . . . , tn ) ¼ P½X(t1 ) x1 , . . . , X(tn ) xn

(19:1:8)

and the nth-order density function as fX (x1 , . . . , xn ; t1 , . . . , tn ) ¼

@n FX (x1 , . . . , xn ; t1 , . . . , tn ) @x1 @xn

(19:1:9)

A random variable is completely deﬁned if its distribution function is known. Similarly, a random process X(t) is completely deﬁned if its nth-order distribution is known for all n. Since this is not feasible for all n, a random process in general cannot be completely deﬁned. Hence, we usually restrict the deﬁnition to second-order distribution function, in which case it is called a second-order process. In a similar manner, we can deﬁne a joint distribution between two different random processes X(t) and Y(t) (Fig. 19.1.10) as given below, where the joint second-order distribution function FXY (x1 , y2 ; t1 , t2 ) for X(t1) and Y(t2) is deﬁned by FXY (x1 , y2 ; t1 , t2 ) ¼ P½X(t1 ) x1 , Y(t2 ) y2

(19:1:10)

and the joint density function fXY (x1 , y2 ; t1 , t2 ) as fXY (x1 , y2 ; t1 , t2 ) ¼

@2 FXY (x1 , y2 ; t1 , t2 ) @x1 @y2

FIGURE 19.1.9

(19:1:11)

19.1

Basic Deﬁnitions

415

FIGURE 19.1.10

The joint nth-order distribution function FXY (x1 , y2 ; t1 , t2 ) for X(t) and Y(t) is deﬁned by FXY (x1 , . . . , xn : y1 , . . . , yn ; t1 , . . . , tn ) ¼ P½X(t1 ) x1 , . . . , X(tn ) xn : Y(t1 ) y1 , . . . , Y(tn ) yn

(19:1:12)

and the corresponding nth-order density function, by fXY (x1 , . . . , xn : y1 , . . . , yn ; t1 , . . . , tn ) ¼

@2n FXY (x1 , . . . , xn : y1 , . . . , yn ; t1 , . . . , tn ) @x1 @xn @y1 @yn

(19:1:13)

Second Order Moments

In Eqs. (19.1.3) and (19.1.4) deﬁned mean and variance for a random process. The second moment of a random process has also been deﬁned in Eq. (19.1.5). Since X(t1) and X(t2) are random variables, various types of joint moments can be deﬁned. Autocorrelation. The autocorrelation function (AC) RX (t1, t2) is deﬁned as the expected value of the product X(t1) and X(t2): ð1 ð1 RX (t1 , t2 ) ¼ E½X(t1 )X(t2 ) ¼ x1 x2 fX (x1 , x2 ; t1 , t2 )dx1 dx2 (19:1:14) 1 1

By substituting t1 ¼ t2 ¼ t in Eq. (19.1.14), we can obtain the second moment or the average power of the random process: ð1 2 x2 fX (x; t)dx (19:1:15) RX (t) ¼ E ½X (t) ¼ 1

Autocovariance. The autocovariance function (ACF) CX (t1, t2) is deﬁned as the covariance between X(t1) and X(t2): CX (t1 , t2 ) ¼ E ½(X(t1 ) mX (t1 ))(X(t2 ) mX (t2 )) ð1 ð1 ¼ (x1 mX (t1 ))(x2 mX (t2 )) fX (x1 , x2 ; t1 , t2 )dx2 dx1 1 1

¼ E½X(t1 )X(t2 ) mX (t1 )mX (t2 )

(19:1:16)

416

Random Processes

Thus, from Eqs. (19.1.15) and (19.1.16) the interrelationships between AC and ACF are CX (t1 , t2 ) ¼ RX (t1 , t2 ) mX (t1 )mX (t2 ) RX (t1 , t2 ) ¼ CX (t1 , t2 ) þ mX (t1 )mX (t2 )

(19:1:17)

Normalized Autocovariance. The normalized autocovariance function (NACF) rX (t1, t2) is the ACF normalized by the variance and is deﬁned by CX (t1 , t2 ) CX (t1 , t2 ) rX (t1 , t2 ) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ s2X (t1 )s2X (t2 ) sX (t1 )sX (t2 )

(19:1:18)

The NACF ﬁnds wide applicability in many problems in random processes, particularly in time-series analysis. The following deﬁnitions pertain to two different random processes X(t) and Y(t): Cross-Correlation. The cross-correlation function (CC) RXY (t1, t2) is deﬁned as the expected value of the product X(t1) and Y (t2): ð1 ð1 RXY (t1 , t2 ) ¼ E½X(t1 )Y(t2 ) ¼ x1 y2 fXY (x1 , y2 ; t1 , t2 )dy2 dx1 (19:1:19) 1

1

By substituting t1 ¼ t2 ¼ t in Eq. (19.1.19), we can obtain the joint moment between the random processes X(t) and Y(t) as ð1 ð1 xy fXY (x, y; t)dydx (19:1:20) RXY (t) ¼ E½X(t)Y(t) ¼ 1

1

Cross-Covariance. The cross-covariance function (CCF) CXY (t1, t2) is deﬁned as the covariance between X(t1) and Y(t2): CXY (t1 , t2 ) ¼ E½(X(t1 ) mX (t1 ))(Y(t2 ) mY (t2 )) ð1 ð1 ¼ (x1 mX (t1 ))(y2 mY (t2 )) fXY (x1 ,y2 ; t1 , t2 )dy2 dx1 1 1

¼ E½X(t1 )Y(t2 ) mX (t1 )mY (t2 )

(19:1:21)

Thus, from Eqs. (19.1.19) and (19.1.21) the interrelationships between CC and CCF are CXY (t1 , t2 ) ¼ RXY (t1 , t2 ) mX (t1 )mY (t2 ) RXY (t1 , t2 ) ¼ CXY (t1 , t2 ) þ mX (t1 )mY (t2 )

(19:1:22)

Normalized Cross-Covariance. The normalized cross-covariance function (NCCF) rXY(t1, t2) is the CCF normalized by the variances and is deﬁned by CXY (t1 , t2 ) CXY (t1 , t2 ) rXY (t1 , t2 ) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 2 2 s X (t1 )sY (t2 ) sX (t1 )sY (t2 )

(19:1:23)

Some Properties of X(t) and Y(t)

Two random processes X(t) and Y(t) are independent if for all t1 and t2 FXY (x, y; t1 , t2 ) ¼ FX (x; t1 )FY (y; t2 )

(19:1:24)

19.1

Basic Deﬁnitions

417

They are uncorrelated if CXY (t1 , t2 ) ¼ RXY (t1 , t2 ) mX (t1 )mY (t2 ) ¼ 0 or RXY (t1 , t2 ) ¼ mX (t1 )mY (t2 )

for all t1 and t2

(19:1:25)

They are orthogonal if for all t1 and t2 RXY (t1 , t2 ) ¼ 0

(19:1:26)

Example 19.1.6 This example is slightly different from Example 19.1.4. A random process X(t) is given by X(t) ¼ A sin(vt þ f) as shown in Fig. 19.1.11, where A is a uniformly distributed random variable with mean mA and variance s2A. We will ﬁnd the mean, variance, autocorrelation, autocovariance, and normalized autocovariance of X(t). Mean: E½X(t) ¼ mX (t) ¼ E ½A sin(vt þ f) ¼ mA sin(vt þ f) Variance: var½X(t) ¼ s2X (t) ¼ E½A2 sin2 (vt þ f) m2A sin2 (vt þ f) ¼ {E ½A2 m2A } sin2 (vt þ f) ¼ s2A sin2 (vt þ f) Autocorrelation. From Eq. (19.1.14) we have RX (t1 , t2 ) ¼ E½X(t1 )X(t2 ) ¼ E ½A2 sin(vt1 þ f) sin(vt2 þ f) 1 ¼ E ½A2 {cos½v(t1 t2 ) cos½v(t1 þ t2 ) þ 2f} 2

FIGURE 19.1.11

418

Random Processes

Autocovariance. From Eq. (19.1.16) we have CX (t1 , t2 ) ¼ RX (t1 , t2 ) mX (t1 )mX (t2 ) ¼ E½A2 sin(vt1 þ f) sin(vt2 þ f) m2A sin(vt1 þ f) sin(vt2 þ f) ¼ s2A sin(vt1 þ f) sin(vt2 þ f) 1 ¼ s2A {cos½v(t1 t2 ) cos½v(t1 þ t2 ) þ 2f} 2 Normalized Autocovariance. From Eq. (19.1.18) we have rX (t1 , t2 ) ¼

CX (t1 , t2 ) s2 sin(vt1 þ f) sin(vt2 þ f) ¼ A2 ¼1 sX (t1 )sX (t2 ) sA sin(vt1 þ f) sin(vt2 þ f)

Example 19.1.7 A random process X(t) with k changes in a time interval t, and its probability mass function is given by p(k; l) ¼ elt ½(lt)k =k!. It is also known that the joint probability Pfk1 changes in t1, k2 changes in t2g is given by P{k1 changes in t1 , k2 changes in t2 } ¼ P{k1 changes in t1 , (k2 k1 ) changes in (t2 t1 )} ¼ P{k1 changes in t1 }P{(k2 k1 ) changes in (t2 t1 )} ¼ elt1

(lt1 )k1 l(t2 t1 ) ½l(t2 t1 )(k2 k1 ) e k1 ! (k2 k1 )!

We have to ﬁnd the mean, variance, autocorrelation, autocovariance, and normalized autocovariance of X(t): Mean: E ½X(t) ¼ mX (t) ¼

1 X

kelt

k¼0

(lt)k ¼ lt k!

Variance: var½X(t) ¼ s2X (t) ¼

1 X

k2 elt

k¼0

(lt)k (lt)2 ¼ lt k!

Autocorrelation. From Eq. (19.1.18) we have RX (t1 , t2 ) ¼ E½X(t1 )X(t2 ) ¼ E{X(t1 )½X(t2 ) X(t1 ) þ X(t1 )} ¼ E ½X 2 (t1 ) þ E ½X(t1 )E½X(t2 ) X(t1 ) (from condition given) ¼ (lt1 )2 þ lt1 þ lt1 l(t2 t1 ) ¼ l2 t1 t2 þ lt1 if t2 . t1 and we have a similar result if t1 . t2: RX (t1 , t2 ) ¼ l2 t1 t2 þ lt2

if

t1 . t2

19.1

Basic Deﬁnitions

419

Combining these two results, we have RX (t1 , t2 ) ¼ l2 t1 t2 þ l min (t1 , t2 ) Autocovariance. From Eq. (19.1.19) we have CX (t1 , t2 ) ¼ RX (t1 , t2 ) mX (t1 )mX (t2 ) ¼ l2 t1 t2 þ l min (t1 , t2 ) l2 t1 t2 ¼ l min (t1 , t2 ) Normalized Autocovariance. From Eq. (19.1.18) we have CX (t1 , t2 ) min (t1 , t2 ) 1 pﬃﬃﬃﬃﬃﬃﬃ ¼ min ¼ rX (t1 , t2 ) ¼ sX (t1 )sX (t2 ) l t1 t2 l

Example 19.1.8

rﬃﬃﬃﬃ rﬃﬃﬃﬃ t1 t2 , t2 t1

Two random processes X(t) and Y(t) are given by X(t) ¼ A sin(vt þ f1 );

Y(t) ¼ B sin(vt þ f2 )

where A and B are two random variables with parameters E[A] ¼ mA, E[B] ¼ mB, var[A] ¼ s2A, var[B] ¼ s2B, cov[AB] ¼ sAB, and correlation coefﬁcient rAB ¼ sAB = (sA sB ). We have to ﬁnd the means and variances of X(t) and Y(t) and their crosscorrelation, cross-covariance, and normalized cross-covariance. The means and variances can be obtained directly from Example 19.1.5. Means: mX (t) ¼ E½A sin(vt þ f1 ) ¼ mA sin(vt þ f1 ) mY (t) ¼ E ½B cos(vt þ f2 ) ¼ mB cos(vt þ f2 ) Variances: s2X (t) ¼ s2A sin2 (vt þ f1 ) s2Y (t) ¼ s2B cos2 (vt þ f2 ) Cross-Correlation: RXY (t1 , t2 ) ¼ E½X(t1 )Y(t2 ) ¼ E½AB sin(vt1 þ f1 ) cos(vt2 þ f2 ) 1 ¼ E ½AB{sin½v(t1 þ t2 ) þ f1 þ f2 þ sin½v(t1 t2 ) þ f1 f2 } 2 Cross-Covariance: CXY (t1 ,t2 ) ¼ RXY (t1 ,t2 ) mX (t1 )mY (t2 ) ¼ E ½ABsin(vt1 þ f1 )cos(vt2 þ f2 ) mA mB sin(vt1 þ f1 )cos(vt2 þ f2 ) ¼ sAB sin(vt1 þ f1 )cos(vt2 þ f2 ) 1 ¼ sAB {sin½v(t1 þ t2 ) þ f1 þ f2 þ sin½v(t1 t2 ) þ f1 f2 } 2 Normalized Cross-Covariance: rXY (t1 , t2 ) ¼

CXY (t1 , t2 ) sAB sin(vt1 þ f1 ) cos(vt2 þ f2 ) sAB ¼ ¼ ¼ rAB sX (t1 )sY (t2 ) sA sB sin(vt1 þ f1 ) cos(vt2 þ f2 ) sA sB

420

Random Processes

B 19.2

STATIONARY RANDOM PROCESSES

The distribution functions of all the random processes except that in Example 19.1.4 were dependent on time. However, many of the random processes such as Example 19.1.4 have the important property that their statistics do not change with time, which is an important step toward obtaining the statistics from a single sample function. The statistics in the time interval (t1, t2) is the same as in the time interval (t1 þ t, t2 þ t). In other words, the probabilities of the samples of a random process X(t) at times t1, . . . , tn will not differ from those at times t1 þ t, . . . , tn þ t. This means that the joint distribution function of X(t1), . . . , X(tn) is the same as X(t1 þ t), . . . , X(tn þ t), or FX (x1 , . . . , xn ; t1 , . . . , tn ) ¼ FX (x1 , . . . , xn ; t1 þ t, . . . , tn þ t)

(19:2:1)

and the corresponding density function may be written as fX (x1 , . . . , xn ; t1 , . . . , tn ) ¼ fX (x1 , . . . , xn ; t1 þ t, . . . , tn þ t)

(19:2:2)

Random processes with the property of Eq. (19.2.1) or (19.2.2) are called nth-order stationary processes. A strict-sense or strongly stationary process is a random process that satisﬁes Eqs. (19.2.1) and (19.2.2) for all n. Analogously, we can also deﬁne lower orders of stationarity. A random process is ﬁrst-order stationary if FX (x; t) ¼ FX (x; t þ t) ¼ FX (x) fX (x; t) ¼ fX (x; t þ t) ¼ fX (x)

(19:2:3)

and the distribution and density functions are independent of time. The random process in Examples 19.1.2 and 19.1.5 is an example of a ﬁrst-order stationary process. A random process is second-order stationary if FX (x1 , x2 ; t1 , t2 ) ¼ FX (x1 , x2 ; t1 þ t, t2 þ t) ¼ FX (x1 , x2 ; t) fX (x1 , x2 ; t1 , t2 ) ¼ fX (x1 , x2 ; t1 þ t, t2 þ t) ¼ fX (x1 , x2 ; t)

(19:2:4)

The distribution and density functions are dependent not on two time instants t1 and t2 but on the time difference t ¼ t1 – t2 only. Second-order stationary processes are also called wide-sense stationary or weakly stationary. Hereafter, stationary means wide-sense stationary, and strict-sense stationary will be speciﬁcally mentioned. In a similar manner, two processes X(t) and Y(t) are jointly stationary if for all n FXY (x1 , . . . , xn ; y1 , . . . , yn ; t1 , . . . , tn ) ¼ FXY (x1 , . . . , xn ; y1 , . . . , yn ; t1 þ t, . . . , tn þ t)

(19:2:5)

fXY (x1 , . . . , xn ; y1 , . . . , yn ; t1 , . . . , tn ) ¼ fXY (x1 , . . . , xn ; y1 , . . . , yn ; t1 þ t, . . . , tn þ t)

(19:2:6)

or

and they are jointly wide-sense stationary if FXY (x1 , y2 ; t1 , t2 ) ¼ FXY (x1 , y2 ; t1 þ t, t2 þ t) ¼ FXY (x1 , y2 ; t)

(19:2:7)

19.2

Stationary Random Processes

421

and the joint density function fXY (x1 , y2 ; t1 , t2 ) ¼ fXY (x1 , y2 ; t1 þ t, t2 þ t) ¼ fXY (x1 , y2 ; t)

(19:2:8)

nth-order stationarity implies lower-order stationarities. Strict-sense stationarity implies wide-sense stationarity. Similar to Eqs. (19.1.24) –(19.1.26), we can enumerate the following properties for stationary random processes X(t) and Y(t). Two random processes X(t) and Y(t) are independent if for all x and y FXY (x, y) ¼ FX (x)FY (y)

(19:2:9)

They are uncorrelated if for all t CXY (t) ¼ RXY (t) mX mY ¼ 0 or RXY (t) ¼ mX mY

(19:2:10)

They are orthogonal if for all t RXY (t) ¼ 0

(19:2:11)

Moments of Continuous-Time Stationary Processes

We can now deﬁne the various moments for a stationary random process X(t). Mean: ð1 E ½X(t) ¼ mX ¼

xf (x)dx

(19:2:12)

1

Variance: 2

E ½X(t) mX ¼

s2X

ð1 ¼

(x mX )2 f (x)dx ¼ E½X 2 (t) m2X

(19:2:13)

1

Autocorrelation: ð1 ð1 RX (t) ¼ E ½X(t)X(t þ t) ¼

x1 x2 f (x1 , x2 ; t)dx1 dx2

(19:2:14)

1 1

Autocovariance: CX (t) ¼ E{½X(t) mX ½X(t þ t) mX } ð1 ð1 ¼ (x1 mX )(x2 mX ) f (x1 , x2 ; t)dx1 dx2 1

(19:2:15)

1

¼ RX (t) m2X Normalized Autocovariance (NACF): rX (t) ¼

CX (t) s2X

(19:2:16)

422

Random Processes

White-Noise Process

A zero mean stationary random process X(t) whose autocovariance or autocorrelation is given by CX (t) ¼ RX (t) ¼ s2X d(t)

(19:2:17)

where d(t) is the Dirac delta function, is called a white-noise process. The energy of a white-noise process is inﬁnite since CX (0) ¼RX (0) ¼ E[X 2(t)] ¼ 1. Hence it is an idealization. White-noise processes ﬁnd extensive use in modeling communication systems. The cross-moments of two jointly stationary processes X(t) and Y(t) are deﬁned below: Cross-Correlation: ð1 ð1 RXY (t) ¼ E ½X(t)Y(t þ t) ¼

x1 y2 f (x1 , y2 ; t)dy2 dx1 1

(19:2:18)

1

Cross-Covariance: CXY (t) ¼ E{½X(t) mX ½Y(t þ t) mY } ð1 ð1 ¼ (x1 mX )(y2 mY ) f (x1 , y2 ; t)dy2 dx1 1 1

¼ RXY (t) mX mY

(19:2:19)

Normalized Cross-Covariance: rXY (t) ¼

CXY (t) sX sY

(19:2:20)

A stationary random process X(t) is passed through a linear system with impulse response h(t). The input – output relationship will be given by the convolution integral: ð1 ð1 Y(t) ¼ X(t a)h(a)da ¼ X(t)h(t a)da (19:2:21) 1

1

The cross-correlation function RXY (t) between the input and the output can be found as follows: ð1 X(t)X(t þ t a)h(a)da (19:2:22) E½X(t)Y(t þ t) ¼ E 1

or ð1 RX (t a)h(a)da

RXY (t) ¼

(19:2:23)

1

Since the cross-correlation function depends only on t, the output Y(t) will also be a stationary random process. Properties of Correlation Functions of Stationary Processes

Autocorrelation Functions 1. RX (0) ¼ E[X 2(t)] ¼ average power 0 2. RX (t) ¼ RX (t). Or, RX (t) is an even function. RX (t) ¼ E½X(t)X(t þ t) ¼ E ½X(t þ t)X(t) ¼ RX (t)

(19:2:24)

19.2

Stationary Random Processes

423

3. jRX (t)j RX (0). Or, the maximum value of jRX (t)j occurs at t ¼ 0 and jRX (t)j for any t cannot exceed the value RX (0). From E½X(t) + X(t þ t)2 0 we have E ½X 2 (t) þ E ½X 2 (t þ t) + 2E ½X(t)X(t þ t) 0 or RX (0) jRX (t)j

(19:2:25)

4. If a constant T . 0 exists such that RX (T ) ¼ RX (0), then the RX (t) is periodic and X(t) is called a periodic stationary process. From Schwartz’ inequality [Eq. (14.5.13)] we have {E ½g(X)h(X)}2 E ½g2 (X)E½h2 (X) Substituting g(X ) ¼ X(t) and h(X) ¼ ½X(t þ t þ T ) X(t þ t), we can write {E½X(t)½X(t þ t þ T ) X(t þ t)}2 E½X 2 (t)E½X(t þ t þ T ) X(t þ t)2 or {RX (t þ T ) RX (t)}2 2RX (0)½RX (0) RX (T ) Hence, if RX (T ) ¼ RX (0), then, since {RX (t þ T ) RX (t)}2 0, the result RX (t þ T ) ¼ RX (t) follows. 5. E ½X(t þ f)X(t þ t þ f) ¼ RX (t) ¼ E½X(t)X(t þ t) E ½X(t þ f)Y(t þ t þ f) ¼ RXY (t) ¼ E½X(t)Y(t þ t) In the formulation of correlation functions the phase information is lost. 6. If E[X(t)] ¼ mX and Y(t) ¼ a þ X(t), where a is constant, then E[Y(t)] ¼ a þ mX and, RY (t) ¼ E{½a þ X(t)½a þ X(t þ t)} ¼ a2 þ aE ½X(t þ t) þ aE ½X(t) þ E ½X(t)X(t þ t) ¼ a2 þ 2amX þ RX (t) ¼ a2 þ 2amX þ m2X þ CX (t) ¼ (a þ mX )2 þ CX (t)

(19:2:26)

If E [X(t)] ¼ 0, then E[Y(t)] ¼ a, and we can obtain the mean value of Y(t) from a knowledge of its autocorrelation function RY(t).

Cross-Correlation Functions

7. RXY (t) ¼ RYX (t): This is not an even function:

ð19:2:27Þ

424

Random Processes

The result follows from the deﬁnition of cross-correlation: RXY (t) ¼ E½X(t)Y(t þ t) ¼ E½Y(t þ t)X(t) ¼ RYX (t)

but

RXY (0) ¼ RYX (0) ð19:2:28Þ

8. R2XY (t) RX (0)RY (0): This result follows from Schwartz’ inequality: {E ½X(t)Y(t þ t)}2 E ½X 2 (t) E ½Y 2 (t þ t) 9. 2jRXY (t)j RX (0) þ RY (0): The result follows from E ½X(t) + Y(t þ t)2 ¼ RX (0) þ RY (0) + 2RXY (t) 0

(19:2:29)

10. If Z(t) ¼ X(t) þ Y(t), then RZ (t) ¼ E ½Z(t)Z(t þ t) ¼ E{½X(t) þ Y(t)½X(t þ t) þ Y(t þ t)} ¼ RX (t) þ RY (t) þ RXY (t) þ RYX (t)

(19:2:30)

and if X(t) and Y(t) are orthogonal, then RZ (t) ¼ RX (t) þ RY (t). _ 11. If X(t) is the derivative of X(t), then the cross-correlation between X(t) and _X(t) is RX X_ (t) ¼

dRX (t) dt

(19:2:31a)

This result can be shown from the formal deﬁnition of the derivative of X(t). Substituting for X_ ¼ lim1!0 {½X(t þ 1) X(t)=1}, we obtain RX X_ (t) ¼ E½X(t)X(t þ t) X(t þ t þ 1) X(t þ t) ¼ lim E X(t) 1!0 1 X(t)X(t þ t þ 1) X(t)X(t þ t) ¼ lim E 1!0 1 ¼ lim

1!0

RX (t þ 1) RX (t) dRX (t) ¼ 1 dt

_ is 12. The autocorrelation of X(t)

RX_ (t) ¼

d2 RX (t) dt2

(19:2:31b)

19.2

Stationary Random Processes

425

This result can be shown following a procedure similar to that described as in (11) above: _ X(t _ þ t) RX_ (t) ¼ E½X(t) X(t þ t þ 1) X(t þ t) ¼ lim E X(t) 1!0 1 X(t þ d) X(t) X(t þ t þ 1) X(t þ t) ¼ lim lim E d!0 1!0 1 d 2 3 X(t þ t þ 1) X(t þ t) X(t þ d) 1 6 7 1 ¼ lim lim E4 X(t þ t þ 1) X(t þ t) 5 d!0 d 1!0 X(t) 1 1 RX (t d þ 1) RX (t d) RX (t þ 1) RX (t) ¼ lim lim d!0 d 1!0 1 1 1 dRX (t d) dRX (t) d2 RX (t) ¼ lim ¼ d!0 d dt dt dt2 It can be shown that if a random process is wide-sense stationary, then it is necessary and sufﬁcient that the following two conditions be satisﬁed: 1. The expected value is a constant, E [X(t)] ¼ mX. 2. The autocorrelation function RX is a function of the time difference t2 2 t1 ¼ t and not individual times, RX (t1 , t2 ) ¼ RX (t2 t1 ) ¼ RX (t). We will now give several examples to establish conditions for stationarity. The ﬁrst few examples will be running examples that become progressively more difﬁcult.

Example 19.2.1 We will revisit Example 19.1.5 and ﬁnd the conditions necessary for the random process X(t) ¼ A sin(vt þ F) to be stationary where A,v are constants and F is a random variable: 1. Mean: ðb E½X(t) ¼ AE½sin(vt þ F) ¼ A

sin(vt þ f) fF (f)df a

One of the ways the integral will be independent of t is for F to be uniformly distributed in (0,2p), in which case we have 1 2p and E [X(t)] ¼ 0.

ð 2p sin(vt þ f)df ¼ 0

2p 1

cos (vt þ f) ¼ 0 0 2p

426

Random Processes

2. Autocorrelation: E ½X(t1 )X(t2 ) ¼ A2 E ½sin(vt1 þ F) sin(vt2 þ F) ¼ A2 E{cos½v(t2 t1 ) cos½v(t2 þ t1 ) þ 2F} ¼ A2 cos½v(t2 t1 ) A2 E{cos½v(t2 þ t1 ) þ 2F} ð 1 2p E{cos½v(t2 þ t1 ) þ 2F} ¼ cos½v(t2 þ t1 ) þ 2f df 2p 0

2p 1 1

sin½v(t2 þ t1 ) þ 2f ¼ 0 ¼ 0 2p 2 Hence, RX (t1 , t2 ) ¼ A2 cos½v(t2 t1 ) ¼ A2 cos½vt, and X(t) is stationary if F is uniformly distributed in (0,2p). Since RX(t) is periodic, X(t) is a periodic stationary process. Example 19.2.2 We will modify Example 19.2.1 with both A and F as random variables with density functions fA(a) and fF(f). We will now ﬁnd the conditions under which X(t) ¼ A sin(vt þ F) will be stationary. 1. Mean: ðð E ½X(t) ¼ E ½A sin(vt þ F) ¼

a sin(vt þ f) fAF (a, f)df da

The ﬁrst condition for the double integral to be independent of t is for A and F to be statistically independent, in which case we have ðð E ½A sin(vt þ F) ¼ a sin(vt þ f) fA (a) fF df da and the second condition Ð 2p is for F to be uniformly distributed in (0,2p), in which case we have (1=2p) 0 sin(vt þ f) df ¼ 0 and E ½X(t) ¼ 0. 2. Autocorrelation: E ½X(t1 )X(t2 ) ¼ E ½A2 sin(vt1 þ F) sin(vt2 þ F) Since A and F are independent, we have E ½X(t1 )X(t2 ) ¼ E ½A2 E{cos½v(t2 t1 ) cos½v(t2 þ t1 ) þ 2F} ¼ E ½A2 cos½v(t2 t1 ) E ½A2 E{cos½v(t2 þ t1 ) þ 2F} and from the previous example E{cos½v(t2 þ t1 ) þ 2F} ¼ 0. Hence, RX (t1 , t2 ) ¼ E ½A2 cos½v(t2 t1 ) ¼ E ½A2 cos½vt and X(t) is stationary if A and F are independent and if F is uniformly distributed in (0,2p). Since RX(t) is periodic, X(t) a periodic stationary process. Example 19.2.3 We will now examine the conditions for stationarity for X(t) ¼ A sin(Vt þ F) when A, V and F are all random variables with density functions fA(a), fV(v), and fF(f) respectively.

19.2

Stationary Random Processes

427

1. Mean: ððð E ½X(t) ¼ E ½A sin(Vt þ F) ¼

a sin(vt þ f) fAVF (a,v,f)df dv da

This triple integral is a difﬁcult one to evaluate. Hence we resort to conditionalÐ expectations by ﬁxing the variable V ¼ v. Under this condition, E ½X(t) ¼ E ½X(t)jV ¼ v fV (v)dv and from the previous example, if A and F are independent and F is uniformly distributed in (0, 2p), we have E ½X(t)jV ¼ v ¼ 0, and ð

ð E ½X(t)jV ¼ v fV (v)dv ¼ E ½A sin(vt þ F) fV (v)dv ¼ 0

2. Autocorrelation: RX (t1 , t2 jV ¼ v) ¼ E ½X(t1 jV ¼ v)X(t2 jV ¼ v) ¼ E ½A2 sin(vt1 þ F) sin(vt2 þ F) and from the previous example 1 E ½A2 sin(vt1 þ F) sin(vt2 þ F) ¼ E ½A2 cos½v t 2 where t ¼ t2 t1 . Hence ð ð 1 E ½A2 cos½vt fV (v)dv RX (t) ¼ RX (tjV ¼ v) fV (v)dv ¼ 2 If V is uniformly distributed in (0,p), then

RX (t) ¼

1 p

ðp 0

1 E ½A2 sin(p) E ½A2 E ½A2 cos½vtdv ¼ ¼ Sa(p) 2 2p t 2

Example 19.2.4 In this example X(t) ¼ A cos(vt) þ B sin(vt), where A and B are random variables with density functions fA(a) and fB(b). We have to ﬁnd the conditions under which X(t) will be stationary: 1. Mean: E ½X(t) ¼ E ½A cos(vt) þ B sin(vt) ¼ cos(vt)E ½A þ sin(vt)E ½B If E [X(t)] is to be independent of t, then E[A] ¼ E[B] ¼ 0, in which case E [X(t)] ¼ 0.

428

Random Processes

2. Autocorrelation: RX (t1 , t2 ) ¼ E ½X(t1 )X(t2 ) ¼ E{½A cos(vt1 ) þ B sin(vt1 )½A cos(vt2 ) þ B sin(vt2 )} ¼ E{A2 cos(vt1 ) cos(vt2 ) þ B2 sin(vt1 ) sin(vt2 ) þ AB½sin(vt1 ) cos(vt2 ) þ cos(vt1 ) sin(vt2 )} 1 ¼ E ½A2 ½cos(v(t2 t1 )) þ cos(v(t2 þ t1 )) 2 1 þ E ½B2 ½cos(v(t2 t1 )) cos(v(t2 þ t1 )) 2 þ E ½AB sin(v(t2 þ t1 )) The conditions under which this equation will be dependent only on (t2 t1 ) are E[A 2] ¼ E[B 2] and E[AB] ¼ 0. In this case RX (t1 ,t2 ) ¼ E ½A2 ½cos(v(t2 t1 )) ¼ E ½A2 ½cos(vt) We can now summarize the conditions for stationarity: (a) E[A] ¼ E[B] ¼ 0 (b) E[A 2] ¼ E[B 2] (c) E[AB] ¼ 0 Example 19.2.5 A die is tossed, and corresponding to the dots S ¼ f1,2,3,4,5,6g, a random process X(t) is formed with the following time functions as shown in Fig. 19.2.1: X(2;t) ¼ 3, X(4; t) ¼ (2 t), X(6; t) ¼ (1 þ t) X(1; t) ¼ 3, X(3;t) ¼ (2 t), X(5;t) ¼ (1 þ t)

FIGURE 19.2.1

19.2

Stationary Random Processes

429

We have to ﬁnd mX (t), s2X (t), RX (t1 ,t2 ), CX (t1 , t2 ), and rX (t1 , t2 ) and check whether X(t) is stationary: Mean: mX (t) ¼

6 1X Xi (t) ¼ 3 3 þ (2 t) (2 t) þ (1 þ t) (1 þ t) ¼ 0 6 i¼1

The mean value is a constant. Variance: s2X (t) ¼

6 1X 1 2 Xi2 (t) ¼ ½32 þ (2 t)2 þ (1 þ t)2 ¼ ½t2 t þ 7 6 i¼1 3 3

Autocorrelation: 1 1 RX (t1 , t2 ) ¼ ½9 þ (1 þ t1 )(1 þ t2 ) þ (2 t1 )(2 t2 ) ¼ ½14 t2 t1 þ 2t1 t2 3 3 Autocovariance. Since the mean value is zero, CX (t1 , t2 ) ¼ RX (t1 , t2 ). Normalized Autocovariance: CX (t1 , t2 ) ½14 t2 t1 þ 2t1 t2 rX (t1 , t2 ) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ CX (t1 )CX (t2 ) (t12 t1 þ 7)(t22 t2 þ 7) This process is not stationary.

Example 19.2.6 (Random Binary Wave) A sample function of a random binary wave X(t) consisting of independent rectangular pulses p(t), each of which is of duration T, is shown in Fig. 19.2.2. The height H of the pulses is a random variable with constant amplitudes, which are equally likely to be +A. The time of occurrence of X(t) after t ¼ 0 is another random variable Z, which is uniformly distributed in (0,T ). We have to ﬁnd the mean, variance, autocorrelation, autocovariance, and the normalized autocovariance of X(t). The random process X(t) is given by X(t) ¼

1 X

Hp(t kT Z)

k¼1

FIGURE 19.2.2

430

Random Processes

The probabilities of the random variable H are P(H ¼ A) ¼ 12 and P(H ¼ A) ¼ 12. Since Z is uniformly distributed in (0,T ), the following probabilities can be formulated for 0 , t1 , t2 , T: P(t1 , Z , t2 ) ¼

t2 t1 , T

P(0 , Z , t1 ) ¼

t1 , T

P(t2 , Z , T ) ¼ 1

t2 T

Mean. Since X(t) assumes only þA or 2A with equal probability, E ½X(t) ¼

1 1 A þ (A) ¼ 0 2 2

Variance. The variance is given by var½X(t) ¼

s2X

1 2 1 2 ¼ E ½X (t) ¼ A þ (A) ¼ A2 2 2 2

Autocorrelation. Determining RX (t1 , t2 ) is a bit more involved. The product X(t1)X(t2) in various intervals of time (t1, t2) can be found: t1 , Z , t2 T Here t1 and t2 lie in adjacent pulse intervals as shown in Fig. 19.2.3. In Fig. 19.2.3a X(t1) ¼ 2A and X(t2) ¼ þA and X(t1)X(t2) ¼ 2A 2. In Fig. 19.2.3b X(t1) ¼ þA and X(t2) ¼ þA and X(t1)X(t2) ¼ A 2. The values 2A 2 and A 2 will occur with equal probability: 0 , Z , t1 T: In this case, t1 and t2 lie in the same pulse interval as shown in Fig. 19.2.3c. Here, X(t1) ¼ þA and X(t2) ¼ þA and X(t1)X(t2) ¼ A 2. t2 , Z , T: Here also t1 and t2 lie in the same pulse interval as shown in Fig. 19.2.3d. However, X(t1) ¼ 2A and X(t2) ¼ 2A and X(t1)X(t2) ¼ A 2. (t1 2 t2) . T: In this case t1 and t2 lie in different pulse intervals and X(t1)X(t2) ¼ +A 2. We can now ﬁnd the autocorrelation function RX (t1 , t2 ) ¼ E ½X(t1 )X(t2 ). For (t1 2 t2) . T, X(t1) and X(t2) are in different pulse intervals and invoking the independence of the pulses E [X(t1)X(t2)] ¼ E [X(t1)]E [X(t2)] ¼ 0 since E [X(t)] ¼ 0. For (t1 2 t2) T, we have the following, using conditional expectations: E ½X(t1 )X(t2 ) ¼ E ½X(t1 )X(t2 )jt1 , Z , t2 P(t1 , Z , t2 ) þ E ½X(t1 )X(t2 )j0 , Z , t1 P(0 , Z , t1 ) þ E ½X(t1 )X(t2 )jt1 , Z , TP(t1 , Z , T ) The conditional expectations can be determined using the probabilities and the products X(t1)X(t2) found earlier for the various intervals. Since A 2 and 2A 2 occur with equal probability in the interval t1 , Z , t2 T, the ﬁrst conditional expectation term becomes 1 t2 t1 1 t2 t1 E ½X(t1 )X(t2 )jt1 , Z , t2 P(t1 , Z , t2 ) ¼ A2 A2 ¼0 T T 2 2

19.2

FIGURE 19.2.3

Stationary Random Processes

431

432

Random Processes

FIGURE 19.2.4

Hence " RX (t1 , t2 ) ¼

A2

t t2 t2 t1 1 þ1 ¼ A2 1 , T T T

(t2 t1 ) T (t2 t1 ) . T

0,

This equation was derived under the assumption t1 , t2. For arbitrary values of t1 and t2 with t ¼ (t2 2 t1), the autocorrelation function RX(t1, t2) is given by 8 jtj < 2 A 1 , jtj , T RX (t) ¼ T : 0, otherwise The autocorrelation function for the process X(t) is shown in Fig. 19.2.4. Autocovariance. Since the mean value is 0, CX(t) ¼ RX(t). Normalized Autocovariance. The NACF is given by ( jtj CX (t) 1 , jtj , T rX (t) ¼ 2 ¼ T sX 0; otherwise Example 19.2.7 (Random Telegraph Wave) This example is a little different from the previous one. A sample function of a random telegraph wave is shown in Fig. 19.2.5. The wave assumes either of the values 1 or 0 at any instant of time.

FIGURE 19.2.5

19.2

Stationary Random Processes

433

The probability of k changes from 0 to 1 in a time interval t is Poisson-distributed with probability mass function p(k; l) given by (lt)k lt e , k!

p(k; l) ¼

t0

where l is the average number of changes per unit time. We have to ﬁnd the mean, variance, autocorrelation, autocovariance, and normalized autocovariance. Mean. Since X(t) assumes only 1 or 0 with equal probability, we obtain E ½X(t) ¼ mX ¼ 1 P(x ¼ 1) þ 0 P(x ¼ 0) ¼

1 2

Variance. The variance is given by var½X(t) ¼

s2X

2

¼ E ½X (t)

m2X

1 2 1 2 1 1 1 þ 0 ¼ ¼ 2 2 4 4

Autocorrelation. The autocorrelation function can be given in terms of the joint density functions with t2 . t1: RX (t1 , t2 ) ¼ E ½X(t1 ) ¼ 0 X(t2 ) ¼ 0 þ E ½X(t1 ) ¼ 0 X(t2 ) ¼ 1 þ E ½X(t1 ) ¼ 1 X(t2 ) ¼ 0 þ E ½X(t1 ) ¼ 1 X(t2 ) ¼ 1 ¼ (0 0)P½X(t1 ) ¼ 0 X(t2 ) ¼ 0 þ (0:1)P½X(t1 ) ¼ 0 X(t2 ) ¼ 1 þ (1 0)P½X(t1 ) ¼ 1 X(t2 ) ¼ 0 þ (1:1)P½X(t1 ) ¼ 1 X(t2 ) ¼ 1 or RX (t1 , t2 ) ¼ P½X(t1 ) ¼ 1 X(t2 ) ¼ 1 Expressing the joint probability in terms of conditional probabilities, we have RX (t1 , t2 ) ¼ P½X(t2 ) ¼ 1 j X(t1 ) ¼ 1P½X(t1 ) ¼ 1 The conditional probability in this equation is the probability of even number of changes and using the Poisson distribution: RX (t1 , t2 ) ¼

1 1 X ½l(t2 t1 )k l(t2 t1 ) e k! 2 k¼0 k even

( ) 1 1 el(t2 t1 ) 1 X ½l(t2 t1 )k X ½l(t2 t1 )k þ ¼ 2 k! k! 2 k¼0 k¼0 el(t2 t1 ) el(t2 t1 ) þ el(t2 t1 ) ¼ 2 2 1 ¼ {1 þ e2l(t2 t1 ) }, 4

t2 . t1

A similar equation holds good for t2 , t1. Hence, substituting jt2 t1 j ¼ jtj in the equation above, RX(t) can be given by 1 RX (t) ¼ {1 þ e2ljtj } 4

434

Random Processes

FIGURE 19.2.6

The autocorrelation function for the process X(t) is shown in Fig. 19.2.6 for l ¼ 0.5. Autocovariance: CX (t) ¼ RX (t) m2X ¼

e2ljtj 4

Normalized Autocovariance: rX (t) ¼

CX (t) ¼ e2ljtj s2X

Example 19.2.8 (Modulation) A random process Y(t) is given by Y(t) ¼ X(t) cos(vt þ F), where X(t), a zero mean wide-sense stationary random process with autocorrelation function RX (t) ¼ 2e2ljtj is modulating the carrier cos(vt þ F). The random variable F is uniformly distributed in the interval (0,2p), and is independent of X(t). We have to ﬁnd the mean, variance, and autocorrelation of Y(t): Mean. The independence of X(t) and F allows us to write E ½Y(t) ¼ E ½X(t)E ½cos(vt þ F) and with E ½X(t) ¼ 0 and E ½cos(vt þ F) ¼ 0 from Example 19.2.1, E[Y(t)] ¼ 0. Variance. Since X(t) and F are independent, the variance can be given by s2Y ¼ E ½Y 2 (t) ¼ E ½X 2 (t) cos2 (vt þ F) ¼ s2X E ½cos2 (vt þ F) However 1 1 E ½cos2 (vt þ F) ¼ E ½1 þ cos(2vt þ 2FÞ ¼ 2 2 and hence s2Y ¼ s2X =2 ¼ 1.

and s2X ¼ CX (0) ¼ RX (0) ¼ 2

19.2

Stationary Random Processes

435

FIGURE 19.2.7

Autocorrelation: RY (t) ¼ E ½Y(t)Y(t þ t) ¼ E ½X(t) cos(vt þ F)X(t þ t) cos(vt þ vt þ F) 1 ¼ RX (t) E ½cos(vt) þ cos(2vt þ vt þ 2F) 2 RX (t) RX (t) cos(vt) þ E ½cos(2vt þ vt þ 2F) ¼ 2 2 From Example 19.2.1 E ½cos(2vt þ vt þ 2F) ¼ 0, and hence RY (t) ¼

RX (t) cos(vt) ¼ e2ljtj cos(vt) 2

A graph of RY(t) is shown in Fig. 19.2.7 with l ¼ 0.5 and v ¼ 2p. Example 19.2.9 (Cross-Correlation) Two random processes X(t) and Y(t) are given by X(t) ¼ A cos(vt) þ B sin(vt) and Y(t) ¼ A sin(vt) þ B cos(vt), where A and B are random variables with density functions fA(a) and fB(b). We have to ﬁnd the cross-correlation, cross-covariance, and the normalized cross-covariance between X(t) and Y(t). From Example 19.2.4 the processes X(t) and Y(t) are stationary if E [A] ¼ E [B] ¼ 0, E [A 2] ¼ E [B 2], and E [AB] ¼ 0. Hence the mean values mX ¼ mY ¼ 0. The variances of these processes are E ½A2 ¼ E ½B2 ¼ s2 . Cross-Correlation. The cross-correlation function RXY(t1, t2) can be written as RXY (t1 , t2 ) ¼ E ½X(t1 )Y(t2 ) ¼ E{½A cos(vt1 ) þ B sin(vt1 )½A sin(vt2 ) þ B cos(vt2 )} ¼ E{ A2 cos(vt1 ) sin(vt2 ) þ B2 sin(vt1 ) cos(vt2 ) þ AB½cos(vt1 ) cos(vt2 ) sin(vt1 ) sin(vt2 )}

436

Random Processes

Substituting the values E [A 2] ¼ E [B 2] ¼ s2 and E [AB]¼0 in this equation, we obtain RXY (t1 ,t2 ) ¼ s2 sin½v(t1 t2 ) ¼ s2 sin½vt where we have substituted t ¼ (t2 2 t1). The term RXY(t) is shown in Fig. 19.2.8 for s2¼4 and v ¼ 2p. Here RXY(t) is an odd function, unlike the autocorrelation function. Since RXY (t)¼0 for t ¼ 0, we conclude that X(t) and Y(t) are orthogonal. Cross-Covariance. Since mX ¼ mY ¼ 0, CXY (t) ¼ RXY(t). Normalized Cross-Covariance:

rXY (t) ¼

CXY (t) s2 sin½vt ¼ ¼ sin½vt s2 s2

Example 19.2.10 A random telegraph wave X(t) as in Example 19.2.7 is passed through a linear system with impulse response h(t) ¼ e – btu(t), where u(t) is a unit step function. The output of the system is Y(t). It is desired to ﬁnd the cross-correlation function RXY (t). From Example 19.2.7 the autocorrelation function RX (t) ¼ 14 {1 þ e2ljtj }. Hence, from Eq. (19.2.23), we have ð1

1 (1 þ e2ljtaj )eba da 4 0 ð ð 8 1 1 t 2l(ta) ba 1 1 2l(at) ba > > e e da þ (e )e da þ < 4b 4 0 4 t ð1 ¼ > 1 1 > : þ e2l(at) eba da 4b 4 0

RXY (t) ¼

FIGURE 19.2.8

t.0 t0

19.2

Stationary Random Processes

437

Evaluating the integrals and substituting l ¼ 1 and b ¼ 0.5, we obtain 8 lebt e2lt > > ; t.0 > > 2 2 < 4l b 4(2l b) 1 þ RXY (t) ¼ 4b > > e2lt > > : , t0 4(2l þ b) or

8 4 (t=2) 1 2t > > e , > < 15 e 6

1 RXY (t) ¼ þ 2 > > > : 1 e2t , 10

t.0

t0

The cross-correlation function RXY (t) is shown in Fig. 19.2.9. RXY (t) does not possess any symmetry, unlike RX (t). Moments of Discrete-Time Stationary Processes

In actual practice observations are made on the sample function of a stationary random process X(t) at equally spaced time intervals fti, i ¼ 0, +1, . . .gwith corresponding sequence of random variables fXi, i ¼ 0, +1, . . .g. These observation random variables will not be independent. Since fXig are samples of a stationary random process, the means and variances of these samples are the same as in the original process: E ½Xi ¼ mX ; var½Xi ¼ s2X ,

i ¼ 0, +1, . . .

(19:2:32)

Analogous to the continuous case, we can deﬁne the various second moments: Autocovariance: CX (h) ¼ E ½(Xi mX )(Xiþh mX ),

i ¼ 0,+1, . . .

(19:2:33)

Autocorrelation: RX (h) ¼ E ½Xi Xiþh , i ¼ 0, + 1, . . .

FIGURE 19.2.9

(19:2:34)

438

Random Processes

Normalized Autocovariance (NACF): rX (h) ¼

CX (h) CX (h) ¼ CX (0) s2X

(19:2:35)

If fYi, i ¼ 0, +1, . . .g is the sequence obtained from a second stationary random process Y(t) with mean mY and variance s2Y, then we can deﬁne the cross-moments as follows: Cross-Covariance: CXY (h) ¼ E ½(Xi mX )(Yiþh mY i ¼ 0,+1, . . .

(19:2:36)

Cross-Correlation: RXY (h) ¼ E ½Xi Yiþh i ¼ 0,+1, . . .

(19:2:37)

Normalized Cross-Covariance (NCCF): CXY (h) CXY (h) rXY (h) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ sX s Y CX (0)CY (0)

Example 19.2.11

(19:2:38)

A discrete zero mean stationary random process Xi is given by Xi ¼ fXi1 þ vi ,

i ¼ 1, . . .

(19:2:39)

where vi is a zero mean Gaussian random process with variance s2v . We want to ﬁnd the variance of Xi and the NACF rX(h). Multiplying both sides of Eq. (19.2.39) by Xi and taking expectations, we have E ½Xi2 ¼ fE ½Xi Xi1 þ E ½Xi vi or

s2X ¼ fCX (1) þ E ½Xi vi

(19:2:40)

The cross-correlation E ½Xi vi can be computed as follows: E ½Xi vi ¼ E ½(fXi1 þ vi )vi ¼ s2v

(19:2:41)

since vi occurs after Xi21. Hence, substituting Eq. (19.2.41) in Eq. (19.2.40) and dividing throughout by s2X , we obtain 1 ¼ frX (1) þ

s2v s2X

and

s2X ¼

s2v 1 frX (1)

(19:2:42)

Premultiplying both sides of Eq. (19.2.39) by Xih and taking expectations, we have E ½Xih Xi ¼ E ½fXih Xi1 þ E ½Xih vi

(19:2:43)

In Eq. (19.2.43) E ½Xih vi ¼ 0 since vi occurs after Xih for h . 0; hence CX (h) ¼ fCX (h 1),

h.0

(19:2:44)

Dividing Eq. (19.2.44) by CX (0) ¼ s2X , we obtain an equation for the NACF rX (h) rX (h) ¼ frX (h 1),

h.0

(19:2:45)

19.3

Ergodic Processes

439

and solving for rX(h) with initial condition rX(0) ¼ 1, we obtain rX (h) ¼ fh ,

h.0

(19:2:46)

Substituting Eq. (19.2.46) in Eq. (19.2.42), we have s2X ¼

s2v 1 f2

(19:2:47)

The process deﬁned by Eq. (19.2.39) is called an autoregressive process of order 1.

B 19.3

ERGODIC PROCESSES

The ensemble average of a random process X(t) is the mean value mX(t) deﬁned by, ð1 xfX (x; t)dx (19:3:1) mX (t) ¼ 1

Finding the ensemble average mX(t) requires storing a multiplicity of sample functions and ﬁnding the average. In many instances this process may be nontrivial. Given a sample function of any random process X(t), the time average m^ X is deﬁned by m^ X ¼

1 2T

ðT X(t)dt

(19:3:2)

T

A reasonable question to ask is whether the ensemble average can be obtained from a much easier time average. We observe that the ensemble average is not a random variable but is a function of time, whereas the time average is a random variable that is not a function of time. If these averages are to be equal, then the ﬁrst condition that we have to impose is that the random process X(t) be stationary, which removes the time factor in the mean value. Under these conditions, we can write

1 E 2T

ðT

1 X(t)dt ¼ E ½m^ X ¼ 2T T

ðT E ½X(t)dt ¼ mX

(19:3:3)

T

and m^ X is an unbiased estimator of mX. If a random variable is to be equal to a constant, then the second condition from Example 14.1.1 is that the variance of m^ X must tend to zero as T ! 1. Such random processes are said to satisfy the ergodic hypothesis. We will now derive the conditions for a random process to be mean-ergodic and correlation-ergodic. Mean-Ergodic

A stationary random process X(t) is called mean-ergodic if the ensemble average is equal to the time average of the sample function x(t). We will assume that the following conditions are satisﬁed by X(t): X(t) is stationary, implying that it has a constant mean mX, and the autocorrelation function RX (t, t þ t) is a function of t only, or RX (t, t þ t) ¼ RX (t). RX (0) ¼ E [X 2(t)] is bounded, or RX(0) ,1. Hence CX (0) ¼ RX (0) 2 m2X ,1.

440

Random Processes

We have to ﬁnd the conditions under which the variance s2m^ X of the time average m^ X goes to 0. The variance s2m^ X is given by s2m^ x ¼ E ½(m^ X mX )2 ¼ E

(

1 2T

2 )

ðT ½X(t) mX dt T

(

) ð ð 1 2 T T ½X(t) mX ½X(s) mX dt ds ¼E 2T T T 2 ð T ð T 1 CX (t,s)dt ds ¼ 2T T T 2 ð T ð T 1 ¼ CX (s,t)dt ds, X(t) is stationary 2T T T

ð19:3:4Þ

Substitution of t ¼ s 2t and u ¼ t in Eq. (19.3.4), transforms the coordinate system from [s,t] to [t,u]. The Jacobian kJk of the transformation is given by 3

@t

1 @s 7

¼ 5 @s 0

@s

2

@t

6 kJk ¼ 4 @t

@s

@t

1

¼1 1

(19:3:5)

Thus s2m^ X ¼

1 2T

2 ð ð CX (t)dt du

(19:3:6)

The integration has to be carried out over the parallelogram shown in Fig. 19.3.1 in regions I and II. Region I: t 0: Limits for t are from – 2T to 0; limits for u are from – T to T þ t. Region II: t . 0: Limits for t are from 0 to 2T; limits for u are from – T þ t to T.

FIGURE 19.3.1

19.3

Ergodic Processes

Substituting these limits in Eq. (19.3.6), we obtain the variance s2m^ X : 2 ð T ð 2T ð Tþt ð 0 1 2 CX (t)dt du þ CX (t)dt du sm^ X ¼ 2T Tþt 0 T 2T ð 2T ð0 1 CX (t)½2T tdt þ CX (t)½2T þ tdt ¼ 2 4T 0 2T ð 2T ð 2T 1 1 jtj ¼ 2 CX (t)½2T jtjdt ¼ CX (t) 1 dt 4T 2T 2T 2T 2T

441

(19:3:7)

Finally, if the random process is to be mean-ergodic, the following condition has to be satisﬁed: ð 2T 1 jtj 2 CX (t) 1 dt ! 0 (19:3:8) lim s ¼ lim T!1 m^ X T!1 2T 2T 2T Since

ð 2T

ð 2T

ð 2T

jtj jtj

dt C (t) 1 jC (t)j 1 jCX (t)jdt dt X X

2T 2T

2T

2T

(19:3:9)

2T

Ð1 hence, if 1 jCX (t)jdt is bounded, then limT!1 s2m^ X does indeed go to zero. A necessary and sufﬁcient condition for a random process to be mean-ergodic is ð1 jCX (t)jdt , 1 (19:3:10) 1

Even though we have derived necessary and sufﬁcient condition for a mean-ergodic process, we cannot use it to test the ergodicity of any process since it involves prior knowledge of the autocovariance function CX(t). However, if a partial knowledge of the ACF such as jCX(t)j goes to 0 as t ! 1, then we can conclude that the process is ergodic. Although ergodic processes must be stationary, stationary processes need not be ergodic. Since it is rather difﬁcult to show that any arbitrary random process is ergodic, we will assume, unless otherwise stated, that a process that is stationary is also ergodic. Correlation-Ergodic

The time autocorrelation function of a stationary random process X(t) is deﬁned by ð 1 T R^ X (l) ¼ X(t)X(t þ l)dt (19:3:11) 2T T A stationary random process X(t) is correlation-ergodic if ðT ð 1 T ^ 1 RX (l)dt ¼ lim X(t)X(t þ l)dt ¼ RX (l) lim T!1 2T T T!1 2T T

(19:3:12)

for all l. Clearly R^ X (l) is a random variable, but ð T ð T 1 1 X(t)X(t þ l)dt ¼ RX (l)dt ¼ RX (l) E ½R^ X (l) ¼ E 2T 2T T T and R^ X (l) is an unbiased estimator of RX (l). As in the case of mean-ergodic processes, here also the variance of R^ X (l) must tend to 0 as T ! 1. Hence we can form the

442

Random Processes

variance s2R^ l of R^ X (l) for any ﬁxed l: X ( ð 2 ) h i2 T 1 s2R^ l ¼ E R^ X (l) RX (l) ½X(t)X(t þ l) RX (l)dt ¼E X 2T T 2 ð T ð T

1 ¼ E ½ X(t)X(t þ l) RX (l)½ X(s)X(s þ l) RX (l) dt ds 2T T T 2 ð T ð T 1 ¼ CRlX (t,s)dt ds 2T T T 2 ð T ð T 1 CRlX (s t)dt ds ¼ 2T T T

(19:3:13)

Substituting (s – t) ¼ t in Eq. (19.3.13), we obtain a result similar to Eq. (19.3.8), namely ð 2T 1 jtj lim s2R^ l ¼ lim CRlX (t) 1 dt ! 0 (19:3:14) T!1 T!1 2T X 2T 2T and the necessary and sufﬁcient condition for X(t) to be correlation-ergodic is ð1

l

C (t) dt , 1 (19:3:15) RX 1

where CRlX (t) ¼ RRlX (t) R2X (l) RRlX ¼ E ½X(t)X(t þ l)X(t þ t)X(t þ t þ l) Thus the autocorrelation ergodicity involves the fourth-order stationary moments of X(t). As a corollary to this result, we can deﬁne mean-square or power ergodicity as ðT 1 2 X (t)dt ¼ RX (0) (19:3:16) lim T!1 2T T and the necessary and sufﬁcient conditions for mean-square ergodicity are ð 2T 1 jtj CR0 X (t) 1 lim s2R^ 0 ¼ lim dt ! 0 T!1 T!1 2T X 2T 2T

(19:3:17)

or ð1 1

jCR0 X (t)jdt , 1

(19:3:18)

where CR0 X (t) ¼ R0RX (t) R2X (0) R0RX (t) ¼ E{½X(t)X(t þ t)2 } Example 19.3.1 We will revisit Example 19.2.1, where X(t) ¼ A sin(v0t þ F), where A and F are independent and A is uniformly distributed over (0,1) and F is uniformly distributed over (0,2p) and ﬁnd whether it is mean- and power-ergodic.

19.3

Ergodic Processes

443

We have already seen in Example 19.2.1 that E ½X(t) ¼ E ½AE ½sin(v0 t þ F) ¼ 0 RX (t) ¼ E ½X(t)X(t þ t) ¼ E ½A1 E ½sin(v0 t þ F) sin(v0 t þ v0 t þ F) 1 ¼ E ½A2 cos(v0 t) 2 and X(t) is indeed stationary. In fact, it is also strict-sense stationary. We will check ﬁrst whether it is mean-ergodic. Since the mean value is zero, CX (t) ¼ 12 E ½A2 cos(v0 t). The variance of the time average s2m^ X is given by s2m^ X

E ½A2 ¼ 2T

ð 2T jtj cos(v0 t) 1 dt 2T 2 2T

We can use the result from Fourier transform that time multiplication is frequency convolution and evaluate the integral shown above. Substituting the Fourier transforms of the triangular and the cosine functions jtj sin(vT ) 2 cos(v0 t) p 1 and () 2T () ½d(v þ v0 ) þ d (v v0 ) 2T vT 2 2 in the equation for the variance, we obtain ð 1 E ½A2 1 sin(vT ) 2 p ½d(v þ v0 ) þ d(v v0 )dv 2T s2m^ X ¼ 2p 2T 1 vT 2 E ½A2 sin(v0 T ) 2 ¼ 2p v0 T 2 and as T ! 1, sm^ X ! 0, for v0 = 0. Thus X(t) is mean-ergodic. Substituting E ½A2 ¼ 13 in the variance equation, we obtain 1 sin(v0 T ) 2 2 sm^ X ¼ 6p v0 T

To check for power ergodicity, it is simpler to use Eq. (19.3.16) rather than Eq. (19.3.17). Thus

1 lim T!1 2T

ðT

1 X (t) dt ¼ lim T!1 2T T 2

ðT

2

2

A cos (v0 t þ F) dt ¼ T

A2 2

for

v0 = 0

Since this expression does not converge to RX (0) ¼ 16, the process X(t) is neither powerergodic nor correlation-ergodic. Example 19.3.2 A random process X(t) ¼ A, where A is random variable uniformly distributed over (0,1]. Since E ½A ¼ 12 and RA (t) ¼ E ½A2 ¼ 13, this process is stationary. To check for ergodicity in the mean, we apply Eq. (19.3.2) for the time average and check whether it tends to the ensemble average as T tends to 1. Thus ð ð 1 T 1 T X(t)dt ¼ A dt ¼ A as T ! 1 m^ X ¼ 2T T 2T T

444

Random Processes

is not equal to the ensemble average mX ¼ 12. Hence X(t) is neither mean-ergodic nor correlation-ergodic. Example 19.3.3 We now consider a random process X(t) ¼ B þ A sin(v0t þ F), where B, A, and F are independent random variables. A and B are uniformly distributed over (0,1] and (0,2] respectively. We will check whether X(t) is mean-ergodic. We ﬁrst determine whether X(t) is stationary. The mean value mX is given by E ½B þ A sin(v0 t þ F) ¼ mB þ E ½A sin(v0 t þ F) ¼ mB since E ½A sin(v0 t þ F) ¼ 0 from previous examples. The autocorrelation function is RX (t) ¼ E{½B þ A sin(v0 t þ F)½B þ A sin(v0 t þ v0 t þ F)} ¼ E ½B2 þ 0 þ 0 þ E ½A2 E ½sin(v0 t þ F)A sin(v0 t þ v0 t þ F 1 ¼ E ½B2 þ E ½A2 E ½cos(v0 t) cos(2v0 t þ v0 t þ 2F 2 1 ¼ E ½B2 þ E ½A2 cos(v0 t) 2 Hence the process X(t) is stationary. The autocovariance function CX(t) is given by 1 CX (t) ¼ RX (t) m2X ¼ E ½B2 þ E ½A2 cos(v0 t) m2B 2 1 ¼ s2B þ E ½A2 cos(v0 t) 2 and substituting this equation in Eq. (19.3.7), we obtain the variance of the time average as ð 2T 1 1 jtj s2B þ E ½A2 cos(v0 t) 1 dt s2m^ X ¼ 2T 2T 2 2T " 2 # 1 E ½A2 sin(v0 T ) 2 2 2 2T sin(v0 T ) 2 2TsB þ E ½A ¼ ¼ sB þ 2T 2p v0 T 2p v0 t and ( lim s2 T!1 m^ X

¼

lim s2B T!1

) E ½A2 sin(v0 T ) 2 ¼ s2B þ 2p v0 T

if

v0 = 0

We ﬁnd that even though X(t) is stationary, it is not mean-ergodic. Hence the time average is not a good estimator of the ensemble average. We can now substitute E ½A2 ¼ 13 and s2B ¼ 13 in the variance equation and write s2m^ X ¼ s2B þ

E ½A2 sin(v0 T ) 2 1 sin(v0 T ) ¼ 1þ 2p v0 T 3 2pv0 T

Example 19.3.4 The autocovariance of a zero mean Gaussian random process X(t) is given by CX (t) ¼ s2X e2ajtj , where s2X is its variance. We will check whether this

19.4

Estimation of Parameters of Random Processes

445

process is mean-ergodic and correlation-ergodic. The variance of the time average from Eq. (19.3.7) is s2m^ X

ð 2T jtj 1 jtj 2 2ajtj CX (t) 1 s e 1 dt ¼ dt 2T 2T 2T X 2T 2T s2X 1 e4aT 1 ¼ 2aT 4aT

1 ¼ 2T

ð 2T

and limT!1 s2m^ X ¼ 0. Hence the process is mean-ergodic. To determine whether it is correlation-ergodic, the fourth moment RRlX (t) ¼ E ½X(t)X(t þ l)X(t þ t)X(t þ t þ l) is calculated from the following result. If fX1,X2,X3,X4) are zero mean Gaussiandistributed random variables, then E ½X1 X2 X3 X4 ¼ E ½X1 X2 E ½X3 X4 þ E ½X1 X3 E ½X2 X4 þ E ½X1 X4 E ½X2 X3 . Hence E ½X(t)X(t þ l)X(t þ t)X(t þ t þ l) ¼ E ½X(t)X(t þ l)E ½X(t þ t)X(t þ t þ l) þ E ½X(t)X(t þ t)E ½X(t þ l)X(t þ t þ l) þ E ½X(t)X(t þ t þ l)E ½X(t þ l)X(t þ t) ¼ R2X (l) þ R2X (t) þ RX (t þ l)RX (t l) The covariance CRlX (t) ¼ RRlX (t) R2X(l) ¼ R2X (t) þ RX (t þ l)RX (t l), and substituting for the correlation functions, we obtain CRlX ðtÞ as CRlX (t) ¼ s4X e4ajtj þ s4X e2ajtþlj e2ajtlj ¼ 2s4X e4ajtj , t . l Substituting for CRlX (t) in Eq. (19.3.14), we obtain lim s2^ l T!1 RX

1 ¼ lim T!1 2T

ð 2T 2T

jtj 1 dt 2T

1 e8aT 1 ¼ lim !0 T!1 aT 8aT s4X

2s4X e4ajtj

A similar result holds good for t , l. We conclude from the preceding result that the process X(t) is also correlation-ergodic and hence power-ergodic.

B 19.4 ESTIMATION OF PARAMETERS OF RANDOM PROCESSES 19.4.1

Continuous-Time Processes

The ergodic hypothesis detailed in the previous section enables us to deﬁne estimators for the parameters of a random process from its sample realization. If X(t) is a sample realization of a stationary ergodic random process available only for a time duration T, then the estimators for the parameters, mean, variance, autocorrelation, autocovariance, and normalized autocovariance can be deﬁned.

446

Random Processes

Mean: m^ X ¼

1 T

ðT X(t)dt

(19:4:1)

0

We note that m^ X is an unbiased estimator of mX since E ½m^ X ¼ E

ðT ð 1 1 T X(t)dt ¼ E½X(t)dt ¼ mX T 0 T 0

The variance of m^ X is very similar to Eq. (19.3.7) and is given by ð 1 T jtj ¼ CX (t) 1 var½m^ X ¼ dt T T T ð h 2 T ti CX (t) 1 dt , even function ¼ T 0 T s2m^ X

(19:4:2)

Variance: s^ 2X

1 ¼ T

ðT

½X(t) m^ X 2 dt

(19:4:3)

0

If the mean value mX is known, then s^ 2X will be an unbiased estimator. We will determine the conditions under which s^ 2X given by Eq. (19.4.3) can be an unbiased estimator: ðT 1 ½X(t) m^ X 2 dt T 0 ðT 1 ^ mX ) (m^ X mX )2 dt ¼E ½X(t) T 0 ( ð " # ) 1 T (X(t) mX )2 þ (m^ X mX )2 dt ¼E T 0 2(X(t) mX )(m^ X mX ) ( ð )

1 T 2 ¼E ½(X(t) mX ) dt þ E (m^ X mX )2 T 0 s2 s2

E ½s^ 2X ¼ E

X

2 E T2

m^ X

ð T ð T

½X(t) mX ½X(s) mX dsdt

0

0

and from Eq. (19.4.2) we have E ½s^ 2X ¼ s2X þ s2m^ X

2 T

ð T

jtj CX (t) 1 dt T T s2m^

¼

s2X

s2m^ X

X

(19:4:4)

Hence s^ 2X cannot be an unbiased estimator for ﬁnite T with the bias equal to s2m^ X . However, it is asymptotically unbiased.

19.4

Estimation of Parameters of Random Processes

447

Autocovariance. Two estimators for the autocovariance function are given by ð Tt 1 C X (t) ¼ ½X(t) m^ X ½X(t þ t) m^ X dt, 0 , t T (19:4:5) T t 0 ð 1 Tt C^ X (t) ¼ ½X(t) m^ X ½X(t þ t) m^ X dt, 0 , t T (19:4:6) T 0 If the mean mX is known, then CX (t) given by Eq. (19.4.5) is an unbiased estimator since ð Tt 1 E{½X(t) mX ½X(t þ t) mX } dt E ½CX (t) ¼ T t 0 ¼ CX (t)

T t ¼ CX (t), 0 , t T T t

and C^ X (t) is a biased estimator as shown below: ð 1 Tt ^ ½X(t) mX ½X(t þ t) mX dt CX (t) ¼ T 0 h T t ti ¼ CX (t) 1 , 0 , t T ¼ CX (t) T T with bias equal to CX (t)(t=T ). In actual practice only the sample mean m^ X is known, and we will determine whether C X (t) can be an unbiased estimator. Equation (19.4.5) can be rewritten as ð Tt1 1 ½X(t) m^ X ½X(t þ t1 ) m^ X dt E ½C X (t1 ) ¼ E T t1 0

1 ¼E T t1

ð Tt1

½(X(t) mX ) ðm^ X mX )

0

½(X(t þ t1 ) mX ) (m^ X mX )dt

,

0 , t1 T

ÐT and substituting for m^ X ¼ (1=T ) 0 X(s)ds in the equation above, we have ð Tt1 1 2 E ½CX (t1 ) ¼ E ½(X(t) mX )(X(t þ t1 ) mX ) þ (m^ X mX ) dt T t1 0 ð Tt1 1 ½(X(t) mX )(m^ X mX ) E T t1 0 þ ½(X(t þ t1 ) mX )(m^ X mX )dt ð Tt1 ð T 1 CX (s t)ds dt ¼ CX (t1 ) þ s2m^ X T(T t1 ) 0 0 ð Tt1 ð T 1 CX (s t t1 )ds dt, 0 , t1 T T(T t1 ) 0 0 where E ½(m^ X mX )2 ¼ s2m^ X ¼

1 T

jtj CX (t) 1 dt [from Eq. (19:4:2) T T

ðT

(19:4:7)

448

Random Processes

FIGURE 19.4.1

In Eq. (19.4.7) we make the transformation t ¼ (s – t) and u ¼ t. As the Jacobian of the transformation is 1, we can use techniques similar to those employed in Eqs. (19.3.5)– (19.3.7) to determine E ½C X (t1 ). This transformation amounts to ﬁnding the integral in the parallelogram shown in Fig. 19.4.1. The integration along the t axis is from – T to (T – t1). Thus ð ð Tt1 1 E ½CX (t1 ) ¼ CX (t1 ) þ s2m^ X du ½CX (t) þ CX (t t1 )dt (19:4:8) T(T t1 ) T Ð The limits of integration for du are determined as follows in the three regions: Region I: 2T , t 2 t1 jtj du ¼ T þ t ¼ T 1 T

(19:4:9a)

t1 du ¼ T t1 ¼ T 1 T

(19:4:9b)

t þ t1 du ¼ T t t1 ¼ T 1 T

(19:4:9c)

ð tþT=2 T=2

Region II: 2t1 , t 0 ð T=2t1 T=2

Region III: 0 , t (T 2 t1) ð T=2t1 tT=2

Substituting Eqs. (19.4.9) in Eq. (19.4.8), we obtain E ½C X (t1 ) as follows: E½CX (t1 ) ¼ CX (t1 ) þ s2m^ X 9 8 ð t1 jtj > > > > 1 (t) þ C (t t )dt ½C X X 1 > > > > T > > T > > > > ð = T (T t1 ) > t1 > > > > ð > > Tt1 h > > > > t þ t1 i > ; : þ ½CX (t) þ CX (t t1 )dt > 1 T 0 (19:4:10)

19.4

Estimation of Parameters of Random Processes

449

For large values of T or for low values of the lag t1, the terms within braces can be approximated as 2s2m^ X , and the approximation for E ½CX (t1 ) is E ½C X (t1 ) CX (t1 ) s2m^ X ,

0 , t1 T

(19:4:11)

Thus, CX (t) is a biased estimator with the bias B s2m^ X . In an analogous manner the expected value of C^ X (t1 ) given by Eq. (19.4.6) can be evaluated as follows: ð Tt1 1 ^ ½(X(t) mX )(X(t þ t) mX ) þ E(m^ X mX )2 dtg E ½CX (t1 ) ¼ E T 0 ð Tt1 1 E ½(X(t) mX )(m^ X mX ) þ ½(X(t þ t1 ) mX )(m^ X mX )dt T 0 ð ð T t1 1 Tt1 T 2 ½CX (t1 ) þ sm^ X 2 CX (s t)ds dt ¼ T T 0 0 ð ð 1 Tt1 T 2 CX (s t t1 )ds dt, 0 , t1 T T 0 0 (19:4:12) By substituting t ¼ (s – t) and u ¼ t and performing the integration in the parallelogram shown in Fig. 19.4.1, the ﬁnal result similar to Eq. (19.4.10) is T t1 ½CX (t1 ) þ s2m^ X E ½C^ X (t1 ) ¼ T 9 8 ð t1 jtj > > > > 1 (t) þ C (t t ) dt ½C X X 1 > > > > T > > T > > > > ð0 h = T T> t1 > > > > ð Tt1 h > > > > > > t þ t1 i > ; : þ ½CX (t) þ CX (t t1 ) dt > 1 T 0 (19:4:13) For large values of T or for low values of the lag t1, the terms within braces can be approximated as ½(T t1 )=T2s2m^ X , and the approximation for E ½C^ X (t1 ) is h t1 i , 0 , t1 T (19:4:14) E ½C^ X (t1 ) ½CX (t1 ) s2m^ X 1 T Thus, C^ X (t1 ) is also a biased estimator with a larger bias than C X (t) with the bias h t1 t1 i , 0 , t1 T B CX (t1 ) s2m^ X 1 T T We will show later that C^ X (t) is the preferred estimator as it gives better minimum meansquare error. Example 19.4.1 The autocovariance function of a random process X(t) is CX (t) ¼ eajtj , with a ¼ 0.5. The data interval T ¼ 50. We will ﬁnd the variance of the estimated mean

450

Random Processes

value m^ X as a function of T, compare the true value CX(t) to E½CX ðtÞ given by Eq. (19.4.10) and E ½C^ X (t) given by Eq. (19.4.13), and ﬁnally compare the approximations to E ½C^ X (t) given by Eq. (19.4.14). Variance s2m^ X (T ): From Eq. (19.4.2), we obtain

s2m^ X (T )

2 ¼ T ¼

ð T

2½e

ð h ti 2 T at h ti CX (t) 1 dt ¼ e 1 dt T T 0 T

0 aT

þ aT 1 a2 T 2

A graph of s2m^ X (T ) is shown in Fig. 19.4.2 for a ¼ 0.5, and the estimate of the mean m^ X is asymptotically unbiased since limT!1 s2m^ X (T ) ! 0. Means of Estimated Autocovariances. C X (t) and C^ X (t). From Eq. (19.4.10) the mean value of C X (t) is given by 2 e50a þ 50a 1 E CX (t) ¼ e þ 2500a2 ðt ab 1 jbj e þ ea(bt) 1 db 50 t 50 50 ð0 ab 1 t e þ ea(bt) 1 db 50 t t 50 ð 50t ab 1 bþt e þ ea(bt) 1 db, 50 t 0 50

at

FIGURE 19.4.2

0 , t 50

19.4

Estimation of Parameters of Random Processes

451

and from Eq. (19.4.13) the mean value of C^ X (t) is given by h i 50 t 2 e50a þ 50a 1 at ^ e þ E CX (t) ¼ 50 2500a2 ðt ab 1 jbj a(bt) 1 e þe db 50 50 50 ð 1 0 ab t e þ ea(bt) 1 db 50 t 50 ð 1 50t ab bþt e þ ea(bt) 1 db, 50 0 50

0 , t 50

In graphing the estimated autocovariance functions, it is usual to plot the lag length t upto 10– 20% of the length T. Expected values E ½C X (t) and E ½C^ X (t) are shown along with CX (t) in Fig. 19.4.3 for t ¼ 0– 10 and tabulated in Table 19.4.1. The bias as shown in the ﬁgure is approximately equal to B s^ 2mX (T ) ¼ 0:0768.

FIGURE 19.4.3

TABLE 19.4.1. t

CX (t)

E[C X (t)]

CX (t)

ˆ X (t)] E[C

CˆX (t)]

0 1 2 3 4 5 6 7 8 9 10

1 0.606531 0.367879 0.22313 0.135335 0.082085 0.049787 0.030197 0.018316 0.011109 0.006738

0.9232 0.529171 0.291129 0.147612 0.061327 0.009859 20.020499 20.038005 20.047664 20.052531 20.054435

0.9232 0.529731 0.291079 0.14633 0.058535 0.005285 20.027013 20.046603 20.058484 20.065691 20.070062

0.9232 0.518588 0.279484 0.138755 0.056421 0.008873 20.018039 20.032684 20.040038 20.044075 20.043548

0.9232 0.519136 0.279436 0.13755 0.053852 0.004756 20.023771 20.040078 20.049127 20.053867 20.05605

452

Random Processes

The ﬁgure also shows that there is very little difference between E ½C X (t) and E ½C^ X (t). Table 19.4.1 also shows the approximation E ½C X (t) given by Eq. (19.4.11) and the approximation E ½C^ X (t) given by Eq. (19.4.14). Covariance of ACF Estimators. Before ﬁnding the covariance of the estimator C X (t) at two different lags t1 and t2 , we will ﬁrst ﬁnd the covariance cov ½X(t)X(tþt1 ), X(t)X(s þ t2 ): cov½X(t)X(t þ t1 ), X(s)X(s þ t2 ) ¼ E{½(X(t) mX )(X(t þ t1 ) mX )½(X(s) mX )(X(s þ t2 ) mX )} ¼ E ½(X(t) mX )(X(t þ t1 ) mX )(X(s) mX )(X(s þ t2 ) mX ) E ½(X(t) mX )(X(t þ t1 ) mX )E ½(X(s) mX )(X(s þ t2 ) mX ) (19:4:15) Expressing E ½(X(t) mX )(X(t þ t1 ) mX )(X(s) mX )(X(s þ t2 ) mX ) of Eq. (19.4.15) in terms of the fourth cumulant k4 from Eq. (11.6.8), we obtain E ½(X(t) mX )(X(t þ t1 ) mX )(X(s) mX )(X(s þ t2 ) mX ) ¼ k4 (s t, t1 , t2 ) þ CX (t1 )CX (t2 ) þ CX (s t)CX (s t þ t2 t1 ) þ CX (s t þ t2 )CX (s t t1 ) and substituting in Eq. (19.4.15), we obtain cov½X(t)X(t þ t1 ), X(s)X(s þ t2 ) ¼ k4 (s t, t1 , t2 ) þ CX (s t)CX (s t þ t2 t2 ) þ CX (s t þ t2 )CX (s t t1 ) (19:4:16) which is an exact expression. The presence of the fourth cumulant k4 in Eq. (19.4.16) renders evaluation of the covariance intractable. However, if X(t) is a Gaussian random process, then the fourth cumulant k4 is zero and the covariance of [X(t)X(t þ t1), X(s)X(s þ t2)] becomes cov½X(t)X(t þ t1 ), X(s)X(s þ t2 ) ¼ CX (s t)CX (s t þ t2 t2 ) þ CX (s t þ t2 )CX (s t t1 )

(19:4:17)

Assuming without loss of generality that 0 t1 t2, the covariance of the estimator C X (t) at two different lags t1 and t2 can be given by cov½C X (t1 ), CX (t2 ) ð Tt1 ð Tt2 1 1 X(t)X(t þ t1 ) dt X(s)X(s þ t2 ) ds ¼ cov (T t1 ) 0 (T t2 ) 0 ð Tt2 ð Tt1 1 cov½X(t)X(t þ t1 )X(s)X(s þ t2 )dt ds ¼ (T t1 )(T t2 ) 0 0 (19:4:18)

19.4

Estimation of Parameters of Random Processes

453

FIGURE 19.4.4

Substituting for the covariance in Eq. (19.4.18) from Eq. (19.4.16), we obtain cov½CX (t1 ), C X (t2 ) ¼

1 (T t1 )(T t2 )

ð Tt2 ð Tt1 0

0

CX (s t)CX (s t þ t2 t1 ) þ CX (s t þ t2 )CX (s t t1 )dt ds (19:4:19)

We now make the transformation t ¼ (s 2 t) and u ¼ t, and as the Jacobian of the transformation is 1, we can use techniques similar to those employed in Eq. (19.4.8) to determine the covariance. This transformation amounts to ﬁnding the integral in the parallelogram shown in Fig. 19.4.4. The integration along the t axis is from 2(T 2 t2) to (T 2 t1). Thus, for a Gaussian process X(t), we obtain cov½CX (t1 ), CX (t2 ) ð ð Tt1 CX (t)CX (t þ t2 t1 ) 1 du ¼ (T t1 )(T t2 ) (Tt2 ) þ CX (t þ t2 )CX (t t1 )d t (19:4:20) Ð The integral du in Eq. (19.4.20) is calculated in three regions as shown in Fig. 19.4.4 in a manner similar to that used in Eqs. (19.4.9): Region I: 2(T 2 t2) , t 0 ð tþT=2t2 T=2

jtj t2 du ¼ T þ t t2 ¼ T 1 T

(19:4:21a)

Region II: 0 (t2 2 t1) ð tþT=2t2 tT=2

t2 du ¼ T t2 ¼ T 1 T

(19:4:21b)

454

Random Processes

Region III: (t2 2 t1) , t (T 2 t1) ð T=2t1 tT=2

t þ t1 du ¼ T t t1 ¼ T 1 T

(19:4:21c)

Substituting Eqs. (19.4.21) into Eq. (19.4.20), the covariance between adjacent values t1 and t2 of the sample autocovariance for a Gaussian process X(t) is given by 1 (T t1 )(T t2 ) 8 ð0 (T jtj t2 )½CX (t)CX (t þ t2 t1 ) > > > > > þCX (t þ t2 )CX (t t1 )dt > Tt2 > > < ð t2 t1 (T t2 )½CX (t)CX (t þ t2 t1 ) þ > þCX (t þ t2 )CX (t t1 )dt 0 > > ð Tt1 > > (T t t1 )½CX (t)CX (t þ t2 t1 ) > > > :þ þCX (t þ t2 )CX (t t1 )dt t2 t1

cov½CX (t1 ), C X (t2 ) ¼

9 > > > > > > > > = > > > > > > > > ;

(19:4:22)

Although Eq. (19.4.22) was derived with the condition 0 t1 t2, it can be shown that it holds good for all t1 and t2. The covariance Ð is generally difﬁcult to evaluate. For large T, (T 2 t1) T (T 2 t2), and the integral du T in Eq. (19.4.21). With these approximations the distinction between CX (t) and C^ X (t) becomes moot and Eq. (19.4.22) becomes cov½CX (t1 ), C X (t2 ) cov½C^ X (t1 ) C^ X (t2 ) ð 1 1 ½CX (t)CX (t þ t2 t1 ) þ CX (t þ t2 )CX (t t1 )dt T 1

(19:4:23)

The variance of the sample autocovariance function is obtained by substituting t1 ¼ t2 ¼Ð s in Eqs. (19.4.20) and (19.4.22). With this substitution Eqs. (19.4.21) become du ¼ T 2 s 2 jtj, and the general form for the variance is var½C X (s) ¼ ð Ts

1 (T jsj)2

(T s jtj)½CX2 (t) þ CX (t þ s)CX (t s) dt,

s,T

(19:4:24)

(Ts)

and the corresponding variance for the estimator C^ X (t) is obtained by substituting T 2 for (T jsj)2 in the denominator of Eq. (19.4.24). The estimator C^ X (t) is more often used since it yields a minimum mean-square estimator as shown in Example 19.4.2. The approximations for the variances CX (s) and C^ X (t) corresponding to Eq. (19.4.16) for large T are ð 1 1 ^ ½C 2 (t) þ CX (t þ s)CX (t s)dt (19:4:25) var½CX (s) var½CX (s) T 1 X Example 19.4.2 We will continue Example 19.4.1 assuming that X(t) is a zero mean Gaussian random process with autocovariance function CX(t) ¼ e – ajtj. Since we are

19.4

Estimation of Parameters of Random Processes

455

given that X(t) is zero mean, C X(t) will be an unbiased estimator. We have to ﬁnd the ˆ X(t) and calculate the minimum meanvariances of the two estimators C X(t) and C – ajtj in Eq. (19.4.24), the variance of C X(t) square error for both. Substituting CX(t) ¼ e is given by

var CX (s) ¼

1 ðT jsjÞ2

ð Ts

(T s jtj) e2ajtj þ eajtþsj eajtsj dt

(19:4:26)

(Ts)

Since the integrand is even, Eq. (19.4.26) can be rewritten as

var C X (s) ¼

2 ðT sÞ2

ð Ts

(T s t) e2at þ eajtþsj eajtsj dt

(19:4:27)

0

where we have removed the absolute sign from the quantities where it is clear that t is always positive. Equation (19.4.27) can integrated numerically to obtain the variance of the unbiased estimator. However, it is instructive to ﬁnd a closed-form solution. The absolute value signs in the product term e – ajtþsj e – ajt2sj has to be carefully removed. In Fig. 19.4.5a, s , T/2 and the range of integration for t contains two segments (0,s) and (s, T 2 s). In Fig. 19.4.5b, s . T/2 and the range of integration for t contains only one segment (0, T – s). Hence we can write

0,s e

T 2

ajtþsj ajtsj

e

(

ea(tþs) ea(ts) ¼ e2as ,

0,ts

ea(tþs) ea(ts) ¼ e2at ,

s,tT s

¼

T ,sT 2 eajtþsj eajtsj ¼ ea(tþs) ea(ts) ¼ e2as ,

0,tT s

(19:4:28)

Substituting Eq. (19.4.28) in Eq. (19.4.27), the variance of the unbiased estimator CˆX(s) can written as follows: 9 8 ðs

2at > > 2as > > dt (T s t) e þe > > > > > > 0 > > > > ð < Ts

2at T= 2 2at dt , 0 , s þ (T s t) e þe (19:4:29) var C X (s) ¼ 2> ðT sÞ2 > s > > > > ð Ts > > > >

T > > > ; : (T s t) e2at þ e2ats dt, ,sT> 2 0

456

Random Processes

FIGURE 19.4.5

Integrating Eq. (19.4.29), we obtain the variance of the estimator CX(s) as var CX (s) 9 8 2a(Ts) e þ 2a(T s) 1 þ e2a(Ts) > > > > > > < þe2as 2a(T 2s) 1 þ 2a2 s(2T 3s), 0 , s T2 = 1 ¼ 2 > 2a2 ðT sÞ2 > > > > e2at(Ts) þ 2a(T s) 1 þ e2as T s , T , s T > ; : T 2 ð19:4:30Þ

The variance of the biased estimator CˆX(s) obtained by replacing (T 2 s)2 by T 2 in the denominator of Eq. (19.4.30). The minimum mean-square errors for both the estimators

19.4

Estimation of Parameters of Random Processes

457

FIGURE 19.4.6

are given by 2 1 2U (s) ¼ E C X (s) CX (s) ¼ var CX (s) h i2 h i 1 2 (s) ¼ E C^ X (s) CX (s) ¼ var C^ X (s) þ B2

(19:4:31)

B

ˆ X(s) is the same as its variance. The mean-square error for the unbiased estimator C ˆ X(s) as shown earlier for a zero mean process is B ¼ 2CX(t)fjtj/T], The bias B for C and substituting for B in Eq. (19.4.31), the mean-square error for the biased estimator ˆ X(s) is C h i t 2 1 2B (s) ¼ var C^ X (s) þ e2as T

(19:4:32)

The mean-square errors and the variances are shown in Fig. 19.4.6 for T ¼ 4, a ¼ 0.9. It is seen from the ﬁgure that the mean-square error for a biased estimator CˆX(s) is less than that of the unbiased one CˆX(s). Autocorrelation. If m ˆ X is zero in Eqs. (19.4.5) and (19.4.6), we obtain two corresponding estimators for the autocorrelation function RX(t): ð Tjtj 1 X(t)X(t þ t)dt, 0 t T (19:4:33a) Unbiased: RX (t) ¼ T jtj 0 ð 1 Tjtj ^ Biased: RX (t) ¼ X(t)X(t þ t)dt (19:4:34a) T 0 RX(t) is an unbiased estimator since ð Tjtj 1 E RX (t) ¼ E X(t)X(t þ t)dt T jtj 0 ð Tjtj 1 ¼ E½X(t)X(t þ t)dt ¼ RX (t) T jtj 0

(19:4:33b)

458

Random Processes

and E [RˆX(t)] is a biased estimator as shown below: ð i T jtj Tjtj jtj ^ E RX (t) ¼ E X(t)X(t þ t)dt ¼ 1 RX (t) T T 0 h

(19:4:34b)

However, RˆX(t), although biased, gives a smaller mean-square error as shown in Example 19.4.2 and hence is the preferred estimator. Normalized Autocovariance: r^ X (t) ¼

C^ X (t) C^ X (0)

(19:4:35)

rˆX(t) will not be an unbiased estimator.

19.4.2 Discrete-Time Processes In actual practice observations are made on the sample function of a stationary ergodic random process X(t) at equally spaced time intervals t1, t2, . . . , tn with corresponding random variables X1, X2, . . . , Xn. These observation random variables will not be independent and will be used in the estimation of mean and autocovariance of the random process. Since fXig are samples of a stationary random process, the means and variances of these samples are the same as those of the original process: E ½Xi ¼ mX : var½Xi ¼ s2X , i ¼ 1, . . . ; n

(19:4:36)

Estimation of the Mean

In the previous chapter we estimated the mean m ˆ X of a random variable by minimizing its variance from a set of independent observation random variables fX1, . . . , Xng. We can use the same techniques to estimate the mean m ˆ X of the random process X(t) except that the set fX1, . . . , Xng is not independent. We deﬁne the estimated mean mˆX as the weighted sum of the dependent observations fX1, . . . , Xng: m^ X ¼

n X

ai Xi ¼ aT X ¼ XT a

(19:4:37)

i¼1

We have to determine the weight vector a if possible under the criteria of unbiasedness and minimum variance. The unbiasedness criterion yields the condition n X

a i ¼ aT 1 ¼ 1

(19:4:38)

i¼1

where 1 is a unity vector deﬁned by 1 ¼ [1, . . . ,1]T. The variance of the estimator mˆX is given by var(m^ X ) ¼ s2m^ X ¼ E aT (X mX )(X mX )T a ¼ aT CX a

(19:4:39)

19.4

Estimation of Parameters of Random Processes

459

where CX is the autocovariance matrix of the vector random variable X, deﬁned by CX ¼ E (X mX )(X mX )T 2 3 CX (0) CX (1) CX (2) CX (n 1) 6 CX (1) CX (0) CX (1) CX (n 2) 7 6 7 6 7 C (2) C (1) C (0) C (n 3) 6 7 ð19:4:40Þ X X X X ¼6 7 .. .. .. .. 6 7 4 5 . . . . CX ½(n 1) CX ½(n 2) CX ½(n 3) CX (0) with CX(h) deﬁned as the discrete autocovariance given by CX (h) ¼ E (Xi mX )(Xiþh mX

(19:4:41)

and CX(h) ¼ CX( –h) since CX is a real symmetric matrix. Equation (19.4.40) has to be minimized subject to the constraint of Eq. (19.4.38). The unconstrained function to be minimized after adjoining with the Lagrange multiplier l is given by J ¼ aT CX a þ l(aT 1 1)

(19:4:42)

where 1 is an n vector of ones given by 1 ¼ [1, 1, . . . , 1]T. Differentiating with respect to the vector a, we obtain @J @ T ¼ a CX a þ l(aT 1 1) ¼ 2CX a þ l1 ¼ 0 @a @a Hence l 1 a ¼ C1 2 X

(19:4:43)

Solving for l from the constraint equation [Eq. (19.4.38)] and substituting this result in Eq. (19.4.43), we obtain a¼

C1 X 1 1T C1 X 1

(19:4:44)

The solution for the coefﬁcients fai, i ¼ 1, . . . , ng will not be the same for every n. If the Xi values are independent, then CX ¼ I and Eq. (19.4.44) reduces to 1 I 1 1 s2X 1¼ T 1¼ 1 a¼ 1 n 1 1 1T 2 I:1 sX

(19:4:45)

and all coefﬁcients have the same weight equaling 1/n, a result already obtained in Eq. (18.3.9). As it is, each ai in Eq. (19.4.44) will be different and the solution becomes cumbersome. To make the solution intuitively appealing we will assume that all the coefﬁcients faig have the same weight ¼ 1/n and the estimator is given by n 1X m^ X ¼ Xi (19:4:46) n i¼1 The estimator given by Eq. (19.4.46) will be unbiased but will not be minimum variance. The variance of the estimator m ˆ X can be obtained from Eq. (19.4.39) 1 (19:4:47) var(m^ X ) ¼ s2mX ¼ 2 1T CX 1 n

460

Random Processes

or, in expanded form var(m^ X ) ¼

s2mX

" # n1 X 1 h ¼ 1 CX (h) CX (0) þ 2 n n h¼1

(19:4:48)

The variance given by Eq. (19.4.48) is not minimum. For independent Xis, CX(h) ¼ 0, h = 0, and the variance s2m^ X , given by s2m^ X ¼

1 T nCX (0) s2X 1 CX ¼ ¼ 2 n n n2

is a minimum, a result that has been derived previously [in Eq. (18.3.11)].

Example 19.4.3 The autocovariance function of a random process X(t) is CX (h) ¼ eajhj , with a ¼ 0.5 as in Example 19.4.1. The number of data points n ¼ 50. We will ﬁnd the variance of the estimated value m^ x . From Eq. (19.4.48) we can write the following equation for the estimated variance: " # n1 X 1 h ah 2 1 e 1þ2 var(m^ X ) ¼ sm^ X (n) ¼ n n h¼1 A closed-form solution for the summation given above can be obtained as a

s2m^ X (n) ¼

1 (n þ 2)e2a þ (n 2)e ne3a 2ea(n2) þ 2ea(n1) n n2 (1 ea )(1 2ea þ e2a )

and substituting a ¼ 0.5 in this equation, we obtain s2m^ X (n) ¼

pﬃﬃﬃ 1 ½7:3258e(n=2) (e e) þ 4:083n 7:83541 n2

and this is graphed in Fig. 19.4.7 along with the value CX(0)/n as the variance of the estimated mean for independent observations. The variance for n ¼ 50 is s2m^ X (50) ¼ 0:0785. We can see from the ﬁgure that the estimated mean m^ X is asymptotically unbiased since the variance s2m^ X (n) ! 0 as n ! 1.

FIGURE 19.4.7

19.4

Estimation of Parameters of Random Processes

461

Estimation of the Autocovariance

The autocovariance of a stationary ergodic random process X(t) has already been deﬁned [Eq. (19.2.15)] as CX (h) ¼ E{½X(t) mX ½X(t þ h) mX }

(19:4:49)

where t has been replaced by h for convenience. The estimator CX (h) for the ACF from the observed random variables set {X1 , . . . , Xn } will be of the form CX (h) ¼

nh X

ai aiþh (Xi m^ X )(Xiþh m^ X )

(19:4:50)

i¼1

where m^ X is the unbiased estimator of the mean from Eq. (19.4.46). We have to choose, if possible, the coefﬁcients {ai , i ¼ 1, . . . , n} such that E ½CX (h) ¼ CX (h). Equation (19.4.50) can be rewritten as follows: CX (h) ¼

nh X

ai aiþh ½(Xi mX ) (m^ X mX )½(Xiþh mX ) (m^ X mX )

i¼1

Expanding this equation and taking expectations, we obtain " # nh X (Xi mX )(Xiþh mX ) þ (m^ X mX )2 ai aiþh E E ½CX (h) ¼ (Xi mX )(m^ X mX ) (Xiþh mX )(m^ X mX ) i¼1 ¼

nh X

ai aiþh {CX (h) þ s2m^ X }

i¼1

" ¼

ai aiþh E ½(Xi mX )(m^ X mX )

i¼1

nh X

ai aiþh E(Xiþh mX )(m^ X mX )

i¼1 nh X

nh X

# ai aiþh (CX (h) þ

s2m^ X )

(S1 þ S2 )

(19:4:51)

i¼1

where the summations S1 and S2 have been deﬁned as follows: S1 ¼

nh X

ai aiþh E ½(Xi mX )(m^ X mX )

i¼1

S2 ¼

nh X

(19:4:52) ai aiþh E ½(Xiþh mX )(m^ X mX )

i¼1

Substituting for m^ X from Eq. (19.4.46) in the summation S1 in Eq. (19.4.52), we can write S1 ¼

nh X n nh X n 1X 1X ai aiþh E ½(Xi mX )(X j mX ) ¼ ai aiþh CX ( j i) n i¼1 j¼1 n i¼1 j¼1

(19:4:53)

This equation represents the sum of every term of the (n 2 h) n matrix shown in Fig. 19.4.8.

462

Random Processes

FIGURE 19.4.8

The summation is carried out in three stages. The shaded parallelogram in region II consists of the sum of (n 2 h) (h þ 1) terms, and the shaded triangles in regions I and III consists of the remaining (n 2 h) (n 2 h 2 1) terms. Equation (19.4.53) can be rewritten in terms of these summations as follows:

S1 ¼

8 h X nh X > > < ai aiþh CX ( j) 1 n> > :

j¼0 i¼1

nh1 nh X X

þ

ai aiþh CX ( j)

j¼1 i¼jþ1

Region II

nh1 X nhj X

þ

j¼1

9 > > ai aiþh CX ( j þ h)=

i¼1

Region I

Region III

> > ; (19:4:54)

An equation similar to Eq. (19.4.53) for S2 is

S2 ¼

¼

nh X n 1X ai aiþh E ½(Xiþh mX )(X j mX ) n i¼1 j¼1 nh X n 1X ai aiþh CX ( j i h) n i¼1 j¼1

(19:4:55)

A summation matrix similar to that in Fig. 19.4.8 is shown in Fig. 19.4.9. From Fig. 19.4.9 we can write the summation in much the same way as we did in Eq. (19.4.54):

S2 ¼

8 h nh 9 nh1 nh1 nh X nhj X X X XX > > > = ai aiþh CX ( j) þ ai aiþh CX ( j) þ ai aiþh CX ( j þ h)> 1< n> > :

j¼0 i¼1

j¼1 Region II

i¼1

j¼1 i¼jþ1 Region III

Region I

> > ; (19:4:56)

19.4

Estimation of Parameters of Random Processes

463

FIGURE 19.4.9

Substituting Eqs. (19.4.55) and (19.4.56) in Eq. (19.4.51), we can write E ½CX (h) as " !# nh h X 2X 2 ai aiþh CX (h) þ sm^ X CX ( j) E ½CX (h) ¼ n j¼0 i¼1 " #" # nhj nh1 X 1 X (ai aiþh þ aiþj aiþhþj ) CX ( j) þ CX (h þ j) n i¼1 j¼1 (19:4:57) where the variance of the sample mean m^ X is as follows: " # n1 X 1 h CX (0) þ 2 1 CX (h) s2m^ X ¼ n n h¼1

(19:4:48)

The analysis becomes intractable ﬁnding the coefﬁcients faig such that E[CX(h)] ¼ CX (h). In the spirit of Eqs. (19.4.5) and (19.4.6), two estimators for the ACF CX (h) can be deﬁned by substituting ai aiþh ¼ ½1=(n h) or ai aiþh ¼ (1=n) in Eq. (19.4.50): CX (h) ¼

nh 1 X (Xi m^ X )(Xiþh m^ X ) n h i¼1

1 C^ X (h) ¼ n

nh X

(Xi m^ X )(Xiþh m^ X )

(19:4:58)

(19:4:59)

i¼1

The expected value E ½C X (h) is obtained by substituting ai aiþh ¼ ½1=(n h) in Eq. (19.4.57) " # h 2X 2 CX ( j) E ½CX (h) ¼ CX (h) þ sm^ X n j¼0 " # nh1 X 2 (n h j)½CX ( j) þ CX (h þ j) (19:4:60) n(n h) j¼1

464

Random Processes

and the expected value E ½C^ X (h) is obtained by substituting ai aiþh ¼ (1=n) in Eq. (19.4.57) "

!# h X h 2 E ½C^ X (h) ¼ 1 CX ( j) CX (h) þ s2m^ X n n j¼0 " # X 2 nh1 (n h j)½CX ( j) þ CX (h þ j) 2 n j¼1

(19:4:61)

where s2m^ X in these equations is as given by Eq. (19.4.48): s2m^ X

" # n1 X 1 h ¼ 1 CX (h) CX (0) þ 2 n n h¼1

(19:4:48)

Example 19.4.4 We will follow Example 19.4.3 and ﬁnd E ½C X (h) and E ½C^ X (h) for the random process whose ACF CX (h) ¼ eajhj with a ¼ 0.5 and the number of data points n ¼ 50. In Example 19.4.3, we have calculated s2m^ X for 50 data points as s2m^ X (50) ¼ 0:0785. E ½C X (h) for n ¼ 50 is obtained from Eq. (19.4.60) as " # h X 2 eaj E ½C X (h) ¼ eah þ s2m^ X (50) 50 j¼0 " # 4X 9h 2 aj a(hþj) (50 h j)½e þ e a ¼ 0:5 50(50 h) j¼1 Similarly, E ½C^ X (h) is obtained from Eq. (19.4.61): # " h h 2 X ah 2 aj ^ E ½CX (h) ¼ 1 þ sm^ X (50) e e 50 50 j¼0 " # 9h 2 4X aj a(hþj) (50 h j)½e þ e a ¼ 0:5 2 50 j¼1 E ½C X (h), E ½C^ X (h) and the true value CX (h) are shown in Fig. 19.4.10 for lags h ¼ 10 (10% of data points). It is seen that both estimators are biased but there is very little difference between them for the 10 lags shown here.

Variance of Autocovariance Estimators

Finding the variances of C X (h) and C^ X (h) is extremely tedious and is not worth the effort, particularly when the number of data points n is large. Hence the sample mean m^ X can be replaced by the true mean mX in Eqs. (19.4.58) and (19.4.59). The unbiased estimator corresponding to Eq. (19.4.58) is given by C X (h) ¼

nh 1 X (Xi mX )(Xiþh mX ) n h i¼1

(19:4:62)

19.4

Estimation of Parameters of Random Processes

465

FIGURE 19.4.10

and E ½C X (h) ¼ CX (h). The biased estimator corresponding to Eq. (19.4.59) is CX (h) ¼

nh 1X (Xi mX )(Xiþh mX ) n i¼1

(19:4:63)

As a ﬁrst step toward ﬁnding the variance C X (h), we will deﬁne a random variable Zi as Zi ¼ (Xi mX )(Xiþh mX ), and Eq. (19.4.62) becomes CX (h) ¼

nh 1 X Zi n h i¼1

From Eq. (19.4.48) the variance of C X (h) can be written as follows: " # nh1 X 1 g var½CX (h) ¼ Cz (0) þ 2 CZ (g) 1 nh nh g¼1

(19:4:64)

(19:4:65)

The mean value of Zi is CX (h) for all i and the autocovariance of Zi can be deﬁned by CZ (g) ¼ E ½(Zi CX (h))(Ziþg CX (h)) ¼ E ½Zi Ziþg CX2 (h)

(19:4:66)

E ½Zi Ziþg ¼ E ½(Xi mX )(Xiþh mX )(Xiþg mX )(Xiþgþh mX )

(19:4:67)

where

Since E(Xk mX ) ¼ 0 for all k, we can write E ½Zi Ziþg in terms of the fourth cumulant k4 from Eq. (11.6.8): E ½Zi Ziþg ¼ k4 (h,g) þ CX2 (h) þ CX2 (g) þ CX (g þ h)CX (g h)

(19:4:68)

and substituting this equation in Eq. (19.4.66), we obtain CZ (g) ¼ k4 (h,g) þ CX2 (g) þ CX (g þ h)CX (g h)

(19:4:69)

CZ (0) ¼ k4 (h,0) þ CX2 (0) þ CX2 (h)

(19:4:70)

466

Random Processes

Substituting Eqs. (19.4.69) and (19.4.70) in Eq. (19.4.65), we obtain var½CX (h) 9 8 k4 (h,0) þ CX2 (0) þ CX2 (h) > > > > > > = 1 < nh1 ¼ X g n h> > > 1 ½k4 (h,g) þ CX2 (g) þ CX (g þ h)CX (g h) > > > ; : þ2 n h g¼1 (19:4:71) For a Gaussian process the fourth cumulant k4 is zero, and hence Eq. (19.4.71) becomes 9 8 2 CX (0) þ CX2 (h) > > > > > > = < 1 nh1 var½CX (h) ¼ X g n h> > > ½CX2 (g) þ CX (g þ h)CX (g h) > 1 > >þ2 ; : n h g¼1 (19:4:72) Or, more compactly 1 var½CX (h) ¼ (n h)2

(

nh1 X

) (n h

jgj)½CX2 (g)

h,n

þ CX (g þ h)CX (g h) ,

g¼(nh1)

(19:4:73) Equation (19.4.73) is the discrete counterpart of the continuous process of Eq. (19.4.24) and represents the variance of an unbiased estimator C X (h) of a Gaussian process with known mean value mX. In a similar manner the variance of the biased estimator C^ X (h) of a Gaussian process with known mean value mX given by Eq. (19.4.63) can be obtained as follows: 9 8 2 CX (0) þ CX2 (h) > > > > > > = < n h nh1 var½C^ X (h) ¼ 2 X g n > > > ½CX2 (g) þ CX (g þ h)CX (g h) > 1 > > þ2 ; : n h g¼1 (19:4:74) Example 19.4.5 We will continue the same way as in Example as 19.4.4 and ﬁnd the variances of the unbiased estimator CX (h) and the biased estimator C^ X (h). We are given CX (h) ¼ e 2ajhj with a ¼ 0.5 and n ¼ 50. From Eq. (19.4.73), var½CX (h) is given by 9 8 1 þ e2ah > > = < 1 nh1 X g var½CX (h) ¼ 2ag a(jgþhj) a(jghj) ½e 1 þe e > n h> : þ2 ; nh g¼1

As in Example 19.4.2, the absolute value signs in the product term eajgþhj eajghj have to be carefully removed. Following the procedure in that example, we can write

19.4

0h

n1 : 2 eajgþhj eajghj ¼

Estimation of Parameters of Random Processes

ea(gþh) ea(gh) ¼ e2ah , ea(gþh) ea(gh) ¼ e2ag ,

467

0gh h,gnh1

n1 , h n: 2 eajgþhj eajghj ¼ ea(gþh) ea(gh) ¼ e2ah ,

0gnh1

The variance equation CX (h), after substituting the equations above, is var½C X (h) 9 8 h X g 2ag > > > > 2ah 2ah > > ½e þ2 1 þe 1þe > > > > n h > > > > g¼1 > > > > > > nh1 < X g n1= 1 2ag 2ag þ2 ½e 1 þe , 0 h ¼ nh 2 > n h> > > g¼hþ1 > > > > > > > > nh1 X > > g n 1 > > 2ag 2ah > > > 1 þe , ½e ,hn > ; : 2 n h 2 g¼1 The variance of the biased estimator C^ X (h) is calculated by substituting (n h)=n2 for 1=(n h) in the preceding equation. Both variances are shown in Fig. 19.4.11. The result is very similar to the one for the continuous case given in Fig. 19.4.6, and the biased estimator gives a better minimum mean-square error. Estimation of Autocorrelation Functions

The two estimators for the autocorrelation function are obtained by substituting mˆ ¼ 0 in the two estimators for the autocovariance function given by Eqs. (19.4.58) and (19.4.59): R X (h) ¼

nh 1 X Xi Xiþh n h i¼1

FIGURE 19.4.11

(19:4:75a)

468

Random Processes

RX (h) is an unbiased estimator since

E ½RX (h) ¼

nh nh 1 X 1 X E ½Xi Xiþh ¼ RX (h) ¼ RX (h) n h i¼1 n h i¼1

1 R^ X (h) ¼ n

nh X

Xi Xiþh

(19:4:75b)

(19:4:76a)

i¼1

R^ X (h) is biased estimator since nh nh 1 X 1 ^ E ½Xi Xiþh ¼ 1 RX (h) E ½RX (h) ¼ n n h i¼1 n

(19:4:76b)

with the bias B ¼ {½RX (h)=n}. Further analyses of these estimators are the same as those of the ACF estimators C X (h) and C^ X (h) with the estimated mean m^ X set equal to 0.

Estimation of Normalized Autocovariance Functions (NACFs)

Estimation of Mean As mentioned earlier, the normalized autocovariance rX (h) (NACF) plays a key role in modeling time series. The NACF for a stationary random process for discrete time can be deﬁned by an equation analogous to Eq. (19.2.16) as follows: rX (h) ¼

CX (h) CX (0)

and CX (h) ¼ rX (h)CX (0)

(19:4:77)

The unbiased estimator C X (h) for large n is deﬁned in Eqs. (19.4.62). The estimated NACF r X (h) for large sample n can be deﬁned as r X (h) ¼

C X (h) C X (0)

(19:4:78)

resulting in a ratio of random variables, and the analysis becomes rather involved. We will now determine the mean of r X (h). Since CX (h) is an unbiased estimator, we can deﬁne its 0 0 variations about the mean value CX (h) as C X (h) with E ½C X (h) ¼ 0 and write CX (h) ¼ 0 CX (h)þ C X (h): " # 0 CX (h) CX (h) þ CX (h) E ½rX (h) ¼ E ¼E 0 CX (0) CX (0) þ CX (0) 8 " #1 9 0 > > > > C (0) 0 X > > > > > > = > CX (0) > > > > > > > > ; :

(19:4:79)

19.4

Estimation of Parameters of Random Processes

469

0

Assuming that the variations C X (h) are small, we can expand Eq. (19.4.79) in Taylor series, and neglecting second-order terms, we have " # 0 1 CX (h)C X (0) CX (h) 0 E CX (h) þ CX (h) ¼E E ½rX (h) ¼ rX (h) CX (0) CX (0) CX (0)

(19:4:80) 0

Thus, we have an unbiased estimator under the assumption of small variations in C X (h). Estimation of Covariance of NACF Finding the covariance between adjacent values of the estimated NACF r X (h) is very involved and is detailed elsewhere in the literature [8,24,31]. The ﬁnal result for a Gaussian process X(t) is as follows:

cov½rX (h), r X (h þ g) 9 8 1 > > > ½rX (k)rX (k þ g) þ rX (k þ h þ g)rX (k h) > > > > > nh > > > > > > 2 > > > > > r (h)r (k)r (k h g) > X X X = < 1 X nhg ¼ > > 2 r (h þ g)r (k)r (k h) > k¼1> > > X X > > > > nh X > > > > > > > > 2 > > ; : þ r2 (k)r (h)r (h þ g) X X X n

(19:4:81)

The variance of r X (h) is obtained by substituting g ¼ 0 in Eq. (19.4.81). The result is 8 9 1 > 1 > < = ½r2X (k) þ rX (k þ h)rX (k h) X nh (19:4:82) var½rX (h) ¼ 4 2 2 > > 2 ; k¼1: r (h)rX (k)rX (k h) þ rX (k)rX (h) nh X n Large-Lag Approximations

For most stationary random processes the NACF becomes negligibly small after some lag q, that is, rX (h) 0, jhj . q. Hence, under the assumption rX(h) ¼ 0 for jhj . q in Eq. (19.4.81), we can write the large-lag covariance as cov½rX (h), r X (h þ g)

q 1 X r (k)rX (k þ g), n h k¼q X

jhj . q

(19:4:83)

Similarly, the large-lag variance after substituting rX(h)¼0 for jhj . q in Eq. (19.4.82) is var½rX (h)

q 1 X 2 r (k), n h k¼q X

jhj . q

(19:4:84)

If the estimator for ACF is C^ X (h), then the divisor (n 2 h) in Eqs. (19.4.83) and (19.4.84) is replaced by the divisor n. Thus the variance equation [Eq. (19.4.84)] becomes var½^rX (h)

q 1X 2 r (k), n k¼q X

jhj . q

(19:4:85)

470

Random Processes

It has been shown [3,25] that if the true NACF rX(h) is zero for jhj . q, then the estimated NACF rX(h) is normally distributed with zero mean and variance given by Eq. (19.4.84). This property is called Anderson’s theorem.

Discrete White-Noise Process

A zero mean stationary discrete white-noise process fXig has the following property: E ½Xi Xj ¼ dji where the Kro¨necker delta dji ¼ 1, j ¼ i and dji ¼ 0, j = i, an equation analogous to Eq. (19.2.17) for continuous white noise. As a consequence, the NACF rX(h) has the property 1, h ¼ 0 rX (h) ¼ (19:4:86) 0, h = 0 Hence the variance of the estimated r X (h) for white noise is obtained by substituting rX (k) ¼ 0 for k = 0 in Eq. (19.4.84) to yield var½rX (h)

1 nh

(19:4:87)

According to Anderson’s theorem, the estimated NACF r X (h) of a stationary discrete white-noise process is Gaussian-distributed with zero mean and variance 1=(n h). As a result, we can construct a test for white noise from knowledge of r X (h). Since the pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ variance is 1=(n h), the 95% conﬁdence interval (+2s) is +2= n h. If the estimated NACF is given by r^ X (h) ¼

C^ X (h) C^ X (0)

(19:4:88)

then, from Eq. (19.4.85), the variance of the estimator r^ X (h) is 1 n 2 and the corresponding conﬁdence interval is + pﬃﬃn. var½^rX (h)

(19:4:89)

Example 19.4.6 We have generated 100 points of a zero mean unit variance Gaussian white noise fXig by the computer. The plot is shown in Fig. 19.4.12. The estimated NACFs r X (h) and r^ X (h) of this sequence were determined from Eqs. (19.4.78) and (19.4.88) and plotted in Fig. 19.4.13 along with their conﬁdence intervals for 50 lags. In the plot of the estimator r X (h), only two points out of 50 (4%) exceed the conﬁdence limits, whereas in the estimator r^ X (h), no points exceed the conﬁdence limits. Both estimators pass the white-noise test at the 5% signiﬁcance level comfortably, and hence we conclude that the process fXig is indeed a white-noise sequence. Example 19.4.7 We will generate 100 points of the discrete-time process for f ¼ 0.8 and 20.8 in Example 19.2.11 with s2X ¼ 1 Xi ¼ fXi1 þ ni , i ¼ 1, . . . , 100 and estimate the NACF and compare it to the true NACF. The computer-generated random sequence is shown in Fig. 19.4.14 for f ¼ 0.8 and 20.8. The ﬁgure shows that for f ¼ 20.8, the process is more oscillatory. The

19.4

Estimation of Parameters of Random Processes

FIGURE 19.4.12

FIGURE 19.4.13

FIGURE 19.4.14

471

472

Random Processes

FIGURE 19.4.15

estimated NACFs r X (h) and r^ X (h) are determined from Eqs. (19.4.78) and (19.4.88) respectively. They are shown in Fig. 19.4.15 for f ¼ 0.8 and 20.8 for 10 lags. Here also the NACFs for f ¼ 20.8 show oscillatory behavior. For the 10 lags shown,there is very little difference between the two estimators r X (h) and r^ X (h). However, the discrepancy between the true NACF and the estimators for higher lags is to be expected.

B 19.5

POWER SPECTRAL DENSITY

19.5.1 Continuous Time In signal analysis power spectra are associated with Fourier transforms that transform signals from the time domain to the frequency domain. The same concept is also applicable to stationary random processes. The correlation functions represent stationary processes in the time domain. We can transform them to the frequency domain by taking their Fourier transforms. The power spectral density (psd) function SX (v) of a real stationary random process X(t) is deﬁned as the Fourier transform of the autocorrelation function: ð1 RX (t)ejvt dt (19:5:1) SX (v) ¼ FT½RX (t) ¼ 1

From the Fourier inversion theorem we can obtain the autocorrelation function from the power spectral density: ð 1 1 SX (v)e jvt dv (19:5:2) RX (t) ¼ IFT½SX (v) ¼ 2p 1 Equations (19.5.1) and (19.5.2) are called the Wiener –Khinchine theorem. Since RX(0) ¼ E [X 2(t)], the average power in the random process, we obtain the following from Eq. (19.5.2): RX (0) ¼ E ½X 2 (t) ¼

1 2p

ð1

ð1 SX (v)dv ¼

1

SX ( f )df 1

(19:5:3)

19.5

Power Spectral Density

473

Thus, SX ( f ) represents the average power per hertz, and hence the term power spectral density. Since RX (t) is an even function, we can rewrite Eq. (19.5.1) as ð1 SX (v) ¼

ð1 RX (t)½cos(vt) þ j sin(vt)dt ¼

RX (t) cos(vt)dt

1

(19:5:4)

1

and SX (v) is also an even function. Hence Eq. (19.5.2) can be rewritten as RX (t) ¼

1 2p

ð1 SX (v) cos(vt)dv

(19:5:5)

1

The cross-spectral density (csd) SXY(v) of two real stationary random processes X(t) and Y(t) is deﬁned as the Fourier transform of the cross-correlation function, RXY(t) ð1

RXY (t)ejvt dt

SXY (v) ¼ FT½RXY (t) ¼

(19:5:6)

1

and the inverse Fourier transform of SXY(v) gives the cross-correlation function: RXY (t) ¼ IFT½SXY (v) ¼

1 2p

ð1

SXY (v)e jvt dv

(19:5:7)

1

The cross-spectral density SXY(v) will be complex, in general, even when the random processes X(t) and Y(t) are real. The Fourier transforms can also be obtained from the tables in Appendix A. Example 19.5.1 The power spectral density (psd) of the random binary wave of Example 19.2.6 is to be determined. The autocorrelation of the random process X(t) is given by 8 <

jtj A 1 , RX (t) ¼ T : 0, 2

jtj , T otherwise

The psd SX (v) given by ð0 SX (v) ¼

ðT t t A2 1 þ ejvt dt þ A2 1 ejvt dt T T T 0

A2 ½(1 e jvT þ jvT ) þ (1 ejvT jvT ) v2 T sin(vT=2) 2 ¼ A2 T vT=2

¼

is shown in Fig. 19.5.1. As we can see from ﬁgure, the psd is an even function. Example 19.5.2 We will ﬁnd the psd from the autocorrelation RX (t) ¼ 14½1 þ e2ljtj of the random telegraph wave given in Example 19.2.7. Taking the Fourier transform, the psd

474

Random Processes

FIGURE 19.5.1

is given by SX (v) ¼

1 4

ð1 1

ejvt dt þ

1 4

ð0

e2lt ejvt dt þ

1

1 4

ð1

e2lt ejvt dt

0

1 4l p l ¼ 2pd(v) þ 2 ¼ d(v) þ 2 2 4 2 4l þ v 4l þ v2 The impulse function (p=2)d(v) represents the direct-current (dc) value of RX (t) ¼ 14. The psd is shown in Fig. 19.5.2 for l ¼ 1. Example 19.5.3 We will ﬁnd the psd from the autocorrelation RX (t) ¼ e2ljtj cos(v0 t) of Example 19.2.8. We can take the Fourier transform of RX(t) directly, but it is easier to

FIGURE 19.5.2

19.5

Power Spectral Density

475

evaluate the FT by using the frequency convolution property of the FT as follows: FT½e2ljtj ¼

4l2 4l2 þ v2

and

FT½cos(v0 t) ¼ p½d(v þ v0 ) þ d(v v0 )

Using the frequency convolution property x(t)y(t) , (1=2p)X(v) Y(v), we have e2ljtj cos(v0 t) () or

1 4l2 p½d(v þ v0 ) þ d(v v0 ) 2p 4l2 þ v2

1 1 þ SX (v) ¼ 2l 4l2 þ (v þ v0 )2 4l2 þ (v v0 )2

and simplifying, we obtain SX (v) ¼

4l(v2 þ v20 þ 4l2 ) v4 2(v20 4l2 )v2 þ (v20 þ 4l2 )2

With l ¼ 1 and v0 ¼ 2p, the psd becomes SX (v) ¼

4(v2 þ 4p2 þ 4l2 ) v4 8(p2 1)v2 þ 16(p2 þ 1)2

The psd SX(v) is shown in Fig. 19.5.3. Example 19.5.4 (Bandlimited Process) The autocorrelation function RX (t) ¼ sin(v0 t)=pt of a stationary random process X(t) is shown in Fig. 19.5.4 for v0 ¼ 2p. We have to ﬁnd the psd SX(v). From item 2 in the FT table, the Fourier transform of RX (t) can be obtained as follows: sin(v0 t) () pv0 (v), hence SX (v) ¼ pv0 (v) pt The psd SX(v) is shown in Fig. 19.5.5 for v0 ¼ 2p. The random process that has a psd as in Fig. 19.5.5 is called a bandlimited signal since the frequency spectrum exists only between –2p and 2p.

FIGURE 19.5.3

476

Random Processes

FIGURE 19.5.4

FIGURE 19.5.5

Example 19.5.5 (Bandlimited Process) RX (t) ¼

The autocorrelation function RX(t)

sin½v0 (t t0 ) sin½v0 (t þ t0 ) þ p(t t0 ) p(t þ t0 )

of a bandlimited random process X(t) is shown in Fig. 19.5.6 for v0 ¼ 2p and t0 ¼ 3.

FIGURE 19.5.6

19.5

Power Spectral Density

477

Since sin(v0 t)=pt , pv0 (v) using the time-shifting property x(t + t0 ) , X(v)e+jvt0 of the FT, we have sin½v0 (t t0 ) () pv0 (v)ejvt0 , p(t t0 )

sin½v0 (t þ t0 ) () pv0 (v)e jvt0 p(t þ t0 )

or sin½v0 (t t0 ) sin½v0 (t þ t0 ) þ () pv0 (v)½e jvt0 þ ejvt0 p(t t0 ) p(t þ t0 ) and SX (v) ¼ 2pv0 (v) cos(vt0 ) The psd is shown in Fig. 19.5.7 for v0 ¼ 2p and t0 ¼ 3.

Example 19.5.6 (Bandpass Process) RX (t) ¼

The autocorrelation function RX(t) 2 sin(v0 t) cos(vc t) pt

of a bandpass random process X(t) is shown in Fig. 19.5.8 for v0 ¼ 2p and vc ¼ 16p. We will ﬁnd the psd using the frequency convolution property of the FT: xðtÞyðtÞ ()

1 XðvÞ YðvÞ 2p

Since 2 sin(v0 t)=pt , 2pv0 (v) and cos(vc t) , p½d(v þ vc ) þ d(v vc Þ, we have 2 sin(v0 t) 1 cos(vc t) () 2pv0 (v) p½d(v þ vc ) þ d(v vc ) pt 2p

FIGURE 19.5.7

478

Random Processes

FIGURE 19.5.8

and the psd SX (v) ¼ pv0 (v þ vc ) þ pv0 (v vc ) The function SX(v) is shown in Fig. 19.5.9 for v0 ¼ 2p and vc ¼ 16p. Example 19.5.7 (White Noise) A white-noise process has the autocorrelation function RX (t) ¼ s2X d(t) given in Eq. (19.2.17). The psd of white noise is given by SX (v) ¼ s2X ,

1 , v , 1

and has a ﬂat spectrum. Since the process has all frequencies with equal power, it is called “white Ðnoise,” analogously to white light. It has inﬁnite energy since 1 RX (0) ¼ (1=2p) 1 s2X dv ¼ 1, and hence it is an idealization. Example 19.5.8 The cross-correlation function of two random processes X(t) ¼ A cos(v0 t) þ B sin(v0 t) and Y(t) ¼ A sin(v0 t) þ B cos(v0 t) in Example 19.2.9 is RXY (t) ¼ s2 sin(v0 t), E[A 2] ¼ E[B 2] ¼ s2. We will ﬁnd the cross-spectral density SXY(v). Taking FT of s2 sin(v0 t), from tables, we have SXY (v) ¼ jps2 ½d(v þ v0 ) d(v v0 ) and this is shown in Fig. 19.5.10. Note that SXY(v) is neither even nor real.

FIGURE 19.5.9

19.5

Power Spectral Density

479

FIGURE 19.5.10

Example 19.5.9 The cross-spectral density is to be found for the cross-correlation function RXY(t) derived in Example 19.2.10: 8 > 4 (t=2) 1e2t , t . 0 1 < 15 e 6 RXY (t) ¼ þ 1 2 > 2t : e , t0 10 Taking FT of each term in RXY(t), we have 4 (t=2) 4 1 e ; () 15 15 1=2 þ jv 1 () pd(v); 2

1 2t 1 1 e () 6 6 2 þ jv

1 2t 1 1 e () 10 10 2 jv

The cross-spectral density SXY(v) is obtained by adding the FT terms shown above: SXY (v) ¼ pd(v) þ

4 1 1 1 1 1 1 8 þ 2v2 þ 3jv þ pdðvÞ þ þ ¼ 15 12 þ jv 6 2 þ jv 10 2 jv 3 (1 þ 2jv)(4 þ v2 )

The cross-spectral density SXY(v) is complex and possesses no symmetry, unlike the power spectral density. The constant term 12 in RXY(t) gives rise to the impulse function in the frequency domain. The amplitude jSXY (v)j and the phase argfSXY(v)g spectra are graphed in Fig. 19.5.11. Properties of Power Spectral Densities of Stationary Random Processes 1. SX(v) is a real function. In general, the Fourier transform X(v) of any function X(t) will be complex. However, the ACF RX(t) is an even function and satisﬁes the relation RX(t) ¼ RX(2t). Hence, from the deﬁnition of psd, we obtain ð1 ð1 jvt RX (t)e dt ¼ RX (t)½cos(vt) þ j sin(vt)dt SX (v) ¼ 1

1

ð1

RX (t) cos(vt)dt

¼ 1

Ð1 since the imaginary part 1 RX (t) j sin(vt)dt ¼ 0 because an even function multiplying an odd function results in an odd function and the integral of an odd function over (21,1) is zero. 2. From property 1 the psd SX(v) is an even function and, hence it is a function of v2. (19.5.8) As a consequence SX(v) ¼ SX(2v). (19.5.9) 3. SX(v) 0: The psd is a nonnegative function of v.

480

Random Processes

FIGURE 19.5.11

4. From the Fourier transform properties: ð ð1 1 1 SX (v)dv ¼ SX ( f )df ¼ E ½X 2 (t) RX (0) ¼ 2p 1 1 ð1 SX (0) ¼ RX (t)dt

(19:5:10)

1

5. If Z(t) ¼ X(t) þ Y(t), then from Eq. (19.2.30), we have RZ (t) ¼ RX (t) þ RY (t) þ RXY (t) þ RYX (t) SZ (v) ¼ SX (v) þ SY (v) þ SXY (v) þ SYX (v) If X(t) and Y(t) SZ (v) ¼ SX (v) þ SY (v)

are

orthogonal,

then

Eq.

(19.5.11)

(19:5:11) reduces

to

Alternate Form for Power Spectral Density

The psd can also be obtained directly from the stationary random process X(t). We will truncate X(t) in the interval (2T, T ) and deﬁne the random process XT (t) as X(t), T t T XT (t) ¼ (19:5:12) 0, otherwise ÐT The FT XT (v) of XT (t), given by XT (v) ¼ T X(t)ejvt dt, is a random variable. The quantity ST (v), deﬁned by

2

ð

jXT (v)j2 XT (v)XT (v) 1

T jvt

¼ ST (v) ¼ (19:5:13) X(t)e dt

¼ 2T

2T T 2T is called the periodogram. The periodogram represents the power of the sample function X(t) at the frequency v. Equation (19.5.13) can be expanded as follows: ð ð 1 T T X(t)X(s)ejv(ts) dt ds (19:5:14) ST (v) ¼ 2T T T We can make the transformation t ¼ t 2 s and u ¼ s in Eq. (19.5.14) and perform the integration in the (t, u) plane. The Jacobian k J k of the transformation from Eq. (19.3.5) is 1,

19.5

Power Spectral Density

481

and Eq. (19.5.14) can be transformed as follows: ST (v) ¼

1 2T

ð ð

X(u)X(u þ jtj)du ejvt dt

(19:5:15)

We can obtain the limits of integration from Fig. 19.3.1 and write Eq. (19.5.15) as ð 2T ð T 1 ST (v) ¼ X(u)X(u þ t)du ejvt dt 2T Tþt 0 ð 0 ð Tjtj 1 X(u)X(u þ t)du ejvt dt þ 2T 2T T ð 2T ð0 R^ X (t)ejvt dt þ R^ X (t)ejvt dt ¼ 0

2T

ð 2T

R^ X (t)ejvt dt

¼

(19:5:16)

2T

where we have used the deﬁnition for R^ X (t) as in Eq. (19.4.34a). Using Eq. (19.4.34b), the expected value of ST(v) can be written as ð 2T h ð 2T i jtj jvt jvt ^ E½ST (v) ¼ E RX (t) e dt ¼ RX (t) 1 dt (19:5:17) e 2T 2T 2T Taking the limit of E½ST (v) in Eq. (19.5.17) as T ! 1, we have ð 1

jtj jvt RX (t) 1 dt lim E½ST (v) ¼ lim e T!1 T!1 2T 1 jtj ¼ lim FT RX (t) 1 T!1 2T

(19:5:18)

Using the frequency convolution property of the FT, we obtain 1 1 X(v) Y(v) ¼ x(t)y(t) () 2p 2p

ð1 X(p)Y(v p)dp 1

and with RX (t) , SX (v) and jtj sin2 vT 1 , 2T 2T (vT )2 Eq. (19.5.18) can be rewritten as follows: jtj lim E½ST (v) ¼ lim FT RX (t) 1 T!1 T!1 2T ð1 2 sin (v p)T SX (p)T dp ¼ lim T!1 1 p½(v p)T 2

(19:5:19)

482

Random Processes

From Ref. 36, lim T

T!1

sin2 (vT ) ! d(v) and p(vT )2

lim T

T!1

sin2 (v p)T ¼ d(v p) p½(v p)T 2

and substituting this result in Eq. (19.5.19), we have ð1 lim E½ST (v) ¼

T!1

SX (p)d(v p)dp ¼ SX (v)

(19:5:20)

1

Thus, the expected value of the periodogram as T ! 1 yields the psd SX(v).

Estimation of Power Spectral Density

We have deﬁned an estimator R^ X (t) for the autocorrelation function in Eq. (19.4.34a). It is intuitive to express the estimator S^ X (v) for the psd as the FT½R^ X (t) as follows: S^ XT ðvÞ ¼

ðT

R^ X ðtÞejvt dt : S^ X ðvÞ ¼ lim S^ XT ðvÞ T!1

T

(19:5:21)

This equation corresponds to Eq. (19.5.16). The periodogram is an asymptotically unbiased estimator for SX(v) since limT!1 E½S^ X (v) ¼ SX (v). We may be tempted to conclude that since R^ X (t) is a consistent estimator of RX(t), the Fourier transform of R^ X (t) will also be a consistent estimator of SX(v). This is not true because S^ X (v) fails to converge to SX(v) for T ! 1. We will show heuristically that var½S^ XT (v) does not approach 0 as T ! 1. From Eq. (19.5.14) the second moment E ½S^ XT (v)2 is given by ð ð ð ð h i2 1 T T T T E ½X(t)X(s)X(v)X(u) E S^ XT (vÞ ¼ 2 4T T T T T ejv(tsþvu) dt ds dv du

(19:5:22)

If the process X(t) is Gaussian, then from Eq. (11.6.8) or Example 19.3.4, we obtain E ½X(t)X(s)X(v)X(u) ¼ RX (t s)RX (v u) þ RX (t v)RX (s u) þ RX (t u)RX (s v)

(19:5:23)

Substituting Eq. (19.5.23) into Eq. (19.5.22), we have 1 E ½S^ XT (v)2 ¼ 2 4T

ðT ðT ðT ðT ½RX (t s)RX (v u) T

T

T

T

þ RX (t u)RX (s u) þ RX (t u)RX (s v) ejv(tsþvu) dt ds dv du

(19:5:24)

19.5

Power Spectral Density

483

Rearranging Eq. (19.5.24), we obtain 1 E ½S^ XT (v)2 ¼ 2T

ðT ðT T

1 2T þ þ

1 2T 1 2T 1 2T

1 2T

RX (t s)ejv(ts) dt ds

T

ðT ðT T

RX (v u)ejv(vu) dv du

T

ðT ðT T

T

T

T

T

T

T

T

ðT ðT ðT ðT

ðT ðT

RX (t u)ejv(tu) dt du RX (v s)ejv(vs) dv ds RX (t v)ejv(tþv) dt dv RX (s u)e jv(sþu) ds du

(19:5:25)

Using Eq. (19.5.21), we can simplify Eq. (19.5.25) as follows: (

2 )

ð T ð T

1

E ½S^ XT (v)2 ¼ 2{E ½S^ XT (v)}2 þ RX (t v)ejv(tþv) dt dv

4T 2 T T

(19:5:26)

As T ! 1, the second term on the righthand side of Eq. (19.5.26) tends to zero. Hence E ½S^ XT (v)2 2{E ½S^ XT (v)}2 ,

v=0

Subtracting {E ½S^ XT (v)}2 from both sides of this equation, we obtain E ½S^ XT (v)2 {E ½S^ XT (v)}2 {E ½S^ XT (v)}2 ,

v=0

or var½S^ XT (v) {E ½S^ XT (v)}2 , v = 0

(19:5:27)

Substitution of Eq. (19.5.20) limT!1 E ½SXT (v) ¼ SX (v), in Eq. (19.5.27) results in lim var½S^ XT (v) S2X (v),

T!1

v=0

(19:5:28)

Thus, as T ! 1, var½S^ XT (v) does not go to zero but is approximately equal to the psd S2X (v), and hence S^ X (v) is not a consistent estimator. To make S^ X (v) consistent, spectral windowing [25] is employed. However, with spectral windowing the asymptotically unbiased nature of the estimator is lost.

484

Random Processes

19.5.2 Discrete Time As in Eq. (19.5.1), we can also deﬁne power spectral density for a discrete-time stationary ergodic random process fXi ¼ X(ti), i ¼ 0, +1, . . .g, where the intervals are equally spaced. The autocorrelation function RX(h) has been deﬁned in Eq. (19.2.34). The psd SX(v) is the Fourier transform of RX(h) and is given by 1 X SX (v) ¼ FT½RX (h) ¼ RX (h)ejvh (19:5:29) h¼1

Since the discrete-time Fourier transforms are periodic, the psd for discrete-time random process Xi as given by Eq. (19.5.29) is periodic with period 2p. The inverse FT of SX(v) is the autocorrelation function RX(h), given by ð 1 p SX (v)e jvh dv (19:5:30) RX (h) ¼ IFT½SX (v) ¼ 2p p Example 19.5.10 (Discrete Analog of Example 19.5.1) The autocorrelation RX(h) of the discrete-time process fXig corresponding to the continuous-time process X(t) of Example 19.5.1 is given by 8 jhj < 2 A 1 , jhj , n RX (h) ¼ n : 0, otherwise and is shown in Fig. 19.5.12 for A ¼ 1 and n ¼ 10. The psd is obtained from Eq. (19.5.29) for ﬁnite n as n1 X jhj jvh 2 SX (v) ¼ A 1 e n h¼(n1) " # 1 n1 X h jvh X h jvh 2 ¼A 1þ 1þ e þ 1 e n n h¼(n1) h¼1 " # " # n1 n1 X X h h ¼ A2 1 þ 1 1 e jvh þ ejvh ¼ A2 1 þ 2 cos(vh) n n h¼1 h¼1 A2 1 cos(nv) A2 sin(nv=2) 2 ¼ ¼ n 1 cos(v) n sin(v=2) and SX (0) ¼

A 2 n2 ¼ A2 n n

The psd is graphed in Fig. 19.5.13 for A ¼ 1 and n ¼ 5 for values of v between – 2p and 2p, and the periodic nature of the psd is evident. Figure 19.5.13 is similar to Fig. 19.5.1 except for the periodicity. Example 19.5.11 (Discrete Analog of Example 19.5.2) The discrete AC function RX(h) corresponding to Example 19.5.2 is given by RX (h) ¼ s2X 1 þ e2ljhj , 1 , h , 1

19.5

Power Spectral Density

485

FIGURE 19.5.12

where 1 is a discrete unity function. The psd is determined as follows:

SX (v) ¼

1 X

s2X 1 þ e2ljhj ejvh

h¼1

"

¼

1 X

s2X

h¼1

( ¼

s2X (

¼

s2X

1 X h¼1

1:e

jvh

þ

1 X

e

2lh jvh

e

h¼1

þ

1 X

# e

2lh jvh

e

h¼1

1 X 2pd(v 2ph) þ 1 þ eh(2lþjv) þ eh(2ljv) h¼1

1 X

1 e4l 2pd(v 2ph) þ 4l 1þe 2e2l cos(v) k¼1

)

)

where d(v) is the Dirac delta function. The delta function train corresponds the discrete constant s2X ¼ 14 in the frequency domain. The plot of SX(v) is shown in Fig. 19.5.14

FIGURE 19.5.13

486

Random Processes

FIGURE 19.5.14

with s2X ¼ 15 and l ¼ 15. The graph is very similar to Fig. 19.5.2 except that it is periodic with period equal to 2p. Alternate Form for Power Spectral Density

The discrete psd can also be obtained from the stationary discrete-time random process fXkg. We will truncate this process in the interval (0, N – 1) and write 0k N1 Xk , XkN ¼ (19:5:31) 0, otherwise PN1 Xk ejvk , which is a The discrete-time FT of the sequence fXkNg is given by XN (v) ¼ k¼0 random variable. We now deﬁne the quantity SN(v) as

2

1

jX (v)j2 X (v)X (v) 1

NX N N jvk

N SN (v) ¼

¼ Xk e (19:5:32)

¼

N N k¼0 N which is the periodogram for the discrete random process. Equation (19.5.32) can be expanded as follows: SN (v) ¼

N 1 N 1 X 1X Xk Xm ejv(km) N m¼0 k¼0

(19:5:33)

Substituting h ¼ (k 2 m) and j ¼ m in Eq. (19.5.33), the summation is carried out along the diagonal with limits found by referring to Fig. 19.4.1, resulting in " # (N1)=2 N1 X X 1 SN (v) ¼ Xm Xmþh ejvh N m¼½(N1)=2þh h¼0 (19:5:34) " # ½(N1)=2jhj 0 X X 1 jvh þ Xm Xmþh e N m¼½(N1)=2 h¼(N1) The estimated autocorrelation R^ X (h) from Eq. (19.4.76) is X 1 Nh1 Xi Xiþh R^ X (h) ¼ N i¼0

(19:5:35)

19.5

Power Spectral Density

487

and substituting Eq. (19.5.35) in Eq. (19.5.34), we obtain SN (v) ¼

N1 X h¼0

R^ X (h)ejvh

h¼(N1)

N 1 X

¼

0 X

R^ X (h)ejvh þ

R^ X (h)ejvh

(19:5:36)

h¼(N1)

Taking expectations and substituting Eq. (19.4.76b) in Eq. (19.5.36), we obtain E ½SN (v) ¼

N1 X

E ½R^ X (h)ejvh ¼

h¼(N1)

N1 X jhj 1 RX (h)ejvh N h¼(N1)

(19:5:37)

Taking the limit as of Eq. (19.5.37) as N ! 1, we have jhj 1 lim E ½SN (v) ¼ lim RX (h)ejvh N!1 N!1 N h¼(N1) jhj ¼ lim DTFT RX (h) 1 N!1 N N 1 X

(19:5:38)

We use the frequency convolution property of the DTFT ð 1 1 p X(p)Y(v p)dp X(v) Y(v) ¼ xn yn () 2p 2p p with RX (h) , SX (v) and from Example 19.5.10 jhj 1 sin(vN=2) 2 1 () N N sin(v=2) Eq. (19.5.38) can be written as jhj lim E ½SN (v) ¼ lim DTFT RX (h) 1 N!1 N!1 N ðp 1 1 sin½(v p) N=2 SX (p) ¼ lim dp N!1 2p p N sin½(v p)=2

(19:5:39)

Using the result 1 X 1 sin(vN=2) d(v 2kp) ! lim N!1 N sin(v=2) k¼1 in Eq. (19.5.39), we obtain 1 N!1 2p

ðp

lim E ½SN (v) ¼ lim

N!1

SX (p)d(v p)dp ¼ SX (v) p

(19:5:40)

488

Random Processes

Thus, the periodogram SN(v) is an asymptotically unbiased estimator of the psd SX(v). The variance of SN(v) can be obtained by using techniques similar to the continuous case, and we can show a result similar to Eq. (19.5.28): lim var½SN (v) S2X (v), v = 0

N!1

(19:5:41)

This equation shows that the estimator SN(v) is not a consistent estimator of SX(v). Example 19.5.12 To check the validity of Eq. (19.5.41), a computer simulation of discrete white noise of zero mean and unit variance was performed. The power spectral density of the white noise is the variance, or SX(v) ¼ s2X ¼ 1. The number 2m of data points with m ¼ 7,9,10,11 were chosen so that they could ﬁt into the discrete Fourier transform algorithm. The psd S(v) was estimated for N ¼ 128, 512, 1024, and 2048 points using Eq. (19.5.32). Because of the symmetry about N/2, the estimates S128(v),

FIGURE 19.5.15

TABLE 19.5.1. #

PSD

N

Mean

Variance

1 2 3 4 5

SX(v) S128(v) S512(v) S1024(v) S2048(v)

— 128 512 1024 2048

1 1.0045 0.9944 0.9971 0.9987

0 0.9214 0.8908 0.9941 1.0240

19.5

Power Spectral Density

489

S512(v), S1024(v), and S2048(v) are graphed for only half the number of data points in Figs. 19.5.15a –19.5.15d. The estimated psd’s and their variances are shown in Table 19.5.1. The table shows that the estimator SN(v) is unbiased because the means for all N are nearly equal to 1 as expected. However, the variances of the psd for all N are nearly the same, equaling the derived value s4X ¼ 1, which is the expected result according to Eq. (19.5.41). Thus, the simulation of discrete white noise conﬁrms the desired theoretical result.

CHAPTER

20

Classiﬁcation of Random Processes

B 20.1

SPECIFICATIONS OF RANDOM PROCESSES

In the last chapter we discussed the fact that a random process X(t) is a random variable indexed on a time parameter and the random processes were classiﬁed as only nonstationary and stationary. We can also give other speciﬁcations of random processes depending on time t and the state X. If the timepoints are a set of points like {tn , n ¼ 0, +1, +2, . . .}, they are discrete, and if they are intervals of time on the real line such as {t T} or {T1 , t , T2 }, they are continuous. The state of a process is the set of values of a sample function Xi or X(t). If this set is ﬁnite or countably inﬁnite, the state space is discrete and if it is uncountably inﬁnite, the state space is continuous. On the basis of these deﬁnitions, we can specify four sets of random processes as follows.

20.1.1 Discrete-State Discrete-Time (DSDT) Process Example 20.1.1 The random variable Xn denotes the position of a particle at time n. At each time i the particle undergoes a jump of þ1 with probability p and a jump of 21 with probability q and stays in the same position with probability 1 2 p 2 q. The jumps are independent. If Yi is a random variable representing the jump at the ith step, then the sequence {Yi , i ¼ 1, 2, . . .} are independent with P{Yi ¼ 1} ¼ p, P{Yi ¼ 1} ¼ q and P{Yi ¼ 0} ¼ 1 p q. Initially the particle is at X0 ¼ 0. The DSDT process Xi is given by Xi ¼ Xi1 þ Yi ,

i ¼ 1, . . . , n

This process, called random walk, is represented in Fig. 20.1.1.

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

490

(20:1:1)

20.1 Speciﬁcations of Random Processes

491

FIGURE 20.1.1

20.1.2

Discrete-State Continuous-Time (DSCT) Process

Example 20.1.2 A random process N(t) has an increment of unity at each random time tk where the random times {tk } are Poisson-distributed with l as the occurrence of events per unit time. The probability of occurrence of k events in the time interval (t1, t2) is given by P{N(t2 ) N(t1 ) ¼ k} ¼

l(t2 t1 )k l(t2 t1 ) e , k!

k ¼ 0, 1, . . . ,

(20:1:2)

Such a process is known as a Poisson or counting process and is shown in Fig. 20.1.2.

20.1.3

Continuous-State Discrete-Time (CSDT) Process

Example 20.1.3 In Example 20.1.1 if the jumps are continuously distributed with probability density function fX(x) as shown in Fig. 20.1.3, then we have a continuousstate discrete-time process. This process is also a random walk with continuous state space. It also characterizes a sample function discretized at equal intervals of time.

20.1.4

Continuous-State Continuous-Time (CSCT) Process

Example 20.1.4 In this process both the state and the time are continuous. For example, in the extrusion of plastic bags, the thickness of the plastic sheet will be continuously

FIGURE 20.1.2

492

Classiﬁcation of Random Processes

FIGURE 20.1.3

FIGURE 20.1.4

varying with respect to time with the statistics being constant over long periods of time. In the random-walk example (Example 20.1.1), if both the differential time Dt and the differential state Dx tend to zero, and if certain limiting conditions are satisﬁed, then we get a Brownian motion whose sample function is shown in Fig. 20.1.4. We will now discuss different types of random processes.

B 20.2

POISSON PROCESS

A random process {N(t), t 0} represents the total number of events (detection of photons or electrons) that have occurred prior to time t. Denoting the time of occurrence of an event as a random variable T, we make the following assumptions about the process {N(t), t 0}: 1. Pfsingle event T in the interval t, t þ Dtg ¼ P{N(t þ Dt) N(t) ¼ 1} ¼ P{N(Dt) ¼ 1} ¼ lDt þ o(Dt), where any function g(x) is of o(Dt) if limDt!0 ½g(Dt)=Dt ! 0: 2. P{no event in the interval t,t þ Dt} ¼ P{N(t þ Dt) N(t) ¼ 0} ¼ P{N(Dt) ¼ 0} ¼ 1 lDt þ o(Dt): 3. P{more than two events in the interval t,t þ Dt} ¼ P{N(Dt) 2} ¼ o(Dt): 4. Events occurring in nonoverlapping time intervals are statistically independent: P{½N(t2 ) N(t1 )½N(t4 ) N(t3 )} ¼ P{½N(t2 t1 )½N(t4 t3 )} ¼ P{N(t2 t1 )}P{N(t4 t3 )}

for

t1 , t2 , t3 , t4

(20:2:1)

20.2

Poisson Process

493

FIGURE 20.2.1

Processes satisfying Eq. (20.2.1) are called independent increment processes. Given these assumptions, we will ﬁnd the probability of n events occurring in time interval t: P{n events in the time interval t} ¼ P{N(t) ¼ n} ¼ PN (n, t) We will ﬁrst ﬁnd the probability of no event in the interval t þ Dt ¼ PN {0, t þ Dt}. The interval (0,t þ Dt) can be partitioned as (0,t) and (t,t þ Dt) as shown in Fig. 20.2.1. Thus, PN {0, t þ Dt} can be given as probability of no event in (0,t) and probability of no event in (t,t þ Dt). Since these two intervals are nonoverlapping, we can write from assumption 4 PN {0, t þ Dt} ¼ PN {0, t}PN {0, Dt}

(20:2:2)

and from assumption 2 PN (0, t þ Dt) ¼ PN (0, t)(1 lDt) þ o(Dt) or PN (0, t þ Dt) PN (0, t) ¼ PN (0, t)lDt þ o(Dt)

(20:2:3)

Dividing Eq. (20.2.3) by Dt and taking the limit as Dt ! 0, we have dPN (0, t) ¼ lPN (0, t) dt

(20:2:4)

The initial condition for this ﬁrst-order differential equation is obtained from Pfno events in zero timeg equals 1. Hence the solution to Eq. (20.2.4) with initial condition PN(0, 0) ¼ 1 is given by PN (0, t) ¼ elt u(t) which is the probability that no event T occurs in the time interval (0, t). Or P{N(t) ¼ 0} ¼ PN (0, t) ¼ P(T . t) ¼ elt u(t)

(20:2:5)

Hence, the probability of occurrence of an event T in the interval (0, t) is the complement of Eq. (20.2.5) and is written as P(T t) ¼ FT (t) ¼ (1 elt )u(t) and the density function is fT (t) ¼ lelt u(t) Equation (20.2.5) will serve as the initial condition for the solution of PN {n, t}.

(20:2:6)

494

Classiﬁcation of Random Processes

FIGURE 20.2.2

From Fig. 20.2.2 it can be observed that n events occurring in the interval (0, t þ Dt) can be partitioned as n events in (0, t) AND no event in (t, t þ Dt) OR (n 2 1) events in (0, t) AND 1 event in (t, t þ Dt). From assumptions 1 –4, we can write PN (n, t þ Dt) ¼ PN (n, t)½1 l Dt þ o(Dt) þ PN (n 1, t) l Dt þ o(Dt)

(20:2:7)

Rearranging terms in Eq. (20.2.7), dividing throughout by Dt, and taking the limit as Dt ! 0, we obtain a differential – difference equation PN (n, t þ Dt) PN (n, t) ¼ lim ½PN (n, t)l þ PN (n 1, t)l þ o(Dt) Dt!0 Dt!0 Dt lim

or dPN (n, t) ¼ lPN (n, t) þ lPN (n 1, t) dt

(20:2:8)

Equation (20.2.8) is a differential – difference equation and the two initial conditions for solving the are 1. PN (0, t) ¼ elt u(t) 2. PN (n, 0) ¼ 0, n . 0 Equation (20.2.8) can be solved recursively as follows: n ¼ 1:

dPN (1, t) ¼ lPN (1, t) þ lelt dt

n ¼ 2:

dPN (2, t) ¼ lPN (2, t) þ lPN (1, t) or dt

or PN (1, t) ¼ ltelt , PN (2, t) ¼

t0

(lt)2 lt e , 2

t0

Proceeding in a similar manner, we obtain the result for n events occurring in interval t as PN (n, t) ¼

ðlt)n lt e , n!

t0

(20:2:9)

The discrete-state continuous-time (DSCT) random process satisfying Eq. (20.2.9) is called a Poisson random process and has been shown in Fig. 20.1.2.

Solution of Eq. (20.2.7) Using Generating Functions

The differential –difference equation [Eq. (20.2.8)] can also be solved using generating functions. From Eq. (11.3.1) we can deﬁne the generating function for PN (n, t) as G(z, t) ¼

1 X n¼0

PN (n, t)zn

(20:2:10)

20.2

Poisson Process

495

Multiplying Eq. (20.2.8) by z n and summing over all n from 0 to 1, we have 1 1 1 X X dX PN (n, t)zn ¼ l PN (n, t)zn þ l PN (n 1, t)zn dt n¼0 n¼0 n¼0

or d G(z, t) ¼ l(z 1)G(z, t) dt

(20:2:11)

For a ﬁxed z, Eq. (20.2.11) is an ordinary differential equation in t. The initial condition G(z, 0) is obtained as follows: G(z, 0) ¼

1 X

PN (n, 0)zn ¼ PN (0, 0) ¼ 1;

since PN (n, 0) ¼ 0 for n 1

n¼0

With this initial condition, the solution to Eq. (20.2.11) is G(z, t) ¼ elt(z1) ,

t0

(20:2:12)

The power-series expansion of Eq. (20.2.12) yields lt (lt)2 (ltÞ n þ þ zn þ G(z, t) ¼ elt 1 þ z þ z2 2! n! 1! Hence, PN(n, t) is the coefﬁcient of zn in the preceding equation and is given by PN (n, t) ¼ P{N(t) ¼ n} ¼ elt

(lt)n , n!

t0

(20:2:13)

The statistics of the Poisson process as derived in Example 19.1.7 are Mean: E½N(t) ¼ mN (t) ¼ lt Variance: var½N(t) ¼ s2N (t) ¼ lt Autocorrelation: E½N(t1 )N(t2 ) ¼ RN (t1 , t2 ) ¼ l2 t1 t2 þ lmin (t1 , t2 ) Autocovariance: E{½N(t1 ) mN (t1 )½N(t2 ) mN (t2 )} ¼ CN (t1 , t2 ) ¼ lmin (t1 , t2 ) Normalized Autocovariance: CN (t1 , t2 Þ 1 ¼ min rN (t1 , t2 ) ¼ sN (t1 )sN ðt2 Þ l

rﬃﬃﬃﬃ rﬃﬃﬃﬃ t1 t2 , t2 t1

Since the mean is a function of time t and the autocorrelation is a function of t1 and t2, the Poisson process is a nonstationary process. Distribution of Interarrival Times

We will now ﬁnd the distribution of interarrival times for the Poisson process N(t). Let T1 be the random time of occurrence of the ﬁrst event after time t ¼ 0, T2 the random time

496

Classiﬁcation of Random Processes

FIGURE 20.2.3

between the occurrence of the ﬁrst and second events, and Tn the random time between the occurrence of the (n 2 1)st and nth events as shown in Fig. 20.2.3. The sequence of times fTi, i ¼ 1, . . . , ng is called interarrival times. Let {ti , i ¼ 1, . . . , n} be the realizations of these random interarrival times. Let Sn be the random variable representing the arrival time of the nth event or the waiting time until the nth event: S1 ¼ T1 , S2 ¼ T1 þ T2 , . . . , Sn ¼ T1 þ T2 þ þ Tn ¼

n X

Ti

(20:2:14)

i¼1

If {ti , i ¼ 1, . . . , n} are the realizations of the random arrival times {Si , i ¼ 1, . . . , n}, then t1 ¼ t1 , t2 ¼ t1 þ t2 , . . . , tn ¼ t1 þ t2 þ þ tn ¼

n X

ti

(20:2:15)

i¼1

We will ﬁnd the distributions of these interarrival times {Ti } and the waiting time for the nth event Sn. The Poisson process can also be speciﬁed in terms of the arrival times {Si } as N(t) ¼

1 X

u(t Si )

(20:2:16a)

i¼1

where u(t) is a unit step function. Referring to Fig. 20.2.3, N(t) is the total number of events that have occurred in the time interval t, and each event causes an increment by 1. Hence Eq. (20.2.16a) can be simpliﬁed as N(t) ¼

1 X i¼1

u(t Si ) ¼

N(t) X i¼1

uðt Si Þ ¼

N(t) X

1,

t0

(20:2:16b)

i¼1

By deﬁnition, the ﬁrst event {T1 . t} occurs after time t, and hence there are no events occurring in the interval (0, t). Hence from Eq. (20.2.5), we obtain P{T1 . t} ¼ P{N(t) ¼ 0} ¼ PN (0, t) ¼ elt u(t)

(20:2:17)

We will show that the interarrival times T1,T2, . . . , Tn are independent identically distributed (i.i.d.) random variables with density function elt u(t). The second event

20.2

Poisson Process

497

{T2 . t} occurs after the ﬁrst event T1 ¼ t1: P{T2 . t j T1 ¼ t1 } ¼ P{T2 . t j S1 ¼ t1 } ¼ P ½no events in (t1 , t1 þ t) = S1 ¼ t1 (20:2:18) From the independence of nonoverlapping intervals (assumption 4) of the Poisson process, Eq. (20.2.18) can be written as P{T2 . t j S1 ¼ t1 } ¼ P{T2 . t} ¼ P{no events in (t1 , t1 þ t)} ¼ elt u(t)

(20:2:19)

Hence T1 and T2, are independent. In a similar manner, for the nth arrival time Tn, we can write P{Tn . t j Tn1 ¼ tn1 , . . . , T1 ¼ t1 } ¼ P{Tn . t j Sn1 ¼ tn1 , . . . , S1 ¼ t1 } (20:2:20) Again, from the independent increment property, Eq. (20.2.20) can simpliﬁed to P{Tn . t j Sn1 ¼ tn1 , . . . , S1 ¼ t1 } ¼ P{Tn . t} ¼ P{no events in (tn , t} ¼ elt u(t)

(20:2:21)

Thus the interarrival times T1, T2, . . . , Tn are i.i.d. random variables with FT (t) ¼ P(Tn the probt) ¼ (1 elt )u(t) and density function fT (t) ¼ lelt u(t). As a consequence, P ability density function for the waiting time for the nth event Sn ¼ ni¼1 Ti , n 1 is an n-fold convolution of lelt u(t), yielding an Erlang density with n-degrees of freedom similar to Eq. (7.4.1): fSn (t) ¼

(lt)n1 lelt , (n 1)!

l . 0, t . 0, n: integer . 0

(20:2:22)

Example 20.2.1 A message sending station sends two independent bitstreams X and Y with Poisson rates l and m. The receiving station receives the composite stream W ¼ X þ Y. We have to ﬁnd (1) the rate of the composite stream and (2) the probability that stream Y arrives ﬁrst. The density functions of the bitstreams X and Y are fX (x) ¼ elx

(lx)k (my)m u(x) and fY (y) ¼ emy u(y) k! m!

1. We will ﬁnd the rate of W ¼ X þ Y using generating functions. The generating functions for X and Y are PX (z) ¼ elðz1Þ

and

PY (z) ¼ em(z1)

Since X and Y are independent, the pdf fW (w) is the convolution of fX(x) and fY (x), and the generating function PW (z) for fW (w) is given by the product PX(z)PY (z), or PW (z) ¼ e(lþm)(z1) . Hence fW (w) ¼ eðlþm)w

½(l þ m)wk u(w) k!

2. Let the times of arrival of streams X and Y be the random variables S and T. From Eq. (20.2.6) the interarrival times are exponentially distributed and hence fS (s) ¼ lels u(s) and

fT (t) ¼ memt u(t)

498

Classiﬁcation of Random Processes

Since the streams are independent, the arrival times are also independent, and the joint density of the arrival times S and T is fST (s, t) ¼ lme(ltþms) ,

s 0, t 0

The event that stream Y arrives before stream X is the same as the arrival time of Y is before the arrival time of X. Hence P{Y X} ¼ P{T S}. We will ﬁnd P(T S). The region {T S} is shown in Fig. 20.2.4: 1 ð1 ð1 ð1 lt e lme(ltþms) dt ds ¼ lmems P(T S) ¼ l 0 s 0 s ð1 m ¼m e(lþm)s ds ¼ lþm 0 Hence P(Y X) ¼

m lþm

Example 20.2.2 A Poisson process N(t) with rate l consists of a subprocess N1(t) with probability p and rate l1 and another subprocess N2(t) with probability (1 2 p) and rate l2. We have to ﬁnd the probabilities P{N1 (t) ¼ n} and P{N2 (t) ¼ m} and their rates l1 and l2 . We are given the following: N(t) elt

(lt)k , k!

N1 (t) el1 t

(l1 t)k (l2 t)k , N2 (t) el2 t k! k!

From the statement of the problem N(t) ¼ N1 (t) þ N2 (t), and if N1 (t) ¼ n and N2 (t) ¼ m, then N(t) ¼ n þ m. The joint probability P{N1 (t) ¼ n, N2 (t) ¼ m} can be given in terms of conditional probabilities as P{N1 (t) ¼ n, N2 (t) ¼ m} ¼ P{½N1 (t) ¼ n, N2 (t) ¼ m j N(t) ¼ m þ n}elt

(lt)mþn (m þ n)!

where P{N(t) ¼ m þ n} ¼ elt (lt)mþn =½(m þ n)! has been substituted in the equation above. The event ½N1 (t) ¼ n, N2 (t) ¼ m j N(t) ¼ m þ n} can be considered as (m þ n)

FIGURE 20.2.4

20.2

Poisson Process

499

Bernoulli trials with success deﬁned as N1 (t) ¼ n with probability p and failure as N2 (t) ¼ m with probability (1 2 p). Thus, using the binomial distribution [Eq. (4.2.2)], we have mþn n P{½N1 (t) ¼ n, N2 (t) ¼ m j N(t) ¼ m þ n} ¼ p (1 p)m n and P{N1 (t) ¼ n, N2 (t) ¼ mg can be given by P{N1 (t) ¼ n, N2 (t) ¼ m} ¼

(m þ n)! n (lt)mþn p (1 p)m elt (m þ n)! n!m!

¼ eltp elt(1p)

(ltp)n ½lt(1 p)m n! m!

The marginal distribution P{N1 (t) ¼ n} can be given as follows: P{N1 (t) ¼ n} ¼ eltp

1 (ltp)n X ½lt(1 p)m (ltp)n ¼ eltp elt(1p) n! m¼0 m! n!

Similarly P{N2 (t) ¼ m} ¼ elt(1p)

½lt(1 p)m m!

Hence, the Poisson rates for N1(t) and N2(t) are l1 ¼ lp and l2 ¼ l(1 2 p). Let S1n and S2m be the waiting times for the nth and mth events of the Poisson subprocesses N1(t) and N2(t) respectively. We will ﬁnd the probabilities of (1) the waiting time for the ﬁrst event in N1(t) is greater than the waiting time for the ﬁrst event in N2(t), or P{S11 . S21 }, and (2) the waiting time for the second event in N1(t) is less than the waiting time for the ﬁrst event in N2(t), or P{S12 , S21 }. From Eq. (20.2.14) the waiting times can be given in terms of the interarrival times, or S11 ¼ T11, S21 ¼ T21 and S12 ¼ T11 þ T12. 1. From Example 20.2.1 P{S11 . S21 } ¼ P{T11 . T21 } ¼

l1 lp ¼ ¼p l1 þ l2 lp þ l(1 pÞ

2. Since S11 , S12, we can write P{S12 , S21 } ¼ P{S12 , S21 ; S11 , S21 } or P{S12 , S21 } ¼ P{S12 , S21 jS11 , S21 }P{S11 , S21 } From Example 20.2.1 PfS11 , S21 g ¼ l 2 =ðl1 þ l2 Þ and PfS12 , S21 j S11 , S21 g ¼ PfT11 þ T12 , T21 j T11 , T21 g ¼ PfT12 , T21 g ¼

l2 l1 þ l2

Hence

PfS12

l2 , S12 g ¼ ¼ ð1 pÞ2 l 1 þ l2

Example 20.2.3 Advance booking for a World Series baseball game is open on weekdays only from 10:00 a.m. to 3:00 p.m. During the lunch hour from 12:00 to 1:00 p.m.,

500

Classiﬁcation of Random Processes

customers arrive at the rate of l ¼ 8 h21 (8 per hour) and for the rest of the time at the rate of l ¼ 4h21. The arrival time N(t) of customers to the booking counter is a random process given by P{N(t) ¼ k} ¼ eLt

(Lt)k k!

where the rate L is a binary random variable assuming values of 4 and 8 with probabilities 0.2 and 0.8 respectively. We have to ﬁnd E[N(t)], var[N(t)] and determine the variance to mean ratio s2 =m. If this ratio equals 1, we can conclude that N(t) is Poisson. We note that the distribution of N(t) conditioned on either L¼8 or 4 is Poisson. Hence E½N(t) j L ¼ 8 ¼ 8 and E½N(t) j L ¼ 4 ¼ 4. This result is used in the following equation: E½N(t)j ¼ E½NðtÞ j L ¼ 8P(L ¼ 8) þ E½N(t) j L ¼ 4P(L ¼ 4) ¼ 8 0:2 þ 4 0:8 ¼ 1:6 þ 3:2 ¼ 4:8 In a similar manner, we can ﬁnd the second moment E[N 2(t)] with conditional second moments E½N 2 (t) j L ¼ 8 ¼ l þ m2 ¼ 8 þ 64 ¼ 72 and E½N 2 (t) j L ¼ 4 ¼ 4 þ 16 ¼ 20. From the conditional second moments, E[N 2(t)] can be given as follows: E½N 2 (t) ¼ E½N 2 (t) j L ¼ 8P(L ¼ 8) þ E½N 2 (t) j L ¼ 4P(L ¼ 4) ¼ 72 0:2 þ 20 0:8 ¼ 30:4 Hence the variance of N(t) is given by var(N(t)) ¼ E½N 2 (t) {E½N(t)}2 ¼ 30:4 4:82 ¼ 7:36 The variance : mean ratio s2 =m ¼ 7:36=4:8 ¼ 1:53. Since this ratio is not equal to 1, N(t) is not a Poisson process. Example 20.2.4 Customers arrive at a checkout counter at a Poisson rate of l per minute. The service time is a random variable Y in minutes. We consider two cases: (1) Y is a constant ¼ k minutes and (2) Y is exponentially distributed with density function fY (y) ¼ memy u(y). For both cases we will ﬁnd the probabilities that (1) a customer does not have to wait and (2) the mean value of a customer’s waiting time. From Eq. (20.2.6) the interarrival time T of customers is distributed as fT (t) ¼ lelt u(t), and from Eq. (20.2.19) the interarrival times are independently distributed. The service time Y is distributed as fY (y) ¼ memy u(y). If the interarrival time T is greater than the service time Y, then there will be no waiting time for the customer. 1. Service time is a constant: Y ¼ k minutes (a) Pfno waiting time for the customerg ¼ PfT . kg. Hence 1 ð1 lt le lelt dt ¼ P{T . k} ¼ ¼ elk l k k

(b) If T k, then the customer has to wait (k 2 T ) minutes and the average waiting time is given by ðk 1 E½k T ¼ (k t)lelt dt ¼ k (1 ekl ) l 0

20.2

Poisson Process

501

2. Service time Y exponentially distributed: fY (y) ¼ memy u(y) (a) Again, P fno waiting time for the customerg ¼ PfT . Yg and ð1 P{T . Y j Y ¼ t}fY (t)dt P{T . Y} ¼ 0

ð1 ¼

elt memt dt ¼

0

m lþm

(b) Here also, if T Y, then the customer has to wait (Y 2 T ) minutes. We have to ﬁnd E[Y 2 T ]. In terms of conditional expectation, we can write ð1 E½Y T j Y ¼ y fY (y)dy E½Y T ¼ 0

and from the result 1b (above) the conditional expectation E½Y T j Y ¼ y is ðy 1 E½Y T j Y ¼ y ¼ (y t)lelt dt ¼ y (1 ely ) l 0 Substituting this equation into the previous equation, we obtain ð1 1 l ly y (1 e ) m emy dy ¼ E½Y T ¼ l m(l þ m) 0 Poisson White Noise

The Poisson process given by Eq. (20.2.16) can be further generalized by associating with each event T a random weight A. The weights are i.i.d. random variables with density function fA(a) and are also independent of the arrival times S. The weighted Poisson process X(t) is deﬁned by X(t) ¼

1 X

Ai u(t Si ),

t0

(20:2:23a)

i¼1

which can be simpliﬁed from Eq. (20.2.16.b) as follows: X(t) ¼

1 X

Ai u(t Si ) ¼

i¼1

N(t) X

Ai u(t Si ) ¼

i¼1

N(t) X

Ai ,

t0

(20:2:23b)

i¼1

The pdf fX (x; t) can be expressed in terms of the conditional expectation as X fX ½x; tjN(t)P{N(t)} fX (x; t) ¼

(20:2:24)

N(t)

The conditional probability fX (x; t j N(t) ¼ i can be evaluated from Eqs. (20.2.23): i ¼ 1:

fX ½x; t j N(t) ¼ 1 ¼ fA (a)

i ¼ 2: fX ½x; t j N(t) ¼ 2 ¼ fA (a1 þ a2 )

(20:2:25) (20:2:26)

Since the Ai terms are i.i.d. random variables, fA (a1 þ a2 ) in Eq. (20.2.26) is a convolution of fA (a) with itself; or fX ½x; t j N(t) ¼ 2 ¼ fA (a) fA (a) Proceeding in this fashion, fX ½x; t j N(t) ¼ i is an i-fold convolution of fA (a) and fX ½x; t j N(t) ¼ i ¼ fA (a) fA (a) fA (a) ¼ fA(i) (a)

(20:2:27)

502

Classiﬁcation of Random Processes

In Eq. (20.2.4), P{N(t) ¼ i} is a Poisson distribution given by Eq. (20.10.13): P{N(t) ¼ i} ¼ PN (i, t) ¼ elt

(lt)i i!

(20:2:28)

Substituting Eqs. (20.2.27) and (20.2.28) in Eq. (20.2.24) we obtain the pdf of X(t) as fX (x; t) ¼ elt

1 X

fA(i) (a)

i¼1

(lt)i i!

(20:2:29)

The mean and autocorrelation of the generalized Poisson process X(t) can be obtained from the properties of expectations as follows: Mean: E½X(t) ¼ E{E½X(t) j N(t) ¼ n} ( " #) N(t)¼n 1 N(t)¼n X X X ¼E E Ai E½Ai PN (n, t) ¼ i¼1

¼

1 X

i¼1

n¼1

1 X (lt) lt (lt)n e ¼ mA elt n! (n 1)! n¼1 n

mA n

n¼1

¼ mA ltelt ¼

1 X (lt)n m¼0

m!

¼ mA lt ¼ mX (t),

t0

(20:2:30)

Autocorrelation: E½X(t1 )X(t2 ) ¼ E{X(t1 )½X(t2 ) X(t1 ) þ X(t1 )} ¼ E½X 2 (t1 ) þ E½X(t1 )E½X(t2 ) X(t1 ) ¼ E{E½X 2 (t1 ) j N(t1 )} þ E½X(t1 ) j N(t1 )E½(X(t2 ) X(t1 )) j N(t2 t1 )

(20:2:31)

We will evaluate E{E½X 2 (t1 ) j N(t1 )}: ( ) n X n X 2 E½Ai A j E{E½X (t1 ) j N(t1 ) ¼ n} ¼ E i¼1 j¼1

¼E

8 > > n > : i¼1

E½A2i þ

n X n X

E½Ai E½A j

i¼1 j¼1 i=j

9 > > = > > ;

¼ E{nE½A2 þ n(n 1)m2A } ¼

1 X

E½A2 nelt1

n¼1

¼ E½A2 lt1 elt1

1 (lt1 )n X (lt1 )n þ m2A n(n 1)elt1 n! n! n¼1

1 X (lt1 )m m¼0

m!

þ m2A l2 t12 elt1

1 X (lt1 )m m¼0

m!

¼ E½A2 lt1 þ m2A l2 t12 (20:2:32)

20.2

Poisson Process

503

Substituting Eqs. (20.2.30) and (20.3.32) in Eq. (20.2.31) and E½A2 ¼ s2A þ m2A , we obtain E½X(t1 )X(t2 ) ¼ E½A2 l(t1 ) þ m2A l2 t12 þ mA lt1 ½mA lt2 mA lt1 ¼ ½s2A þ m2A lt1 þ m2A l2 t1 t2 ¼ RX (t1 , t2 )

(20:2:33a)

Equation (20.2.33a) was obtained on the premise that t1 , t2. We can derive a similar equation for t2 , t1. Hence the autocorrelation function RX(t1, t2) is given by RX (t1 , t2 ) ¼ ½s2A þ m2A l min(t1 , t2 ) þ m2A l2 t1 t2 , t1 , t2 , 0

(20:2:33b)

The autocovariance function CX (t1, t2) is given by CX (t1 , t2 ) ¼ (s2A þ m2A )l min(t1 , t2 ), t1 , t2 0

(20:2:34)

Example 20.2.5 The events of a Poisson process with l ¼ 12 are weighted with i.i.d. random variables {Ai }, resulting in a weighted Poisson process X(t). The weights are uniformly distributed in the interval (0,2). We will ﬁnd the mean, autocorrelation, and autocovariance of X(t). The mean value of the weights mA ¼ 1. The variance of the weights s2A ¼ 13. X(t) given by Eq. (20.2.23) is shown in Fig. 20.2.5. Mean. From Eq. (20.2.30) the mean value of X(t) is mX (t) ¼ mA lt ¼ 1 12 t ¼ t=2. Autocorrelation. The second moment of A is E½A2 ¼ s2A þ m2A ¼ 13 þ 12 ¼ 43. Hence, from Eq. (20.2.33), the autocorrelation function is given by 2 41 2 1 2 1 t1 t2 ¼ min(t1 , t2 ) þ t1 t2 min(t1 , t2 ) þ 1 RX (t1 , t2 ) ¼ 32 2 3 4 Autocovariance. From Eq. (20.2.34) the ACF is CX (t1 , t2 ) ¼ 23 min(t1 , t2 ): Formal Derivative of Poisson Process

The generalized Poisson process given by Eq. (20.2.23) shown in Fig. 20.2.5 consists of a series of step functions occurring at Poisson-distributed random times with random

FIGURE 20.2.5

504

Classiﬁcation of Random Processes

weights. The formal derivative of X(t) yields a Poisson impulse process V(t) consisting of a series of Dirac delta functions: 1 1 X d dX Ai u(t Si ) ¼ Ai d(t Si ), X(t) ¼ dt dt i¼1 i¼1

t0

(20:2:35)

The mean of V(t) can be calculated as mV (t) ¼

dmX (t) dmA lt ¼ ¼ mA l, dx dt

t0

(20:2:36)

The autocorrelation function of V(t) is RV (t1 , t2 ) ¼

@2 RX (t1 , t2 ) @2 ¼ {½s2A þ m2A l min(t1 , t2 ) þ m2A l2 t1 t2 } @t1 @t2 @t1 @t2

¼ ½s2A þ m2A ld(t2 t1 ) þ m2A l2 ¼ RV (t2 t1 ) ¼ RV (t)

(20:2:37)

Since mV is independent of time and RV (t1 , t2 ) is a function of the time difference t2 2 t1, we conclude that V(t) is a stationary process. The autocovariance function is CV (t1 , t2 ) ¼ ½s2A þ m2A ld(t2 t1 ) ¼ CV (t2 t1 ) ¼ CV (t)

(20:2:38)

The FT SV 0 (v) of the acf CV (t) of Eq. (20.2.38) is given by SV 0 (v) ¼ FT{½s2A þ m2A ld(t2 t1 )} ¼ l½s2A þ m2A

(20:2:39)

This is the psd of the process V 0 (t) ¼ V(t) mA l. Since the spectrum is constant over the entire frequency range, the process V 0 (t) is called Poisson white noise. Example 20.2.6 We will determine the derivative process V(t) of Example 20.2.5, where l ¼ 12 and the weight process A . 0 is uniformly distributed in the interval (0,2). The process V(t) is shown in Fig. 20.2.6. The mean mV is obtained from Eq. (20.2.36) as mV ¼ mA l ¼ 1 12 ¼ 12. The autocorrelation function from Eq. (20.2.37) is 4 1 RV (t) ¼ ½s2A þ m2A d(t2 t1 ) þ m2A l2 ¼ d(t) þ 3 4

FIGURE 20.2.6

20.3

Binomial Process

505

and the autocovariance from Eq. (20.2.38) is CV (t) ¼ 43 d(t). The psd equals FT½RV ðtÞ given by SV (v) ¼ 43 þ ðp=2Þd(v).

B 20.3

BINOMIAL PROCESS

Bernoulli Process

In Section 4.1 we discussed Bernoulli trials as repeated functionally and statistically i.i.d. trials, each of which consists of only two events s and f with probabilities p and q ¼ (1 2 p) respectively. Consider a countably inﬁnite sequence {Xn , n ¼ 0, 1, . . .} of random variables deﬁned by

1 for success s in the nth trial Xn ¼ 0 for failure f in the nth trial with probabilities P(Xn ¼ 1) ¼ p P(Xn ¼ 0) ¼ 1 p

(20:3:1)

Thus Xn represents a Bernoulli process, shown in Fig. 20.3.1. We will now ﬁnd the statistics of this process: Mean: E½Xn ¼ mX ¼ 1 P(Xn ¼ 1) þ 0 P(XN ¼ 0) ¼ p

(20:3:2)

Variance: var½Xn ¼ s2X ¼ E½X 2 m2X ¼ 12 p þ 02 (1 p) p2 ¼ p(1 p)

(20:3:3)

Autocorrelation: ( E½Xn1 Xn2 ¼ RX (n1 , n2 ) ¼

12 p þ 02 (1 p) ¼ p, 2

E½Xn1 E½Xn2 ¼ p ,

n1 n 2 ¼ 0

(20:3:4)

n 1 n2 = 0

Autocovariance: CX (n1 ; n2 ) ¼ RX (n1 ; n2 ) m2X ( 12 p þ 02 (1 p) p2 ¼ p(1 p), ¼ E½Xn1 E½Xn2 ¼ p2 ¼ 0, n1 n2 = 0

FIGURE 20.3.1

n1 n 2 ¼ 0

(20:3:5)

506

Classiﬁcation of Random Processes

Normalized Autocovariance: rX (n1 , n2 ) ¼

n1 n2 ¼ 0 n1 n 2 = 0

1, 0,

(20:3:6)

Since the mean is independent of time n and the autocorrelation depends only on the time difference (n1 – n2), the process Xn is stationary. Binomial Process

We now deﬁne a new process Yn of the sum of the random variables {Xi } in a Bernoulli process as discussed earlier. This is called the binomial process, given by Yn ¼

n X

Xi ,

n.0

(20:3:7)

i¼1

This process, corresponding to the Bernoulli process of Fig. 20.3.1, is shown in Fig. 20.3.2 and is a discrete-state discrete-time (DSDT) process. The probability of k successes in n trials has already been derived in Eq. (4.2.2) and is given by a binomial distribution: n k Probability{k successes in any sequence in n trials} ¼ P(Yn ¼ k) ¼ p (1 p)nk k The statistics of the binomial process are: Mean. Since the probability of each success is p, the mean value of Yn can be obtained using the linearity of the expectation operator: " # n n X X Xi ¼ E ½Xi ¼ np, n . 0 (20:3:8) E ½Yn ¼ E i¼1

i¼1

Variance. From the independence of Xi, the second moment of Yn can be given by " #2 n n n X n X X X 2 E½Yn ¼ E Xi ¼ E½Xi2 þ E ½Xi X j ¼ np þ n(n 1)p2 , n . 0 i¼1

s2Y

¼

E ½Yn2

i¼1

i¼1 j¼1

2

{E ½Yn } ¼ np þ n(n 1)p2 n2 p2 ¼ np(1 p),

n.0 (20:3:9)

FIGURE 20.3.2

20.4

Independent Increment Process

507

Autocorrelation: For n1 n2, we have " # n2 X n1 X Xi X j E ½Yn1 Yn2 ¼ RY (n1 , n2 ) ¼ E j¼1 i¼1

" ¼E

n1 X i¼1

# Xi2

þ

n1 X n1 X

E ½Xi X j þ

j¼1 i¼1 i=j

nX n1 2 n1 X

E ½Xi X j

j¼1 i¼1 i=j

¼ n1 p þ n1 (n1 1)p2 þ n1 (n2 n1 )p2 or RY (n1 , n2 ) ¼ n1 p(1 p) þ n1 n2 p2 ;

for

n1 n 2

(20:3:10)

Similarly for n2 n1 we have RY (n1 , n2 ) ¼ n2 p(1 p) þ n1 n2 p2 ;

for

n2 n 1

(20:3:11)

n1 , n2 . 0

(20:3:12)

Combining Eqs. (20.3.10) and (20.3.11), we obtain RY (n1 , n2 ) ¼ p(1 p) min(n1 , n2 ) þ n1 n2 p2 , Autocovariance: CY (n1 , n2 ) ¼ p(1 p) min (n1 , n2 ), n1 , n2 . 0

(20:3:13)

Unlike the Bernoulli process, the mean of the binomial process is dependent on time n and the autocorrelation function is dependent on both n1 and n2 and hence is a nonstationary process.

B 20.4

INDEPENDENT INCREMENT PROCESS

A random process X(t) is called an independent increment process if for a sequence of times t1 , t2 , . . . , tn the increments X(t1 ), {X(t2 ) X(t1 )}, . . . , {X(tn ) X(tn1 )} of the process X(t) are a sequence of independent random variables: E{½X(ti ) X(ti1 )½X(tiþ1 ) X(ti )} ¼ E ½X(ti ) X(ti1 ) E ½X(tiþ1 ) X(ti )

(20:4:1)

Further, if all order distributions of the increments fX(t) 2 X(s)g, t , s are dependent only on the time difference t ¼ t s, then X(t) is called a stationary independent increment process. We will show that a stationary independent increment process is not a stationary process, and its mean value and variance are given by E ½X(t) ¼ m0 þ m1 t

(20:4:2a)

where m0 ¼ E ½X(0) and m1 ¼ E ½X(1) E ½X(0) and var½X(t) ¼ s2X(t) ¼ s20 þ s21 t where s20 ¼ E ½X(0) m0 2 and s21 ¼ E ½X(1) m1 2 s20 .

(20:4:2b)

508

Classiﬁcation of Random Processes

We will verify Eq. (20.4.2a). Deﬁning g(t) ¼ E ½X(t) X(0), for any t and t, we have g(t þ t) ¼ E ½X(t þ t) X(0) ¼ E{½X(t þ t) X(t) þ ½X(t) X(0)} ¼ E ½X(t þ t) X(t) þ E ½X(t) X(0)

(20:4:3)

In Eq. (20.4.3) E ½X(t) X(0) ¼ g(t) by definition

(20:4:4)

and from the stationary independent increment property we can write E ½X(t þ t) X(t) ¼ E ½X(t) X(0) ¼ g(t)

(20:4:5)

Substituting Eqs. (20.4.5) and (20.4.4) in Eq. (20.4.3), we obtain g(t þ t) ¼ g(t) þ g(t)

(20:4:6)

Differentiating both sides of Eq. (20.4.6) independently with respect to t and t, we have d d g(t þ t) ¼ g(t); dt dt

d d g(t þ t) ¼ g(t) dt dt

(20:4:7)

Hence g0 (t) ¼ g0 (t) and in particular at t ¼ 1, g(t)jt¼1 ¼ E ½X(1) X(0) ¼ k1 . Substituting k1 for g0 (t) and solving the differential equation g0 (t) ¼ k1 , we obtain the solution g(t) ¼ k1t þ k0. Substitution of the initial condition g(0) ¼ E ½X(0) X(0) ¼ 0 in the solution results in g(t) ¼ k1t, or E ½X(t) X(0) ¼ E ½X(1) X(0)t E ½X(t) ¼ m0 þ m1 t

(20:4:8) (20:4:2a)

thus verifying Eq. (20.4.2a). In a similar manner we can verify Eq. (20.4.2b). Hence, a random process with stationary independent increments is nonstationary since the mean and variance are functions of time.

Uncorrelated and Orthogonal Increment Processes

A random process X(t) is called an uncorrelated increments process if for a sequence of times t1 , t2 , t3 , t4 the increments fX(t2) 2 X(t1)g and fX(t4) 2 X(t3)g of the process X(t) are uncorrelated random variables, or E{½X(t2 ) X(t1 )½X(t4 ) X(t3 )} ¼ E ½X(t2 ) X(t1 )E ½X(t4 ) X(t3 )

(20:4:9)

Similarly, the random process X(t) is called an orthogonal increments process if E{½X(t2 ) X(t1 )½X(t4 ) X(t3 )} ¼ 0

(20:4:10)

The term independent increment process implies an uncorrelated increment process, but the converse is not true. If a process has uncorrelated increments, it need not necessarily have independent increments. Also, if a process X(t) has uncorrelated increments, then the process Y(t) ¼ X(t) E ½X(t) has orthogonal increments. Example 20.4.1 (Poisson Process) A Poisson process N(t) with random arrival times fSk , k ¼ 1, . . . , n, . . .} and random interarrival times {Tk , k ¼ 1, . . . , n, . . .} is an

20.4

Independent Increment Process

509

independent increment process since P{Tk1 s, Tk t} ¼ P{½N(Sk1 ) N(Sk2 ) ¼ 1, ½N(Sk ) N(Sk1 ) ¼ 1} ¼ P ½N(Sk1 ) N(Sk2 ) ¼ 1 P ½N(Sk ) N(Sk1 ) ¼ 1 ¼ (1 elt )ð1 elt Þu(s)u(t) The independent increments are stationary since P ½N(t) N(s) ¼ 1 ¼ l(t s)el(ts) depends only on the time difference (t 2 s) and not the individual times t and s. The mean of the Poisson process can be obtained by substituting in Eq. (20.4.2a) m0 ¼ E ½N(0) ¼

1 X

k el:0

k¼0

(l:0)k ¼0 k!

and with

E½N(1) ¼

1 X

k el

k¼0

lk ¼l k!

m1 ¼ l 0 ¼ l and hence E½N(t) ¼ lt, a result that has been obtained previously. The variance of the process can be obtained by substituting in Eq. (20.4.2b) s20 ¼ 0 and s21 ¼

1 X

k2 el

k¼0

lk (lt)2 ¼ l k!

Example 20.4.2 (Binomial Process) Eq. (20.3.7) as Yn ¼

and hence var ½N(t) ¼ lt

A binomial process has been deﬁned in

n X

Xi ,

n.0

i¼1

where n is the number of trials and Xi represents either 1 for success or 0 for failure at count i. Since each Xi is independent of Xj, j = i, we conclude that the process Yn is an independent increment process. The independent increments are stationary because k nm k P ½(Yn Ym ¼ k) ¼ p (1 p)(nm) , n . m k is dependent only on the count difference (n 2 m) and not on individual counts n and m. In the binomial process m0 ¼ E½Y0 ¼ 0 by deﬁnition and E ½Y1 ¼ E ½X1 ¼ p. Substitution of E ½Y1 in m1 results in m1 ¼ p 0 ¼ p, and hence, substituting m0 ¼ 0, m1 ¼ p, and t ¼ n in Eq. (20.4.2a), we obtain E ½Yn ¼ np, a result obtained previously. In a similar manner the variance of the process Yn can be obtained by substituting in Eq. (20.4.2b) s20 and s21 ¼ p(1 p), resulting in var ½Yn ¼ npð1 pÞ, another result obtained previously. Elementary Random-Walk Process

The Bernoulli process considered earlier is now modiﬁed as shown in Fig. 20.4.1 so that the failure is now represented by 21 instead of 0. Under this modiﬁcation we deﬁne a random variable Zn Zn ¼

1 1

for success in the nth trial for failure in the nth trial

(20:4:11a)

510

Classiﬁcation of Random Processes

FIGURE 20.4.1

with probabilities P(Zn ¼ 1) ¼ p P(Zn ¼ 1) ¼ 1 p ¼ q

(20:4:11:b)

If p ¼ 12, this random process is called a symmetric random walk. The mean and variance of Zn are Mean: mZ ¼ E ½Zn ¼ 1 P(Xn ¼ 1) 1 P(XN ¼ 1) ¼ p (1 p) ¼ 2p 1 (20:4:12) Second Moment: E ½Zn2 ¼ 1 p þ 1 (1 p) ¼ 1

(20:4:13)

Variance: s2Z ¼ var½Zn ¼ 12 p þ (1)2 (1 p) (2p 1)2 ¼ 4p(1 p)

(20:4:14)

In much the same way as we did in Eq. (20.3.7), we deﬁne a new process Wn as the sum of the random variables fZig in the modiﬁed Bernoulli process. This is called an elementary random-walk process, given by Wn ¼

n X

Zi ,

n.0

(20:4:15)

i¼1

This process, shown in Fig. 20.4.2, is also a discrete-state discrete-time (DSDT) process. The probability that there are k successes and (n 2 k) failures in n trials is again a binomial

FIGURE 20.4.2

20.4

Independent Increment Process

511

distribution given by n k Probability{k successes in any sequence in n trials} ¼ P(Wn ¼ k) ¼ p (1 p)nk k The statistics of this elementary random walk are Mean: " mW ¼ E ½Wn ¼ E

n X

# Zi ¼

i¼1

n X

E ½Zi ¼ n(2p 1),

n.0

(20:4:16)

i¼1

Variance. From the independence of Zi, the second moment of Wn can be given by " #2 n n n X n X X X 2 E ½Wn ¼ E Zi ¼ E ½Zi2 þ E ½Zi Z j i¼1

i¼1

i¼1 j¼1 2

¼ n þ n(n 1)(2p 1)

n.0

Hence s2W ¼ E ½Wn2 {E ½Wn }2 ¼ n þ n(n 1)(2p 1)2 n2 (2p 1)2 ¼ 4np(1 p),

n.0

(20:4:17)

Autocorrelation. For n1 n2, we obtain " # n2 X n1 X E ½Wn1 Wn2 ¼ RW (n1 , n2 ) ¼ E Zi Z j j¼1 i¼1

" ¼E

n1 X i¼1

# Zi2 þ

n1 X n1 X j¼1 j¼1 i=j

E ½Zi Z j þ

nX n1 2 n1 X

E ½Zi Z j

(20:4:18)

j¼1 i¼1 i=j

Substituting for the expectation operators from Eqs. (20.4.12) and (20.4.13), we obtain E ½Wn1 Wn2 ¼ 4p(1 p)n1 þ n1 n2 (2p 1)2 with a similar expression for n2 n1 given by E ½Wn1 Wn2 ¼ 4p(1 p)n2 þ n1 n2 (2p 1)2 . Combining these two equations, we have E ½Wn1 Wn2 ¼ RW (n1 , n2 Þ ¼ 4p(1 p) min(n1 , n2 ) þ n1 n2 (2p 1)2

(20:4:19)

Autocovariance: CW (n1 , n2 ) ¼ 4p(1 p) min(n1 , n2 )

(20:4:20)

Since the mean is a function of time n and the autocorrelation is a function of both n1 and n2, we conclude that the random-walk process is also a nonstationary process. Example 20.4.3 (Random-Walk Process) Since the random-walk process is a takeoff from the Bernoulli process and with distribution function similar to the binomial process, we can conclude that this process is also a stationary independent increment process. We will show that the random-walk process satisﬁes Eqs. (20.4.2a) and (20.4.2b). In this process m0 ¼ E ½W0 ¼ 0 by deﬁnition and

512

Classiﬁcation of Random Processes

m1 ¼ E ½W1 ¼ E ½Z1 ¼ 2p 1. Substituting for m0 ¼ 0, m1 ¼ (2p 2 1) and t ¼ n in Eq. (20.4.2a), we obtain E ½Wn ¼ mW ¼ n(2p 1), which is the same as Eq. (20.4.16). In a similar manner, s20 ¼ E ½W0 m0 2 ¼ 0 and s21 ¼ E ½W12 m21 ¼ 1 (2p 1)2 ¼ 4p(1 p). Hence var½Wn ¼ 4np(1 p), which is the same as Eq. (20.4.17).

B 20.5

RANDOM-WALK PROCESS

An elementary random walk has been discussed in the previous section. We will generalize this concept a little further. A particle is initially at rest at i ¼ 0 with W0 ¼ 0. At times fi ¼ 1,2, . . . , n, . . .g it undergoes jumps fZi, i ¼ 1, 2, . . . , n, . . .g, where fZig is a sequence of independent identically distributed random variables with probabilities 9 P(Zi ¼ þ1Þ ¼ p = i ¼ 1, . . . , n, . . . , P(Zi Z j ) ¼ P(Zi )P(Z j ), i = j (20:5:1) P(Zi ¼ 1Þ ¼ q ; P(Zi ¼ 0Þ ¼ 1 p q If Wn is the position of the particle after n jumps, then it can be written in terms of Zi as Wn ¼

n X

Zi , n . 0, and W0 ¼ 0

(20:5:2)

i¼1

The random processes Zi and Wn are shown in Fig. 20.5.1. If the particle tends to move indeﬁnitely in either the positive or negative direction, then this random walk is called unrestricted. On the other hand, the particle may be restricted to þa in the positive direction and 2b in the negative direction where a and b are positive, in which case a and b are called absorbing barriers. Let the position of the particle at time n be equal to k where k ¼ 0, +1, +2, . . . , +n. We have to ﬁnd the probability P(Wn ¼ kg. We observe that to reach point k, the particle undergoes r1 positive jumps, r2 negative jumps, and r3 zero jumps with the following constraining conditions: r1 þ r2 þ r3 ¼ n,

k ¼ r1 r2 ,

FIGURE 20.5.1

r3 ¼ n r1 r2

(20:5:3)

20.5 Random-Walk Process

513

With r1 as an independent variable P(Wnr1 ¼ k} is a multinomial distribution given by P(Wnr1 ¼ k) ¼

n! pr1 qr2 (1 p q)r3 r1 !r2 !r3 !

(20:5:4)

Substituting the constraints of Eq. (20.5.3) in Eq. (20.5.4), we obtain P(Wnr1 ¼ k) ¼

n! pr1 qr1 k (1 p q)n2r1 þk r1 !(r1 k)!(n 2r1 þ k)!

(20:5:5)

From Eq. (20.5.5), the constraints for r1 are r1 k and r1 ðn þ kÞ=2. In other words, nþk kr (20:5:6) 2

where n þ2 k represents the integer closest to but not exceeding ðn þ kÞ=2. Since the events {Wnr1 ¼ k} for each r1 are mutually exclusive, we can sum over all possible values of r1 in Eq. (20.5.6) and write P(Wn ¼ kg as follows: X P(Wn ¼ k) ¼ P(Wnr1 ¼ k) r1

¼

bðnþkÞ=2c X r1 ¼k

n! r1 !(r1 k)!(n 2r1 þ k)!

pr1 qr1 k (1 p q)n2r1 þk

(20:5:7)

The probability P(Wn ¼ k) can be more easily determined by using the generating function (GF) deﬁned in Eq. (11.3.1). The generating function for the jumps Zi is X PZ (z) ¼ E ½zZi ¼ zk P(Zi ¼ k) ¼ pz þ qz1 þ (1 p q)z0 k

¼

pz2 þ (1 p q)z þ q z

(20:5:8)

Since Wn is the sum of n independent jumps, the GF for Wn is the n-fold multiplication of the GF PZ (z), or 2 n pz þ (1 p q)z þ q n Wn , n.0 (20:5:9) PWn (z) ¼ E ½z ¼ ½PZ (z) ¼ z and since W0 ¼ 0, the GF for W0 can be deﬁned as follows: PW0 (z) ¼ E ½zW0 ¼ ½PZ (z)0 ¼ 1

(20:5:10)

With PW0 (z) ¼ 1 from Eq. (20.5.10), we can take the s transform of PWn (z) ¼ ½PZ (z)n and write G(z, s) ¼

1 X

sn ½PZ (z)n ¼

n¼0

¼

z

s( pz2

1 1 sPZ (z)

z þ (1 p q)z þ q)

for

jsPZ (z)j , 1

The probability P(Wn ¼ k) is the coefﬁcient of s nz k in Eq. (20.5.11).

(20:5:11)

514

Classiﬁcation of Random Processes 3 Example 20.5.1 In a random-walk process p ¼ 12 and q ¼ 10 . We will determine the probability P(Wn ¼ k) of the particle being at k ¼ 3 in n ¼ 10 trials. We will solve this problem using the direct approach of Eq. (20.5.5) and the generating function approach of Eq. (20.5.11).

Direct Approach. In Eq. (20.5.5) the known quantities are n ¼ 10, k ¼ 3, p ¼ 12, and 3 . We have to ﬁnd P(W10 ¼ 3) for all possible values of r1 the positive jumps. q ¼ 10 The values of r1 from Eq. (20.5.6) are 3,4,5,6. From Eq. (20.5.7) P(W10 ¼ 3) is given by P(W10 ¼ 3) ¼

6 X

10! 1r1 3 r1 3 1132r1 r !(r 3)!(13 2r1 )! 2 10 5 r ¼3 1 1 1

or P(W10 ¼ 3) ¼ 0:000192 þ 0:00756 þ 0:0567 þ 0:070875 ¼ 0:135327: 3 Generating Function Approach. Substituting the values of n ¼ 10, p ¼ 12, and q ¼ 10 in Eq. (20.5.11), we obtain z 1 ¼ G(z, s) ¼ 1 2 1 3 s(5z2 þ 2z þ 3) zs z þ zþ 1 2 5 10 10z

The required probability P(W10 ¼ 3) is the coefﬁcient of s 10z 3 in the power-series expansion of G(z, s). The power-series expansion for G(z,s) in terms of s is given by k 1 X s(5z2 þ 2z þ 3) G(z, s) ¼ 10z k¼0 and the coefﬁcient of s 10 is Coefficient of s10 ¼

10 5z2 þ 2z þ 3 10z

This coefﬁcient of s 10 will be expanded as a power series in z. Expanding the preceding equation, the coefﬁcient of z 3 is 135,327 Coefficient of z3 ¼ 1,000,000 135, 327 , agreeing with the previous result obtained from the Hence, P(W10 ¼ 3) ¼ 1,000,000 direct method.

Random Walk with Two Absorbing Barriers a and 2b

We will now consider a random walk with two absorbing barriers. The particle starts at an initial value of j and terminates at either a or 2b with a,b . 0. The initial value of the random walk is W0 ¼ j, b , j , a It can be shown from heuristic arguments that as n ! 1, the ultimate probability of absorption at either a or 2b is certain. We will deﬁne the probability of absorption as Pa (n, j) ¼ P{absorption at a at time n starting from j},

n ¼ 1, 2; . . .

(20:5:12)

20.5 Random-Walk Process

515

The probability Pa(n, j) also represents the ﬁrst passage of the particle into a at time n starting from j, without having entered either 2b or a at any previous times: Pa (n, j) ¼ P{b , W1 , a, . . . , b , Wn1 , a, Wn ¼ a j W0 ¼ j},

n ¼ 1, 2, . . .

(20:5:13)

At n ¼ 0, we have the initial probabilities 1, if j ¼ a (absorption occurs at a at 0 when it starts at a Pa (0, j) ¼ 0, if j = a (absorption cannot occurs at a at 0 not starting at a (20:5:14) and at j ¼ a and j ¼ 2b the end probabilities for n ¼ 1, 2, . . . , are Pa (n, a) ¼ 0, Pa (n, b) ¼ 0,

n ¼ 1, 2, . . . n ¼ 0, 1, 2, . . .

(20:5:15)

We deﬁne for convenience an event An as An ¼ {absorption at a at time n} and P{An jW0 ¼ j} ¼ Pa ðn, j)

(20:5:16)

Starting from j at n ¼ 0, after the ﬁrst step at n ¼ 1, absorption at a at time (n 2 1) will occur for the cases when the state can be at either ( j þ 1) with probability p or ( j 2 1) with probability q, or remain at j with probability (1 2 p 2 q). Since these are mutually exclusive events, we have P{An j W0 ¼ j} ¼ pP{An1 j W0 ¼ j þ 1} þ qP{An1 j W0 ¼ j 1} þ (1 p q)P{An1 j W0 ¼ j}

(20:5:17)

Substituting the deﬁnition of Pa(n, j) from Eq. (20.5.16) in Eq. (20.5.17), we obtain Pa (n, j) ¼ pPa (n 1, j þ 1) þ qPa (n 1), j 1) þ (1 p q)Pa (n 1, j), b , j , a,

n ¼ 1, 2, . . .

(20:5:18)

Equation (20.5.15) is a difference equation in the two variables n and j. We can eliminate the time variable n by taking the z transform of Pa(n, j), or Ga (z, j) ¼

1 X

zn Pa (n, j)

(20:5:19)

n¼0

with the initial probability Pa(0, j ) given by Eq. (20.5.14). Multiplying Eq. (20.5.18) by z n and summing over all n, we obtain Ga (z, j) ¼ z{pGa (z, j þ 1) þ qGa (z, j 1) þ (1 p q)Ga (z, j)}

(20:5:20)

Equation (20.5.20) is a second-order difference equation in j. The end conditions from Eqs. (20.5.14) and (20.5.15) are Ga (z, a) ¼ 1,

Gb (z, a) ¼ 0

(20:5:21)

We can take another z transform for the variable j. Since j is bounded by a, we can solve the difference equation [Eq. (20.5.20)] more easily by assuming a solution of the form Ga(z, j ) ¼ lj(z). Thus l j (z) þ z pl jþ1 (z) þ ql j1 (z) þ (1 p q)l j (z) ¼ 0

516

Classiﬁcation of Random Processes

or z pl2 (z) ½1 z(1 p q)l(z) þ zq ¼ 0 The solutions to the quadratic equation [Eq. (20.5.22)] are pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 z(1 p q) þ ½1 z(1 p q)2 4z2 pq l1 (z) ¼ 2zp pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 z(1 p q) ½1 z(1 p q)2 4z2 pq l2 (z) ¼ 2zp

(20:5:22)

(20:5:23)

with the relationship between the roots l1 (z)l2 (z) ¼ (zq=zp) ¼ (q=p) and l1 (z) . l2 (z). We will assume that z is positive and real. If l1(z) and l2(z) are to be real, then the discriminant in Eq. (20.5.23) must be positive. Hence we have the following inequality ½1 z(1 p q)2 4z2 pq . 0 or z2 ½(1 p q)2 4pq 2z(1 p q) þ 1 . 0

(20:5:24)

The factors of the preceding quadratic equation in z are pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2(1 p q) + 4(1 p q)2 4½(1 p q)2 4pq z1 ; z 2 ¼ 2½(1 p q)2 4pq pﬃﬃﬃﬃﬃ 1 p q + 2 pq 1 1 ¼ ¼ pﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃ pﬃﬃﬃ (1 p q)2 4pq 1 p q + 2 pq 1 ( p + q)2 or "

1 z1 pﬃﬃﬃﬃ pﬃﬃﬃ 2 1 ( p q)

#"

# 1 z2 .0 pﬃﬃﬃﬃ pﬃﬃﬃ 1 ( p þ q) 2

(20:5:25)

If z is positive, the inequality of Eq. (20.5.25) will be satisﬁed only when both 1 1 pﬃﬃﬃﬃ pﬃﬃﬃ 2 and z2 , pﬃﬃﬃﬃ pﬃﬃﬃ 2 1 ( p q) 1 ( p þ q) pﬃﬃﬃ pﬃﬃﬃ 2 pﬃﬃﬃ pﬃﬃﬃ 2 Since ( p þ q) . ( p q) , Eq. (20.2.25) is equivalent to z1 ,

0,z,

1 pﬃﬃﬃﬃ pﬃﬃﬃ 1 ( p þ q)2

The general solution of the difference Eq. (20.5.2) can be given by Ga (z, j) ¼ c1 (z)lj1 (z) þ c2 (z)lj2 (z)

(20:5:26)

where c1(z) and c2(z) are arbitrary functions of z. Substituting the initial conditions of Eq. (20.5.21) in Eq. (20.5.26), we can solve for c1(z) and c2(z) as follows: c1 (z) ¼

lb1 (z)lb 2 (z) a b l2 (z) l2 (z)laþb 1 (z)

and

c2 ðzÞ ¼

1 la2 (z)

aþb lb 2 (z)l1 (z)

(20:5:27)

20.5 Random-Walk Process

517

Substituting Eq. (20.5.27) into Eq. (20.5.36), we obtain the solution Ga(z, j) as Ga (z, j) ¼

j j lb1 (z)lb l1jþb (z) l2jþb (z) 2 (z)l1 (z) þ l2 (z) ¼ aþb a b aþb l2 (z) l2 (z)l1 (z) l1 (z) laþb 2 (z)

(20:5:28)

To obtain the probability of absorption at a at time n starting from j, we have to express Eq. (20.5.8) as a power series in z, and the coefﬁcient of z n will yield Pa(n,j ). These involved calculations can be found in Refs. 13 and 17. However, the events of eventual absorption at a or – b are mutually exclusive, and their probabilities can be calculated easily. We will ﬁnd the probability of eventual absorption at a starting from j. Equation (20.5.28) is rewritten as Ga (z, j) ¼

l1jþb (z) l2jþb (z) aþb laþb 1 (z) l2 (z)

(20:5:29)

The probability of eventual absorption at a starting at j will be given by the sum of the probabilities Pa(n, j ) over all n and can be obtained by substituting z ¼ 1 in Eq. (20.5.29): Pa ( j) ¼ P{absorption at a starting at j} ¼

1 X

Pa (n, j)

n¼0

¼ Ga (1, j) ¼

l1jþb (1) l2jþb (1) aþb laþb 1 (1) l2 (1)

(20:5:30)

Substituting z ¼ 1 in the roots l1(z) and l2(z), since l1(z) . l2(z), we have p , q, l1 (1) ¼

( p þ q) ( p q) q ( p þ q) þ ( p q) ¼ . l2 (1) ¼ ¼1 2p p 2p

p . q, l1 (1) ¼

( p þ q) þ ( p q) ( p þ q) ( p q) q ¼ 1 . l2 (1) ¼ ¼ 2p 2p p

(20:5:31)

p ¼ q, l1 (1) ¼ l2 (1) ¼ 1 Substituting Eq. (20.5.31) in Eq. (20.5.30), we obtain Pa ( j) ¼ G(1, j) ¼ paj

p jþb q jþb , paþb qaþb

p=q

(20:5:32a)

Using l’Hoˆpital’s rule when p ¼ q, we have Pa (j) ¼

jþb , aþb

p¼q

(20:5:32b)

In a similar manner we can obtain the solution G2b(z, j) for the absorption at 2b starting from j. The result is Gb (z, j) ¼

l1ja (z) (aþb) l1 (z)

l2ja (z) l2(aþb) (z)

The probability of absorption at 2b starting at j is given by Gb (1, j) ¼ Pb ( j) ¼ 1 Pa ( j) ¼ 1 Ga (1, j)

(20:5:33)

518

Classiﬁcation of Random Processes

or 8 paj qaj > > < qbþj aþb , p qaþb Pb (0) ¼ Gb (1, 0) > aj > : , p¼q aþb

p=q (20:5:34)

Example 20.5.2 (Gambler’s Ruin) This is a classic problem that has wide applications in testing the efﬁcacy of drugs, insurance payoffs, and so on. Two gamblers A and B wager on the ﬂip of a biased coin. If heads show up, A wins 1 dollar from B, and if tails show up, A loses one dollar to B. The probability of heads is p and tails q ¼ 1 2 p. Initially A starts with k dollars and B with N 2 k dollars. The game stops when either A loses all the money or is ruined or A wins all the money and B is ruined. We want to calculate the probabilities of A either winning all the money or losing all the money. This problem can be cast in the random-walk framework with absorbing barriers. We will solve this problem from fundamentals and show that we get the same result from using generating functions. Each toss is an i.i.d. random variable Zi with probabilities

P(Zi ¼ þ1) ¼ p i ¼ 1, . . . , n, . . . , P(Zi Zj ) ¼ P(Zi )P(Zj ), i = j P(Zi ¼ 1) ¼ q ¼ (1 p) and the cumulative sum Wn is given by Eq. (20.5.2) with initial value W0 ¼ k. Or, Wn ¼

n X

Zi , n . 0

and

W0 ¼ k

(20:5:35)

i¼1

The game terminates when Wn ¼ N or Wn ¼ 0. The absorbing barriers are either N or 0. We deﬁne an event An as absorption at N at time n and the probability of absorption at N at time n on condition that the starting state is k, as follows: P{An jW0 ¼ k} ¼ PN (n, k) Starting from k at n ¼ 0, after the ﬁrst step at n ¼ 1, absorption at N at time (n 2 1) will occur for the cases when the state can be at either (k þ 1) with probability p or (k 2 1) with probability q. Since these are mutually exclusive events, we have P{An jW0 ¼ k} ¼ pP{An1 jW0 ¼ k þ 1} þ qP{An1 jW0 ¼ k 1} and in terms of the deﬁned probability PN(n, k) PN (n, k) ¼ pPN (n 1, k þ 1) þ qPN (n 1, k 1),

k ¼ 1, 2, . . . , N 1

with end conditions PN(n, N ) ¼ 1 and PN(n, 0) ¼ 0, for n ¼ 1, 2, . . . , N 2 1. We are interested in the eventual probability of absorption in N and hence we can eliminate n from the equation above and write PN (k) ¼ pPN (k þ 1) þ qPN (k 1),

k ¼ 1, 2, . . . , N 1

with end conditions PN(N ) ¼ 1 and PN(0) ¼ 0, Since p þ q ¼ 1, the preceding equation can be rewritten as PN (k)( p þ q) ¼ pPN (k þ 1) þ qPN (k 1)

20.5 Random-Walk Process

519

and rearranging terms, we have q PN (k þ 1) PN (k) ¼ ½PN (k) PN (k 1), p

k ¼ 1, 2, . . . , N 1

We solve this equation recursively as follows from the initial condition PN(0) ¼ 0 k ¼ 1: k ¼ 2:

q q PN (2) PN (1) ¼ ½PN (1) PN (0) ¼ PN (1) p p 2 q q PN (1) PN (3) PN (2) ¼ ½PN (2) PN (1) ¼ p p

q k ¼ k 1: PN (k) PN (k 1) ¼ ½PN (k 1) PN (k 2) p k1 q ¼ PN (1) p k ¼ N 1:

q PN (N) PN (N 1) ¼ ½PN (N 1) PN (N 2) p N1 q PN (1) ¼ p

Adding the ﬁrst k21 equations, we obtain "

2 k1 # q q q PN (k) PN (1) ¼ PN (1) þ þ þ p p p or "

2 k1 # q q q PN (k) ¼ PN (1) 1 þ þ þ þ p p p k q 1 p ¼ PN (1) q , 1 p PN (k) ¼ PN (1)k,

p=q

p¼q

Substituting k ¼ N in the equation above, we can obtain PN(1) by applying the boundary condition PN(N ) ¼ 1 as shown below: q 1 p PN (1) ¼ N , p = q q 1 p

and

PN (1) ¼

1 , p¼q N

520

Classiﬁcation of Random Processes

Substituting for PN(1) in the equation for PN(k), we obtain k k q q q 1 1 1 p p p PN (k) ¼ N ¼ N , q q q 1 1 p 1 p p ¼

k , N

p¼q

p=q

(20:5:36)

PN(k) is the probability that A starting from k dollars will eventually reach N dollars. The probability of ruin P0(k) of A is Nk p 1 q P0 (k) ¼ 1 PN (k) ¼ N , p = q p 1 q ¼

Nk , N

p¼q

(20:5:37)

The probability of ruin P0(k) can be easily obtained from Eq. (20.5.34) by substituting for the absorbing barriers b ¼ 0, a ¼ N and j ¼ k, which results in Eq. (20.5.37). Discussion of Results. We will investigate what happens when A plays against an inﬁnitely rich adversary B. If p . q, then ðq=pÞ , 1, and as N ! 1 in Eq. (20.5.36), we have k q k 1 q p ¼ 1 .0 (20:5:38) p . q: lim PN (k) ¼ lim N N!1 N!1 p q 1 p This results in a ﬁnite probability that A will get inﬁnitely rich. On the other hand, if p q, then (q=p) . 1, and as N ! 1 in Eq. (20.5.36), we have k q Nk 1 p p p , q: lim PN (k) ¼ lim ¼ lim ! 0 N N!1 N!1 N!1 q q (20:5:39) 1 p p ¼ q:

lim PN (k) ¼ lim

N!1

k

N!1 N

! 0

This results in A getting ruined with probability 1 if p q. The interesting part is that even if it is a fair game with equal probability of winning, a rich player will always ruin a poorer player. On the other hand, if p . q, A can get inﬁnitely rich even against an inﬁnitely rich adversary. The gambling halls all over the world make sure that the probability of any player winning is always less than 0.5! Example 20.5.3 An insurance company earns $1 per day from premiums with probability p. However, on each day it pays out a claim of $2 each day independent of the past with probability q ¼ 1 2 p. When a claim is paid, $2 is removed from the reserve

20.6

Gaussian Process

521

of money of $k. Hence we can assume that on the ith day, the income for that day is a random variable Zi equal to 1 with probability p and 21 with probability q as in Example 20.5.2. The cumulative income for the company is Wn as in Eq. (20.5.35). If it starts initially with a reserve capital of $k, then we have to ﬁnd the probability that it will eventually run out of money. The solution is a reformulation of Eqs. (20.5.38) and (20.5.39). If p . q, then the probability that it will not run out of money is given by 1 (q=p)k . 0. If k is large, then it has very high probability of staying in business. On the other hand, if p q, it will always run out of money as indicated in Eq. (20.5.39).

B 20.6

GAUSSIAN PROCESS

Many of the physically occurring random phenomena can be modeled as a Gaussian process. Further, according to the central limit theorem, random variables characterized individually by arbitrary distributions can be modeled as Gaussian if a sufﬁcient number of them can be added. The distribution of a more commonly used white noise is Gaussian. We have already stated (in Section 6.4) that a random variable X is Gaussiandistributed if its density function is given by fX (x) ¼

2 1 pﬃﬃﬃﬃﬃﬃ eð1=2Þ½(xmX )=sX sX 2p

(20:6:1)

where mX is the mean value and s2X is the variance. A second-order random process X(t) is a Gaussian random process if for any collection of times ft1, t2, . . . , tng, the random variables, X1 ¼ X(t1), X2 ¼ X(t2), . . . , Xn ¼ X(tn) are jointly Gaussian distributed random variables for all n with the joint density given by fX1 X2 ...Xn (x1 , x2 , . . . , xn ) ¼

T 1 1 e(1=2)(xmX ) CX (xmX ) (2p)n=2 jCX j

where the mean vector mX and the covariance 2 2 3 mX (t1 ) CX (t1 ) 6 CX (t2 , t1 ) 6 mX (t2 ) 7 6 6 7 mX ¼ 6 . 7; CX ¼ 6 .. 4 4 .. 5 . CX (tn , t1 ) mX (tn )

(20:6:2)

matrix CX are given by CX (t1 , t2 ) CX (t2 ) .. . CX (tn , t2 )

3 CX (t1 , tn ) CX (t2 , tn ) 7 7 7 .. 5 . CX (tn )

(20:6:3)

Properties 1. A Gaussian process is completely determined if the mean vector mX and the covariance matrix CX are known. 2. Sum of Gaussian processes are also Gaussian. 3. If a Gaussian process is weakly stationary, it is also strictly stationary.

White Gaussian Noise

A white-noise process has been deﬁned in Eq. (19.2.17) as a stationary process whose autocorrelation is a Dirac delta function. The white-noise process that is widely used for

522

Classiﬁcation of Random Processes

FIGURE 20.6.1

modeling communication problems is the Gaussian process. A standard Gaussian whitenoise process V(t) has zero mean and unit variance with probability density 1 2 fV (v) ¼ pﬃﬃﬃﬃﬃﬃ e(v =2) , 1 , v , 1 2p

(20:6:4)

The autocorrelation function and power spectral densities are given by RV (t) ¼ d(t);

SV (v) ¼ 1, 1 , v , 1

(20:6:5)

Similarly, a standard white Gaussian noise sequence fVng has zero mean and unit variance with the sequences coming from a density function given by Eq. (20.6.4). However, its autocorrelation function and power spectral densities are given by RV (h) ¼ dh ;

SV (v) ¼ 1, 1 , v , 1

(20:6:6)

In this study, 10,000 points of a Gaussian white noise fVng was generated from a computer using the Box – Mueller method. The noise sequence was standardized to zero mean and unit variance. This sequence is shown in Fig. 20.6.1a for only 200 points for clarity, and its histogram illustrating the Gaussian nature of the density is shown in Fig. 20.6.1b. Example 20.6.1 [AR(1) Process] In Example 19.2.11 a discrete zero mean stationary random process Xi was given by Xn ¼ fXn1 þ Vn , n ¼ 1, . . . , X0 ¼ V0 where fVn, n ¼ 0, 1. . .g is a Gaussian process. This process is called an autoregressive (1) process. We will investigate this process assuming that Vn is standard Gaussian white noise and the coefﬁcient f ¼ 0.8. We will show that Xn is also Gaussian-distributed. By deﬁning B k as a backward difference operator deﬁned by Bk Xn ¼ Xnk , k 0, we can write the given equation as (1 fB)Xn ¼ Vn or Xn ¼ (1 fB)1 Vn ¼ (1 þ fB þ f2 B2 þ þ fk1 Bk1 )Vn ¼ Vn þ fVn1 þ f2 Vn2 þ þ fk1 Vnkþ1 þ Since Xn is a sum of a weighted Gaussian-distributed random sequence, it is also a Gaussian-distributed continuous-state discrete-time (CSDT) random process. The

20.7

Wiener Process (Brownian Motion)

523

FIGURE 20.6.2

variance of this process as given by Eq. (19.2.47) is s2X ¼ s2X =(1 f2 ) ¼ 1=ð1 0:64Þ ¼ 2:778 with mean mX ¼ 0. This process Xn simulated on a computer with 10,000 points of the Gaussian noise sequence generated previously is shown in Fig. 20.6.2a for only 200 points for clarity. The variance of the simulated process is 2.905, which is very close to the true value of 2.778. The histogram of the data of 10,000 points is shown in Fig. 20.6.2b. On the same histogram plot is shown the true pdf of Xn and the pdf of true white-noise sequence Vn.

B 20.7

WIENER PROCESS (BROWNIAN MOTION)

The random motion of a particle in a ﬂuid subject to collisions and inﬂuence of other particles is called Brownian motion. One of the mathematical models of this motion is the Wiener process. If the following conditions are satisﬁed, W(t) is a Wiener process with parameter s2: 1. W(0) ¼ 0. 2. W(t) is a Gaussian process with sample continuous functions. 3. E[W(t)] ¼ 0. 4. The autocovariance function CW(t, s) ¼ RW(t, s) ¼ E[W(t), W(s)] ¼ s2 min(t, s). 5. The variance of W(t) from (4) is s2W (t) ¼ s2 t. The pdf of the Wiener process is given by 1 2 2 fW (w, t) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e(1=2)(w =s t) 2pst

(20:7:1)

The covariance matrix of the random variable vector WT ¼ ½W(t1 ), W(t2 ), . . . , W(tn )T for times 0 , t1 , t2 , . . . , tn can be given from condition 4 as 2 3 t1 t1 t1 t1 6 t1 t2 t2 t2 7 6 7 T 2 6 t1 t2 t3 t3 7 CW ¼ E ½WW ¼ s 6 (20:7:2) 7 6 .. .. .. .. 7 4. . . 5 . t1 t2 t3 tn

524

Classiﬁcation of Random Processes

and this matrix is positive deﬁnite. Since W(t) is Gaussian, the ﬁnite-dimensional density function using Eq. (20.7.2) is given by fW (w1 , t1 ; w2 , t2 ; . . . ; wn , tn ) ¼

T 1 1 e(1=2)W CW W 1=2 (2p) jCW j

n=2

(20:7:3)

The determinant jCW j ¼ t1 (t2 t1 ) (tn1 tn ) and Eq. (20.7.3) can be rewritten as

n Y 1 1 (wk wk1 )2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp fW (w1 , t1 ; w2 , t2 ; . . . ; wn , tn ) ¼ 2 2 s2 (tk tk1 ) k¼1 2p(tk tk1 )s (20:7:4) with t0 ¼ 0 ¼ w0. The density function of Eq. (20.7.4) shows that the sequence {Wt1 , (Wt2 Wt1 ), . . . , (Wtn Wtn1 )} is a collection of independent random variables for increasing f0 , t1 , t2 , . . . , tng and the density depends only on the time difference, showing that the Wiener process is a process of stationary independent increments. Hence, it is nonstationary and is a continuous-state continuous-time (CSCT) process. Derivation of Wiener Process

We will derive the Wiener process as the limiting form of the elementary randomwalk process discussed in Section 20.4, which will be restated in a form where limits can be taken. A particle starts from origin at t ¼ 0. In a small interval Dt it undergoes a small step +Dw with equal probabilities, thus constituting a symmetric random walk. The number n of Bernoulli trials in the time interval (0, t) is equal to the greatest integer less than or equal to t/Dt, or jtk n¼ Dt where bxc represents the greatest integer less than or equal to x, called the “ﬂoor function.” Similar to the deﬁnition in Eq. (20.4.11), the Bernoulli random variable Zi is deﬁned by þ1 if the ith step of Dw is upward Zi ¼ 1 if the ith step of Dw is downward with probabilities 1 2 1 P(Zi ¼ 1) ¼ 2 P(Zi ¼ þ1) ¼

(20:7:5)

The mean and variance of Zi are given by E ½Zi ¼ 0 and var ½Zi ¼ 1. If the position of the particle at time nDt is W(nDt), then W(nDt) ¼ Dw

n X i¼1

Zi u(t iDt)

(20:7:6)

20.7

Wiener Process (Brownian Motion)

525

The mean and variance of the process W(nDt) are E ½W(nDt) ¼ 0 and

var ½W(nDt) ¼ E ½W 2 (nDt) ¼ Dw2 n ¼ Dw2

jtk Dt

(20:7:7)

We now let Dw and Dt tend to zero in such a way that Eq. (20.7.7) is nontrivial. Hence we have to set pﬃﬃﬃﬃﬃ Dw ¼ s Dt (20:7:8) where s is a positive constant so that as Dt ! 1, W(t) ¼ lim s Dt!0

1 X

pﬃﬃﬃﬃﬃ Zi u(t iDt) Dt

i¼1

and lim Dw2

Dt!0

jtk ! s2 t Dt

(20:7:9)

W(t) is the desired Wiener process, and a sample function is shown in Fig. 20.7.1. The statistics of the Wiener process are as follows: Mean. Since E[Zi] ¼ 0, we have E ½W(t) ¼ mW (t) ¼ 0

(20:7:10)

Variance. From Eq. (20.7.7), we obtain var½W(t) ¼ E ½W 2 (t) ¼ s2W (t) ¼ s2 t

(20:7:11)

Autocorrelation. Since E[W(t)] ¼ 0, RW (t1, t2) ¼ CW (t1, t2). For t1 , t2: RW (t1 , t2 ) ¼ CW (t1 , t2 ) ¼ E½W(t1 )W(t2 ) ¼ E{½W(t2 ) W(t1 ) þ W(t1 )W(t1 )}: From linearity of expectations: RW (t1 , t2 ) ¼ E{½W(t2 ) W(t1 )W(t1 )} þ E ½W 2 (t1 ). From independent increments: RW (t1 , t2 ) ¼ E ½W 2 (t1 ) ¼ s2 t1 . Similarly, for t2 , t1, we have RW (t1 , t2 ) ¼ E ½W 2 (t2 ) ¼ s2 t2 .

FIGURE 20.7.1

526

Classiﬁcation of Random Processes

Hence RW (t1 , t2 ) ¼ CW (t1 , t2 ) ¼ s2 min(t1 , t2 )

(20:7:12)

The Wiener process has the following properties: 1. From the properties of the random walk from which W(t) was constructed, we conclude that for (0 , t , s) the increments W(t) 2 W(0) and W(s) 2 W(t) are independent. 2. From Eq. (20.7.1), for t2 , t1 the density function of the increment W(t2) 2 W(t1) is 2 2 1 2 fW (w1 , t1 ; w2 , t2 ) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ eð1=2Þ½(w2 w1 ) =s (t2 t1 ) 2 2p(t2 t1 )s

(20:7:13)

and since it depends only on the time difference (t2 2 t1), W(t) is a stationary independent increment process. 3. Since W(t) is the limit of a symmetric random walk, it can be conjectured that it is a continuous process for all t . 0. Indeed it can be shown [35,37] that it is continuous everywhere with probability 1. 4. It can be shown [35,37] that W(t) is of unbounded variation, and as a result it is differentiable nowhere in the conventional sense. Property 4 has far-reaching consequences. If we have any function g[W(t)] of another nonrandom function W(t), then g[W(t þ dt)] can be expanded in a Taylor series as g½W(t þ dt) ¼ g½W(t) þ

dg½W(t) 1 d2 g½W(t) dW 2 (t) þ dW(t) þ dw 2 dw2

(20:7:14)

Neglecting higher-order terms Eq. (20.7.14), we can write dg[W(t)] as g½W(t þ dt) g½W(t) ¼ dg½W(t) ¼

dg½W(t) dW(t) dw

(20:7:15)

However, if W(t) is a Wiener process, the term dW 2(t) cannot be neglected because from Eq. (20.7.8) it is of the order of dt. Hence the differential dg[W(t)] can be written only as g½W(t þ dt) g½W(t)

dg½W(t) 1 d2 g½W(t) dW(t) þ dW 2 (t) dw 2 dw2 dg½W(t) 1 d 2 g½W(t) dW(t) þ s2 dt dw 2 dw2

(20:7:16)

neglecting only third and higher-order terms. Thus the differential dg[W(t)] does not obey the ordinary laws of calculus! It is also interesting to note that if W(t) is a deterministic process, then s2 is equal to zero and Eq. (20.7.16) reverts to the usual calculus. For more details on this interesting aspect, see Refs. 35 and 37. Formal Derivative of Wiener Process

We have stated in properties 3 and 4 that the Wiener process is continuous everywhere but differentiable nowhere in the conventional sense. However, we will formally

20.8

Markov Process

527

differentiate it and obtain its autocorrelation function. If Z(t) ¼ W(t) is the formal derivative of the Wiener process, it can be shown from Eq. (19.2.31b) that the autocorrelation function of Z(t) ¼ W(t) is given by RZ (t, s) ¼ RW (t, s) ¼ E ½W(t)W(s) ¼

@2 RW (t, s) @t @s

(20:7:17)

and if s ¼ t þ t, we obtain Eq. (19.2.31b). We substitute RW (t, s) ¼ s2 min(t, s) from Eq. (20.7.12) in Eq. (20.7.17) and obtain RW (t, s) ¼ RZ (t, s) ¼ E ½Z(t)Z(s) ¼

@2 2 s min(t, s) ¼ s2 d(t s) @t @s

(20:7:18)

From Eq. (20.7.18) we conclude that since the autocorrelation of the derivative of the Wiener process yields a Dirac delta function, it exists only in the generalized sense. However, the autocorrelation of Z(t) ¼ W(t) is a function of the time difference (t 2 s). It is also a zero mean Gaussian process. Hence the formal derivative Z(t) of a Wiener process is a zero mean stationary Gaussian process with power spectral density s2 that is constant in the entire frequency range. This is the Gaussian white noise that has been used extensively to model communication and time-series problems. The similarity between the Gaussian and Poisson white noises [Eq. (20.2.39)] is to be noted.

B 20.8

MARKOV PROCESS

It was stated in Section 19.1 that a random process X(t) is completely speciﬁed if the nth-order joint density given by Eq. (19.1.9) for an ordered set of time functions ft1 , t2 , t3 , . . . , tng fX (xn , . . . , x1 ; tn , . . . , t1 ) ¼

@n FX (xn , . . . , x1 ; tn , . . . , t1 ) @xn @x1

(20:8:1)

is speciﬁed for all positive integers n. The joint density given by Eq. (20.8.1) can be decomposed into n ﬁrst-order conditional densities for the same set of ordered time functions as shown below: fX (x1 ; t1 ) fX (x2 ; t2 j x1 ; t1 ) fX (x3 ; t3 j x2 , x1 ; t2 , t1 ) fX (xn1 ; tn1 j xn2 , . . . , x2 , x1 ; tn2 , . . . , t2 , t1 )

(20:8:2)

fX (xn ; tn j xn1 , . . . , x2 , x1 ; tn1 , . . . , t2 , t1 ) By repeated application of the deﬁnition of conditional probability, the n-dimensional joint density of Eq. (20.8.1) can be obtained from Eq. (20.8.2). Thus, for the random process to be completely determined, we should have n ! 1 of these ﬁrst-order conditional densities. However, the speciﬁcations for a class of random processes can be considerably simpliﬁed if we can express their conditional densities as fX (xn ; tn jxn1 , . . . , x2 , x1 ; tn1 , . . . , t2 , t1 ) ¼ fX (xn ; tn jxn1 ; tn1 )

(20:8:3)

What Eq. (20.8.3) conveys is that the probabilistic behavior of the present given all past history is dependent only on the immediate past. The property exhibited by Eq. (20.8.3) is called the Markov property, and the class of processes having this property is known as

528

Classiﬁcation of Random Processes

Markov processes. Using the Markov property the n-dimensional density can be given as follows: fX (xn , . . . , x1 ; tn , . . . , t1 ) ¼ f (xn ; tn j xn1 ; tn1 )fX (xn1 , . . . , x1 ; tn1 , . . . , t1 )

(20:8:4)

By applying the Markov property repeatedly to lower-order density functions, we can write the n-dimensional density as fX (xn , . . . , x1 ; tn , . . . , t1 ) ¼ fX (x1 ; t1 )

n Y

f (xi ; ti j xi1 ; ti1 )

(20:8:5)

i¼2

Hence all ﬁnite-dimensional densities of a Markov process are completely determined by the initial marginal density fX (x1 ; t1 ) and the set of ﬁrst-order conditional densities, {f (xi ; ti j xi1 ; ti1 ), i ¼ 2, 3, . . .}. The conditional probability density f (xi ; ti j xi1 ; ti1 ) is called the transition probability density. For a discrete-time Markov process, the equation corresponding to Eq. (20.8.3) can be written as follows: P{X(tn ) xn j X(tn1 ) xn1 , . . . , X(t1 ) x1 } ¼ P{X(tn ) xn j X(tn1 ) xn1 } (20:8:6) What we have shown is that given a Markov process, its n-dimensional density can be expressed by the initial marginal density and a set of ﬁrst-order conditional densities. However, to construct a Markov process from the given ﬁrst-order conditional densities, the following consistency conditions have to be satisﬁed: ð1 1. fX (x2 ; t2 j x1 ; t1 ) fX (x1 ; t1 ) dx2 , t1 , t2 (20:8:7) fX (x2 ; t2 ) ¼ 1

This equation is the usual equation that relates a joint density to its marginal density. ð1 (x ; t j x ; t ) ¼ fX (x3 ; t3 j x2 ; t2 ) fX (x2 ; t2 j x1 ; t1 ) dx2 , t1 , t2 , t3 (20:8:8) f 2. X 3 3 1 1 1

This equation is called the Chapman– Kolmogorov equation, which all Markov processes must satisfy. We will show that Markov processes satisfy this equation. The joint pdf fX (x3 , x1 ; t3 , t1 ) with t1 , t2 , t3 can be written as ð1 fX (x3 , x2 , x1 ; t3 , t2 , t1 ) dx2 (20:8:9) fX (x3 , x1 ; t3 , t1 ) ¼ 1

Using the property of conditional expectation twice, we can rewrite Eq. (20.8.9) as, ð1 fX (x3 ; t3 j x2 , x1 ; t2 , t1 )fX (x2 , x1 ; t2 , t1 ) dx2 fX (x3 , x1 ; t3 , t1 Þ ¼ 1

ð1

fX (x3 ; t3 j x2 , x1 ; t2 , t1 )fX (x2 ; t2 j x1 ; t1 )fX (x1 ; t1 ) dx2

¼ 1

(20:8:10) Using the Markov property fX (x3 ; t3 j x2 , x1 ; t2 , t1 ) ¼ fX (x3 ; t3 j x2 ; t2 ) in Eq. (20.8.10), we obtain ð1 fX (x3 ; t3 j x2 ; t2 )fX (x2 ; t2 j x1 ; t1 ) dx2 (20:8:11) fX (x3 , x1 ; t3 , t1 ) ¼ fX (x1 ; t1 ) 1

20.8

Markov Process

529

Dividing Eq. (20.8.11) throughout by fX (x1 ; t1 ), for t1 , t2 , t3 , we have fX (x3 , x1 ; t3 , t1 ) ¼ fX (x3 , t3 j x1 ; t1 ) fX (x1 ; t1 ) ð1 fX (x3 ; t3 j x2 ; t2 ) fX (x2 ; t2 j x1 ; t1 ) dx2 ¼

(20:8:8)

1

which is the desired Chapman–Kolmogorov (C–K) equation. This derivation also holds good for discrete-time processes, and the result is X P{X(t3 ) x3 j X(t1 ) x1 } ¼ P{X(t3 ) x3 j X(t2 ) x2 } x2

P{X(t2 ) x2 j X(t1 ) x1}

(20:8:12)

The state fx2, t2) represents the intermediate state, and operation of the C – K equation is shown in Fig. 20.8.1. The C – K equation is a sufﬁcient condition for a Markov process in the sense that nonMarkovian processes may also satisfy this equation. We can also deﬁne a Markov process in terms of conditional expectations. If g() is any function with maxx jg(x)j , 1, then X(t) is a Markov process for an ordered sequence ft1 , t2 , . . . , tng if and only if E{g½X(tn ) j X(tn1 ), X(tn2 ), . . . , X(t1 )} ¼ E{g½X(tn ) j X(tn1 )}

(20:8:13)

If the conditional probability P{X(tn ) xn j X(tn1 ) xn1 } of a Markov process is dependent only on the time difference (tn tn1 ), it is called a homogeneous process. In other words, if (t2 t1 ) ¼ (tn tn1 ), then homogeneity implies P{X(tn ) xn j X(tn1 ) xn1 } ¼ P{X(t2 ) x2 j X(t1 ) x1 } Generally, a homogenous Markov process may not be stationary since PfX(t1) x1g may depend on t1. We have shown earlier that a stationary independent increment process is always nonstationary in light of Eqs. (20.4.2a) and (20.4.2b). We will now show that this process is always a Markov process. Let X(t) be a stationary independent increment process with

FIGURE 20.8.1

530

Classiﬁcation of Random Processes

t0 ¼ 0 and X(t0) ¼ x0 ¼ 0. For all ti , tj, the events {X(ti ) ¼ xi , X(tj ) ¼ xj

and

{X(ti t0 ) ¼ xi x0 , X(tj ti ) ¼ xj xi }

are equal, and hence their joint densities are also equal. As a consequence fX (xj ; tj , xi ; ti ) ¼ fX (xj xi ; tj ti , xi x0 ; ti t0 ),

i,j

(20:8:14)

With x0 ¼ t0 ¼ 0 and using the independent increment property, Eq. (20.8.14) can be rewritten after dropping the time difference (tj 2 ti), due to stationarity: fX (xj ; tj , xi ; ti ) ¼ fX (xj xi ) fX (xi ; ti ),

i,j

(20:8:15)

Analogous to Eq. (20.8.15), for the sequence ftn , tn21 , . . . , t1 , t0 ¼ 0g, we can write fX (xn , . . . ; x1 , x0 ; tn , . . . , t1 , t0 ) ¼ fX (xn xn1 , . . . , x2 x1 , x1 x0 ; tn tn1 , . . . , t2 t1 , t1 t0 ) ¼ fX (xn xn1 ) fX (x2 x1 ) fX (x1 ; t1 )

(20:8:16)

From Eq. (20.8.15) we can substitute fX (xk xk1 ) ¼

fX (xk , xk1 ; tk , tk1 ) , fX (xk1 ; tk1 )

k ¼ n, n 1, . . . , 2

in Eq. (20.8.16) and write fX (xn , . . . x1 ; tn , . . . , t1 ) ¼

fX (xn , xn1 ; tn , tn1 ) fX (x2 , x1 ; t2 , t1 ) fX (xn1 ; tn1 ) fX (x2 ; t2 )

(20:8:17)

Equation (20.8.17) expresses the nth-order joint density of a stationary independent increment process X(t) in terms of the second- and ﬁrst-order densities. From the deﬁnition of conditional density, we can write fX (xn ; tn j xn1 , . . . , x1 ; tn1 , . . . , t1 ) ¼

fX (xn , xn1 x1 ; tn , tn1 , . . . , t1 ) fX (xn1 x1 ; tn1 , . . . , t1 )

(20:8:18)

Applying Eq. (20.8.17) to both numerator and denominator of Eq. (20.8.17), we obtain fX (xn ; tn j xn1 , . . . , x1 ; tn1 , . . . , t1 ) fX (xn , xn1 ; tn , tn1 ) fX (x2 , x1 ; t2 , t1 ) fX (xn1 ; tn1 ) fX (x2 ; t2 ) ¼ fX (xn1 ; tn , tn1 ) fX (x2 , x1 ; t2 , t1 ) fX (xn2 ; tn2 ) fX (x2 ; t2 ) ¼

(20:8:19)

fX (xn , xn1 ; tn , tn1 ) ¼ fX (xn ; tn j xn1 ; tn1 ) fX (xn1 ; tn1 )

which is the deﬁning equation of a Markov process. However, the converse is not true, or, not every Markov process is a stationary independent increment process. Summary of Properties of Markov Processes 1.

fX (xn ; tn j xn1 ; tn1 , . . . , x1 ; t1 ) ¼ fX (xn ; tn j xn1 ; tn1 )

(20:8:20)

20.8

Markov Process

531

Applying the chain rule for conditional expectations from Eq. (20.8.5), we can write fX (xn ; tn j xn1 ; tn1 , . . . , x1 ; t1 ) ¼

fX (xn ; tn , xn1 ; tn1 , . . . , x1 ; t1 ) fX (xn1 ; tn1 , . . . , x1 ; t1 )

¼ fX (xn ; tn j xn1 ; tn1 )

(20:8:21)

2. This rule can be generalized to any sequence of times ftnk . tnk21 . . . . . tn1g: fX (xnk ; tnk j xnk1 ; tnk1 , . . . , xn1 ; tn1 ) ¼ fX (xnk ; tnk j xnk1 ; tnk1 ) 3. A time-reversed Markov process is also a Markov process: fX (xn ; tn j xnþ1 ; tnþ1 , . . . , xnþk ; tnþk ) ¼ fX (xn ; tn j xnþ1 ; tnþ1 )

(20:8:22)

4. Given the present information, the past and the future are conditionally independent. Or, for k , m , n, we obtain fX (xn ; tn , xk ; tk j xm ; tm ) ¼ fX (xn ; tn j xm ; tm ) f (xk ; tk j xm ; tm )

(20:8:23)

We will show that this result is true: fX (xn ; tn , xk ; tk j xm ; tm ) ¼ ¼

fX (xn ; tn , xm ; tm , xk ; tk ) fX (xm ; tm ) fX (xn ; tn j xm ; tm ) fX (xm ; tm j xk ; tk ) fX (xk ; tk ) fX (xm ; tm )

(chain rule)

Substituting for fX (xm ; tm j xk ; tk ) ¼

fX (xk ; tk j xm ; tm ) fX (xm ; tm ) fX (xk ; tk )

in the preceding equation, we obtain Eq. (20.8.23). 5. Taking conditional expectations in property 3, we have E ½X(tn )X(tk ) j X(tm ) ¼ xm ¼ E ½X(tn ) j X(tm ) ¼ xm ¼ E ½X(tk ) j X(tm ) ¼ xm

(20:8:24)

6. The future, conditioned on the past history upto the present, is the future conditioned on the present: E{X(tnþ1 ) j X(tn ), X(tn1 ), . . . , X(t1 )} ¼ E{X(tnþ1 ) j X(tn )} This equation is a restatement of Eq. (20.8.20).

(20:8:25)

532

Classiﬁcation of Random Processes

7. If t1 , t2 , t3, then E ½X(t3 ) j X(t1 ) ¼ E{E ½X(t3 ) j X(t2 )jX(t1 )}

(20:8:26)

This equation is a restatement of the C –K property in terms of conditional expectations.

Example 20.8.1 (Poisson Process) The Poisson process N(t) is an independent increment process from Eq. (20.2.10.). It is also a stationary independent increment process, because the probability of the increment, N(t2) 2 N(t1), given by P{N(t2 ) N(t1 ) ¼ k2 k1 } ¼

½l(t2 t1 )k2 k1 l(t2 t1 ) e (k2 k1 )!

is dependent only on time difference (t2 2 t1). From the result of Eq. (20.8.19), we can conclude that the Poisson process is a homogeneous Markov process. However, it will be instructive to show the Markov property from the deﬁnition of Eq. (20.8.20). The second-order probability function is given by P{N(t2 ) ¼ k2 , N(t1 ) ¼ k1 } ¼ P{N(t2 ) ¼ k2 jN(t1 ) ¼ k1 }P{N(t1 ) ¼ k1 } The conditional probability N(t2 ) ¼ k2 given that N(t1 ) ¼ k1 is the probability that there are k2 2 k1 events occurring in the time interval t2 2 t1: P{N(t2 ) ¼ k2 jN(t1 ) ¼ k1 } ¼ P{N(t2 ) N(t1 ) ¼ k2 k1 } ¼

½l(t2 t1 )k2 k1 l(t2 t1 ) e , (k2 k1 )!

k2 k1

Hence the second-order probability function P{N(t2 ) ¼ k2 , N(t1 ) ¼ k1 } is given by P{N(t2 ) ¼ k2 , N(t1 ) ¼ k1 } ¼

½lt1 )k1 ½l(t2 t1 )k2 k1 lt2 e , k1 !(k2 k1 )!

k2 k1

Proceeding in a similar manner, we can obtain the third-order probability function as P{N(t3 ) ¼ k3 , N(t2 ) ¼ k2 , N(t1 ) ¼ k1 } ¼

½lt1 )k1 ½l(t2 t1 )k2 k1 ½l(t3 t2 )k3 k2 lt3 e , k1 !(k2 k1 )!(k3 k2 )!

k3 k2 k1

The second-order probability function P{N(t3 ) ¼ k3 , N(t2 ) ¼ k2 } can be evaluated as P{N(t3 ) ¼ k3 , N(t2 ) ¼ k2 } ¼ P{N(t3 ) N(t2 ) ¼ k3 k2 }P{N(t2 ) ¼ k2 } ¼

½lt2 )k2 ½l(t3 t2 )k3 k2 lt3 e , k2 !(k3 k2 )!

k3 k2

20.8

Markov Process

533

Combining all these equations, we have

P{N(t3 ) ¼ k3 j N(t2 ) ¼ k2 } ¼

P{N(t3 ) ¼ k3 , N(t2 ) ¼ k2 } P{N(t2 ) ¼ k2 }

½lt2 k2 ½l(t3 t2 )k3 k2 lt3 e ½l(t3 t2 )k3 k2 l(t3 t2 ) k2 !(k3 k2 )! e ¼ ¼ k2 (k3 k2 )! ½lt2 lt2 e k2 ! P{N(t3 ) ¼ k3 j N(t2 ) ¼ k2 , N(t1 ) ¼ k1 } ¼

P{N(t3 ) ¼ k3 , N(t2 ) ¼ k2 , N(t1 ) ¼ k1 } P{N(t2 ) ¼ k2 , N(t1 ) ¼ k1 }

½lt1 )k1 ½l(t2 t1 )k2 k1 ½l(t3 t2 )k3 k2 lt3 e k1 !(k2 k1 )!(k3 k2 )! ¼ ½lt1 )k1 ½l(t2 t1 )k2 k1 lt2 e k1 !(k2 k1 )! ¼

½l(t3 t2 )k3 k2 l(t3 t2 ) e (k3 k2 )!

Hence P{N(t3 ) ¼ k3 j N(t2 ) ¼ k2 , N(t1 ) ¼ k1 } ¼ P{N(t3 ) ¼ k3 j N(t2 ) ¼ k2 }, showing that N(t) is indeed a Markov process. Since the conditional density

P{N(t2 ) ¼ k2 jN(t1 ) ¼ k1 } ¼

½l(t2 t1 )k2 k1 l(t2 t1 ) e , (k2 k1 )!

k2 k1

is a function of the time difference (t2 2 t1), we conclude further that the Poisson process is also homogeneous Markov process.

Example 20.8.2 (Wiener Process) process as given by Eq. (20.7.4) is

The n-dimensional density function of a Wiener

( ) 1 1 (wk wk1 )2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp fW (w1 , w2 , . . . , wn ; t1 , t2 , . . . , tn ) ¼ 2 2 s2 (tk tk1 ) k¼1 2p(tk tk1 )s n Y

with t0 ¼ 0 ¼ w0 and the sequence

{Wt1 , (Wt2 Wt1 ), . . . , (Wtn Wtn1 )}

534

Classiﬁcation of Random Processes

is a collection of independent random variables for increasing f0 , t1 , t2 , . . . , tng. We have to show that fW (wn j wn1 , . . . , w1 ) ¼ fW (wn j wn1 ) or fW (wn , wn1 , . . . , w1 ) fW (wn , wn1 ) ¼ fW (wn1 , . . . , w1 ) fW (wn1 )

1 1 (wk wk1 )2 ﬃ exp k¼1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 s2 (tk tk1 ) fW (wn , wn1 , . . . , w1 ) 2p(tk tk1 )s2 ¼

fW (wn1 , . . . , w1 ) Qn1 1 1 (wk wk1 )2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ p exp k¼1 2 s2 (tk tk1 ) 2p(tk tk1 )s2

1 1 (wn wn1 )2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp 2 s2 (tn tn1 ) 2p(tn tn1 )s2 Qn

1 1 (wn wn1 )2 exp 2p(tn tn1 )(tn1 tn2 )s4 2 s2 (tn tn1 )

1 (wn1 wn2 )2 exp fW (wn , wn1 ) 2 s2 (tn1 tn2 ) ¼

fW (wn1 ) 1 1 (wn1 wn2 )2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp 2 s2 (tn1 tn2 ) 2p(tn1 tn2 )s2

1 1 (wn wn1 )2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp 2 s2 (tn tn1 ) 2p(tn tn1 )s2 which shows that W(t) is a homogeneous Markov process with stationary independent increments. Example 20.8.3 (AR Process) given by

In Example 20.6.1 an autoregressive(1) process was

Xn ¼ fXn1 þ Vn ,

n ¼ 1, . . . , X0 ¼ V0

where fVn, n ¼ 0, 1. . .g is a Gaussian process with zero mean and variance s2V. The conditional density fX (xn j Xn21 ¼ xn21) is also Gaussian-distributed with mean fxn21 and variance s2V. This is a Markov process since the conditional density function depends only on the Xn21 and not any of the previous values. Thus fX (xn j Xn1 ¼ xn1 , . . . , X0 ¼ x0 ) ¼ fX (xn j Xn1 ¼ xn1 ) Example 20.8.4 (Polya Urn Model) Initially at n ¼ 0, an urn contains b blue balls and r red balls. In each trial, a ball is drawn at random and its color recorded. It is then replaced with c balls of the same color. Let Xn be the random variable representing the color of the ball at the nth draw. We will also assume that Xk ¼ 1 if the ball selected at the kth trial is

20.8

Markov Process

535

blue and Xk ¼ 0 if it is red. The probability of X1 ¼ 1 is P{X1 ¼ 1} ¼

b bþr

The probability of X2 ¼ 1 conditioned on X1 ¼ 1 is P{X2 ¼ 1 j X1 ¼ 1} ¼

bþc bþcþr

Hence, the joint probability PfX2 ¼ 1, X1 ¼ 1g P{X2 ¼ 1, X1 ¼ 1} ¼

b bþc bþr bþcþr

The probability of X3 ¼ 1 conditioned on X2 ¼ 1 and X1 ¼ 1 is b þ 2c b þ 2c þ r

P{X3 ¼ 1 j X2 ¼ 1, X1 ¼ 1} ¼ and the joint probability PfX3 ¼ 1, X2 ¼ 1, X1 ¼ 1g is P{X3 ¼ 1, X2 ¼ 1, X1 ¼ 1} ¼

b þ 2c bþc b b þ 2c þ r b þ c þ r b þ r

The realization fX6 ¼ 0, X5 ¼ 1, X4 ¼ 0, X3 ¼ 0, X2 ¼ 1, X1 ¼ 1g can be interpreted as the event of drawing 3 blue balls in 6 trials, and the probability of this event is p(3, 6). Thus P{X6 ¼ 0, X5 ¼ 1, X4 ¼ 0, X3 ¼ 0, X2 ¼ 1, X1 ¼ 1} ¼ p(3, 6) ¼

r þ 2c b þ 2c rþc r bþc b b þ 5c þ r b þ 4c þ r b þ 3c þ r b þ 2c þ r b þ c þ r b þ r

Extending these results, the probability of a sequence of k blue balls and (n 2 k) red balls deﬁned by p(k, n) can be determined as follows: p(k, n) ¼ P{Xn ¼ xn , Xn1 ¼ xn1 , . . . , X1 ¼ x1 } ¼

b(b þ c)(b þ 2c) ½b þ (k 1)cr(r þ c)(r þ 2c) ½r þ (n k 1)c (b þ r)(b þ c þ r)(b þ 2c þ r) ½b þ r þ (n 1)c

This random sequence is an example of a discrete-state discrete-time (DSDT) random process. It is nonstationary because the probability is dependent on time, n. From the discussion above, we have P{X3 ¼ 1jX2 ¼ 1} ¼

bþc bþcþr

but P{X3 ¼ 1jX2 ¼ 1, X1 ¼ 1} ¼

b þ 2c b þ 2c þ r

Since P{X3 ¼ 1jX2 ¼ 1, X1 ¼ 1} = P{X3 ¼ 1jX2 ¼ 1}, the sequence fXng is not a Markov process.

536

Classiﬁcation of Random Processes

B 20.9

MARKOV CHAINS

When the state of a Markov process is discrete, we call it a Markov chain (MC). If the time is also discrete, it is called a discrete-time Markov chain. The binomial process (Example 20.4.2) and the random-walk process (Example 20.4.3) are illustrations of a discrete Markov chain. If the time is continuous, it is called a continuous-time Markov chain. The Poisson process discussed in detail in Section 20.2 is an example of a continuoustime Markov chain. These processes have been extensively studied, and many physical phenomena are modeled by them. We will now discuss discrete Markov chains. For continuous Markov chains, see Refs. 13, 30, and 54.

Discrete Markov Chain

A random process fXn, n ¼ 0, 1, . . .g with discrete state space S is a Markov chain if the following probability relation holds for each f j, in, . . . , i0g [ S: P{Xnþm ¼ j j Xn ¼ in , . . . , X0 ¼ i0 } ¼ P{Xnþm ¼ j j Xn ¼ in } ¼ pi(m) (n) nj

(20:9:1)

The quantity P{Xnþm ¼ j j Xn ¼ in } ¼ pi(m) (n) is the conditional probability of transition nj from state in at time n to state j at time n þ m. It is referred to as the m-step transition probability, which in general will be a function of time n. However, if the transition from state i to state j is dependent only on the time difference m ¼ n þ m 2 n and not on the current time n, then the transition probability is stationary and the Markov chain is homogeneous, or P{Xnþm ¼ j j Xn ¼ i} ¼ P{Xm ¼ j j X0 ¼ i} ¼ pij(m) The one-step transition probability of a homogeneous Markov chain deﬁned as pij ¼ P{Xnþ1 ¼ j j Xn ¼ i} ¼ P{X1 ¼ j j X0 ¼ i}

(20:9:2) p(1) ij

¼ pij can be (20:9:3)

If the number of states N is ﬁnite, the Markov chain is ﬁnite; otherwise it is inﬁnite. The one-step probabilities fpijg can be expressed as either a transition matrix P or a state transition ﬂow diagram. To make these ideas clear, we will consider the modiﬁed Bernoulli process given in Eq. (20.4.11), where the random variable Xn at time n is in state 1 with probability p(n) 0 or in state 21 with probability p(n) 1 as given below: P(Xn ¼ 1) ¼ p(n) 0

(20:9:4)

P(Xn ¼ 1) ¼ p(n) 1

Further, if the particle is in state 1 at time n, then the probability of transition to 21 is a at time (n þ 1), and if it is in state 21 at n, then the probability of transition to 1 is b at the time (n þ 1). We are considering a two-state homogeneous Markov chain. The transition matrix P is shown in Eq. (20.9.5): 1 P¼

1

1 1a a 1 b 1b

The state transition ﬂow diagram is shown in Fig. 20.9.1.

(20:9:5)

20.9

Markov Chains

537

FIGURE 20.9.1

If there are Nþ1 states and the transition probability from state i to state j is pij, then the one-step transition matrix P is given by 2

0 p00

8 0 > > > 6 p10 > > 16 > 6 . > > 6 < 6 .. 6 P¼ 6p > i > 6 i0 > > 6 . > > 6 . > > : 4 . N pN0

1 p01

j p0j

p11 .. . pi1 .. .

p1j .. . pij .. .

pN1

pNj

N 3 p0N p1N 7 7 .. 7 7 . 7 7 piN 7 7 .. 7 7 . 5

(20:9:6)

pNN

The matrix P is called a stochastic matrix and has the property that the row sum equals 1: N X pij ¼ 1, i ¼ 0, 1, . . . , N (20:9:7) j¼0

The m-step transition probability for all i, j deﬁned in Eq. (20.9.2) can also be given in matrix form as follows:

P

(m)

0 8 2 (m) 0 p00 > > > > (m) > 16 > 6 p10 > > 6 > 6 > > < 6 ... 6 ¼ 6 (m) > 6 pi0 > > 6 i > > 6 . > > 6 . > > 4 . > > : (m) N pN0

1 (m) p01 (m) p11

j

(m) p0j (m) p1j

.. .

.. .

(m) pi1

pij(m)

.. .

.. . (m) pN1

(m) pNj

N

(m) p0N (m) p1N

3

7 7 7 .. 7 . 7 7 (m) 7 piN 7 7 .. 7 7 . 5 (m) pNN

(20:9:8)

Let us consider the two-step transition from state i at time 0 through an intermediate state k at time m 2 1 to state j at time m. From the property of conditional expectations we can write X P{Xm ¼ j, Xm1 ¼ k, X0 ¼ i} P{Xm ¼ j, X0 ¼ i} ¼ k

¼

X

P{Xm ¼ j j Xm1 ¼ k, X0 ¼ i}

k

P{Xm1 ¼ k j X0 ¼ i}Pf X0 ¼ i}

(20:9:9)

Dividing Eq. (20.9.9) throughout by PfX0 ¼ ig and using the Markov property, P{Xm ¼ j j Xm1 ¼ k, X0 ¼ i} ¼ P{Xm ¼ j j Xm1 ¼ k}, we obtain the following equation

538

Classiﬁcation of Random Processes

for all i, j: P{Xm ¼ j j X0 ¼ i} ¼ pij(m) ¼

X

P{Xm ¼ j j Xm1 ¼ k}P{Xm1 ¼ k j X0 ¼ i}

k

¼

X

(m1) pkj pik

(20:9:10)

k

This equation is a restatement of the Chapman –Kolmogorov equation [Eq. (20.8.12)]. . Thus, in matrix Similar equations can be written for pij(m1) in terms of pkj and p(m2) ik form P(m) ¼ PP(m1)

P(m) ¼ Pm

or

Eq. (20.9.11) is an important result that shows the m-step transition matrix P mth power of the one-step transition matrix P.

(20:9:11) (m)

as the

State Occupancy Probabilities (n) (n) Let the row vector p(n) ¼ ½p(n) represent the state occupancy 0 , p0 , . . . , p N (0) (0) (0) probabilities at time n and the row vector p ¼ ½p(0) 0 , p1 , . . . , pN represent the initial probability of the states at time 0. Since p is the state transition probability matrix, we can obtain the state occupancy probabilities p (n) of any state at time n from a knowledge of the probability of the state at time n21 from the following compact matrix relation:

p(n) ¼ p(n1) P

(20:9:12)

This equation can be solved recursively, and the probabilities of the states at time n can be related to the initial probabilities of the states at time 0 as given below: p(n) ¼ p(n1) P ¼ p(n2) P2 ¼ ¼ p(0) Pn

(20:9:13)

The next logical question that may arise is whether after a sufﬁciently long time a statistical equilibrium can be established in the sense that the occupancy probabilities of the states are independent of the initial probabilities. To answer that question, we set limn!0 p(n) ¼ p in Eq. (20.9.12) and obtain lim p(n) ¼ lim p(n1) P

n!0

n!0

or

p ¼ pP

or p(I P) ¼ 0

(20:9:14)

The matrix equation p(I P) is a homogeneous system of linear equations and has a solution only if the determinant of (I 2 P) is equal to 0. This is very similar to the eigenvector [Eq. (16.2.2)]. Solution of Eq. (20.9.14) will give the row vector p only P upto a mulitiplicative constant, and it can be made unique by the normalization equation Ni¼0 pi ¼ 1. By utilizing the techniques of Chapter 16, we can obtain the solution for a ﬁnite Markov chain by diagonalizing the matrix P. The eigenvalues of the matrix P are obtained by forming the characteristic equation jlI 2 Pj ¼ lN þ a1lN21 þ . . . þ aN ¼ 0 and solving for the N eigenvalues. If we substitute l ¼ 1 in the eigenequation lI 2 P ¼ 0, we obtain I 2 P ¼ 0. Since the row sum of the stochastic matrix is 1 and all the terms are positive, we conclude that l ¼ 1 is always an eigenvalue of the stochastic matrix P and all the other eigenvalues will be less than 1. We can diagonalize the matrix P assuming that the eigenvalues are distinct and write M1 PM ¼ L

or

P ¼ MLM1

(20:9:15)

20.9

Markov Chains

539

where M is the modal matrix of eigenvectors and L is a diagonal matrix of distinct eigenvalues of P. From Eq. (20.9.15) we can write the limn!1 Pn as follows: P ¼ lim Pn ¼ lim ½MLM1 n ¼ M lim Ln M1 n!1

n!1

(20:9:16)

n!1

As n ! 1, since all except one of the eigenvalues in L are less than 1, they all tend to 0 as shown below: 2 3 2 3 1 0 0 1 0 0 6 1 l2 0 7 6 0 0 0 7 6 7 6 7 lim Ln ¼ lim 6 . . ¼6. . (20:9:17) .. 7 .. 7 n!1 n!14 . .5 . 5 4 .. .. . .. 0

0

0

lN

0

0

It can be shown that for a stochastic matrix with distinct thepeigenvector corpﬃﬃﬃﬃ eigenvalues, pﬃﬃﬃﬃ ﬃﬃﬃﬃ responding to the eigenvalue 1 is given by m ¼ ½1= N , 1= N , . . . , 1= N T . Substituting Eq. (20.9.17) in Eq. (20.9.16) and expanding, we obtain P ¼ lim Pn ¼ P1 n!1

2 1 pﬃﬃﬃﬃ m01 6 N 6 6 1 6 pﬃﬃﬃﬃ m11 6 ¼6 N 6 . .. 6 . . 6 . 4 1 pﬃﬃﬃﬃ mN1 N

m0N

3

72 7 1 0 7 6 m1N 7 76 0 0 76 4 .. 7 7 . 7 0 0 5 mNN

0

32 m 0 00 0 6 m0 7 0 76 10 76 . 56 4 .. 0

m0N0

m001 m011 .. . m0N1

3 m00N m01N 7 7 .. 7 7 . 5 m0NN

(20:9:18) 21

0

where the primes on m denote the elements of the inverse matrix M ¼ M of the modal matrix M. Equation (20.9.18) can be simpliﬁed as 2 2 0 3 3 1 m00 m001 m00N pﬃﬃﬃﬃ p ﬃﬃﬃﬃ p ﬃﬃﬃﬃ p ﬃﬃﬃﬃ 6 N7 6 N N N7 6 6 7 7 6 1 7 6 m0 0 0 7 m01 m0N 7 6 pﬃﬃﬃﬃ 7 6 p00 6 N 7 0 6 ﬃﬃﬃﬃ pﬃﬃﬃﬃ pﬃﬃﬃﬃ 7 0 0 6 7 N N7 P¼6 (20:9:19a) 6 . 7 m00 m01 m0N 6 N 7 6 .. 6 . 7 .. 7 .. 6 . 6 . 7 . 7 . 6 6 0 7 7 4 1 5 4 m00 m001 m00N 5 pﬃﬃﬃﬃ pﬃﬃﬃﬃ pﬃﬃﬃﬃ pﬃﬃﬃﬃ N N N N or

2

p0 6 p0 6 P¼6 . 4 .. p0

p1 p1 .. . p1

3 pN pN 7 7 .. 7 . 5 pN

(20:9:19b)

is the steady-state probability transition matrix. Note that in the P matrix each column has the same elements. Substitution of Eq. (20.9.19b) in Eq. (20.9.14) results in lim p(n) ¼ p(0) P1 ¼ p(0) P ¼ ½ p0

n!1

p1

pN ¼ p

(20:9:20)

This equation shows that the limiting state occupancy probabilities are independent of the initial state probability row vector p(0).

540

Classiﬁcation of Random Processes

Example 20.9.1 A two-state Markov chain with probability transition matrix P as in Eq. (20.9.5) is shown below along with the initial probability row vector p(0): 1a a ¼ ½ p 1 p P¼ ; p(0) ¼ p(0) p(0) 0 1 b 1b We want to ﬁnd the equilibrium probability distributions of the occupancy of states 1, 21 (0) and show that they are independent of the initial occupancy p(0) 0 and p1 . The eigenvalues of P are obtained by solving the characteristic equation given by l 1 þ a a ¼ (l 1)(l 1 a þ b) ¼ 0 jlI Pj ¼ b l 1 þ b and the eigenvalues l1 ¼ 1 and l2 ¼ 1 2 a 2 b. The unnormalized eigenvector c1 corresponding to l1 ¼ 1 is obtained from the eigenequation a a c21 c11 1 ¼ 0 or ¼ c22 c12 b b 1 and the unnormalized eigenvector c2 corresponding to l2 ¼ 1 2 a 2 b is obtained from

b b

a a

c21 c22

¼0

or

c21 c22

"

¼

1 b a

#

The matrices of unnormalized eigenvectors MU and its inverse M21 U are 2 3 b a " # 6bþa bþa7 1 1 6 7 b MU ¼ ¼6 M1 7 U 1 4 a a 5 a bþa bþa The diagonal eigenvalue matrix is L ¼ n

(n)

lim P(n)

0

0 1ab 1 0 0 0 .

, and since 1 2 a 2 b , 1, the limit as

Hence, as n ! 1, P n is given by ¼ 2 3 2 3 b a b a " # 6 7 6 7 1 1 1 0 6 bþa bþa 7 6 bþa bþa 7 b ¼ 6 7¼6 7 0 0 4 a 1 a 5 4 a a 5 a bþa bþa bþa bþa

n ! 1 of L is given by limn!1 L

n!1

1

and 2

(1)

p¼p

(0) 1

¼p P ¼½p

b 6b þ a 6 1 p 6 4 b bþa

a 3 b þ a7 b 7 7¼ bþa a 5 bþa

which is independent of the initial probability vector p (0) ¼ [ p

a bþa

1 2 p].

Example 20.9.2 A book representative visits three universities a, b, and c each week. However, there is a ﬁnite probability that she may stay over for another week in any of the three places. The transition probability matrix P and the initial probability row

20.9

Markov Chains

541

FIGURE 20.9.2

vector p (0) of being in states a, b, and c at time 0 are shown below: 2 a a 0:2 6 P ¼ b 4 0:1

b 0:15 0:4

c 3 0:65 7 0:5 5

c

0:25

0:05

0:7

3T 0:3 7 6 7 6 ¼ 4 p(0) b 5 ¼ 4 0:45 5 0:25 p(0) c 2

p(0)

p(0) a

3T

2

We have to ﬁnd (1) the probabilities of the representative being in the universities a, b, and c after 3 weeks and (2) the steady-state probability of being in any of the universities. The state transition ﬂow diagram for this three-state Markov chain is shown in Fig. 20.9.2. 1. We can apply Eq. (20.9.14) to obtain the probabilities that she will be in the various universities after 3 weeks. Thus 2

p(3) a

3T

2

0:3

3T 2

0:2

0:15

0:65

33

6 7 6 7 6 7 (0) 3 0:4 0:5 5 p(3) ¼ 4 p(3) b 5 ¼ p P ¼ 4 0:45 5 4 0:1 0:25 0:7 0:25 0:05 p(3) c 3 2 3T 2 3 2 0:347 0:3 0:293 0:237 0:47 7 6 7 6 7 6 ¼ 4 0:45 5 4 0:315 0:254 0:431 5 ¼ 4 0:251 5 0:25 2. The eigenvalues of the matrix P are The modal matrix M is: 2 0:57735 M ¼ 4 0:57735 0:57735

0:47

0:262

0:268

0:402

l1 ¼ 1, l2 ¼ 0.23197, l3 ¼ 20.58197. 0:37197 0:89737 0:22574

3 0:56601 0:32757 5 0:75652

and P1 ¼ML1 M1 2 32 32 3 0:57735 0:37197 0:56601 100 0:63437 0:43479 0:66288 6 76 76 7 ¼ 4 0:57735 0:89737 0:32757 54 0 0 0 54 0:65635 0:80072 0:14437 5 0:57735 0:22574 0:75652 000 0:67997 0:0929 0:77287 2 3 0:36626 0:25103 0:38272 6 7 ¼ 4 0:36626 0:25103 0:38272 5 0:36626 0:25103 0:38272

542

Classiﬁcation of Random Processes

FIGURE 20.9.3

and the steady-state probability p ¼ p(1) ¼ p(0)P ¼ [0.36626 0.25103 0.38272]. The evolution of state occupancy probabilities p(n) for all three states are shown in Fig. 20.9.3 for n ¼ 1, 2, ... ,12. From the ﬁgure it is seen that steady-state equilibrium has been obtained in about 9 weeks. Example 20.9.3 A new automobile can be classiﬁed into ﬁve states: 1 ¼ excellent, 2 ¼ good, 3 ¼ fair, 4 ¼ poor, 5 ¼ scrap. The transition probabilities of going from one state to another in each year is given by the transition probability matrix P. Clearly, the initial probability row vector p will be given by p ¼ ½ 1 0 0 0 0 . At the end of the ﬁrst year the probabilities of the automobile being in excellent condition (state 1) ¼ 0.6, in good condition (state 2) ¼ 0.25, in fair condition (state 3) ¼ 0.1 and in poor condition (state 4) ¼ 0.05. Similarly, if the automobile is in scrap condition (state 5), the probabilities of it continuing to be in scrap condition ¼ 0.85, in poor condition ¼ 0.1, and in fair condition ¼ 0.05. We will ﬁnd the probabilities of the condition of the car in (1) 5 years and (2) steady-state equilibrium for the state occupancy probabilities: 2 2 1 1 0:6 0:25 26 6 0:2 0:45 P ¼ 36 6 0:05 0:15 44 0 0:05 5 0 0

3 4 5 3 0:1 0:05 0 0:2 0:1 0:05 7 7 0:4 0:25 0:15 7 7 0:15 0:35 0:45 5 0:05 0:1 0:85

The state transition ﬂow diagram is shown in Fig. 20.9.4.

FIGURE 20.9.4

20.9

Markov Chains

543

1. From Eq. (20.9.13) the state probabilities after n ¼ 5 years is given by p(5) ¼ p(0) P5 , where p(0) is the row vector p(0) ¼ ½ 1 0 0 0 0 . Hence

p(5)

2 3T 2 1 0:2034 6 0 7 6 0:1506 6 7 6 6 7 6 7 6 ¼ p(0) P5 ¼ 6 6 0 7 6 0:0863 6 7 6 4 0 5 4 0:0419 0 0:0204 2 3T 0:2034 6 7 6 0:1984 7 6 7 7 ¼6 6 0:1736 7 6 7 4 0:1502 5 0:2744

0:1984 0:1618

0:1736 0:1502 0:1641 0:1571

0:1090

0:1447 0:1615

0:0656 0:0415

0:1215 0:1572 0:1061 0:1518

3 0:2744 0:3664 7 7 7 0:4985 7 7 7 0:6138 5 0:6802

Thus, after 5 years there is a probability of 0.2034 that the car is still in excellent condition and a probability of 0.2744 that the car is in scrap condition. It is also interesting that with the given transition probabilities, there is a better chance that the car is in excellent condition than in either good, fair, or poor condition. We will now check what happens to these probabilities after a long period of time has elapsed. 2. We follow the procedure of ﬁnding the eigenvalues and eigenvectors of the transition matrix P. The eigenvalue matrix L, the corresponding modal matrix of P, and its inverse are as follows: 2

1 60 6 6 L¼6 60 6 40 0

0 0:4529

0 0

0 0

0 0

0

0:2417

0

0

0 0

0 0

2

0:4472 0:7139 6 0:4472 0:1188 6 6 M¼6 6 0:4472 0:6081 6 4 0:4472 0:2899 0:4472 2 M1

0:1114

6 6 0:4614 6 ¼6 6 0:3828 6 4 0:5697 0:0048

0:1496

0:1555

0:7999 0 0 0:1555

3 7 7 7 7 7 7 5

0:4491 0:8065

0:7891 0:5471

0:2789 0:2566

0:2238 0:0293

0:0651

0:1647

0:2695

0:1919 0:5895 0:7069 0:0846

0:3446 0:5495 0:7773

0:5102

0:2363

0:0127

0:1395

0:6005

0:9562

2

1 60 6 6 1 L ¼6 60 6 40 0 3 0:1265 0:4190 7 7 7 0:7403 7 7 7 0:5099 5 0:0201

1:3551

3

7 7 7 7 7 7 1:3288 5 0:4999 0:8695 0:5378

0 0 0 0

0 0

0 0

0

0 0 0 0

0 0

3 0 07 7 7 07 7 7 05 0

544

Classiﬁcation of Random Processes

Hence from Eq. (20.9.18) we have 2

P ¼ P1 ¼ ML1 M1

0:0499 6 0:0499 6 ¼6 6 0:0499 4 0:0499 0:0499

0:0695 0:0695 0:0695 0:0695 0:0695

0:1205 0:1205 0:1205 0:1205 0:1205

0:1541 0:1541 0:1541 0:1541 0:1541

3 0:6060 0:6060 7 7 0:6060 7 7 0:6060 5 0:6060

and the steady-state probabilities row vector p is p ¼ p(0) P ¼ ½ 0:0499 0:0695

0:1205 0:1541

0:6060

These are more reasonable numbers since the long-term probabilities of being in various states are excellent 4.99%, good 6.95%, fair 12.05%, poor 15.41%, and ﬁnally scrap 60.6%. The conclusion is that there is 60.6% chance that the automobile will be a scrap in the long run with the given probability transition matrix. First-Passage Probabilities and Classiﬁcation of States

A Markov chain is called irreducible if every state can be reached from every other state in a ﬁnite number of steps. In this case, the probability of going from every state i to every other state j in the state space S of the chain will be given by the probability, pij(m) . 0, where m is a ﬁnite integer. In this case, the states are called intercommunicating. On the other hand, if E is a proper subset of S, E , S, and if i and j are states in E, then the subset E is closed if X

pij ¼ 1

for all

i[E

j[E

In other words, no transition is possible from a state i [ E to a state k [ E c. In this case the Markov chain is called reducible. If there are no other proper subsets of E that are closed, then it is called an irreducible sub-Markov chain. If i [ E is the only state, then it is called an absorbing state, in which case pii ¼ 1. Let us consider that the Markov chain is initially in state i at time n. We deﬁne the probability of ﬁrst return to i at time n þ m as fii(m), or fii(m) ¼ P{Xnþ1 = i, . . . , Xnþm1 = i, Xnþm ¼ i j Xn ¼ i}

(20:9:21)

with one-step transition fii(1) ¼ P{Xnþ1 ¼ i j Xn ¼ i} ¼ p(1) ii ¼ pii similar to Eq. (20.9.2). For a homogeneous Markov chain, Eq. (29.9.21) can also be written as fii(m) ¼ P{X1 = i, . . . , Xm1 = i, Xm ¼ i j X0 ¼ i}

(20:9:22)

with one-step transition fii(1) ¼ P{X1 ¼ i j X0 ¼ i} ¼ p(1) ii ¼ pii . In a similar manner we can also deﬁne the probability of ﬁrst passage from state i at time n to state j at time n þ m as fij(m) , or fij(m) ¼ P{Xnþ1 = j, . . . , Xnþm1 = j, Xnþm ¼ j j Xn ¼ i}

(29:9:23)

20.9

Markov Chains

545

with one-step transition fij(1) ¼ P{Xnþ1 ¼ j j Xn ¼ i} ¼ p(1) ij ¼ pij in conformity with Eq. (20.9.2). For a homogeneous Markov chain, Eq. (20.9.23) is equivalent to fij(m) ¼ P{X1 = j, . . . , Xm1 = j, Xm ¼ j j X0 ¼ i}

(20:9:24)

with the corresponding one-step probability fij(1) ¼ P{X1 ¼ j j X0 ¼ i} ¼ p(1) ij ¼ pij . The probabilities of eventual ﬁrst return to state i or the ﬁrst passage to state j starting from state i are given by

fi ¼

1 X

fii(m)

fij ¼

m¼1

1 X

fij(m)

(20:9:25)

m¼1

It is now possible to have more classiﬁcations of the Markov chains depending on the nature of fi. If fi ¼ 1, the state i is called recurrent; if fi , 1, the state i is called transient. If the chain starts at time 0 in state i and returns to state i only at times a, 2a, . . . , with period a . 1, then the state i is called periodic with period a. If a ¼ 1, then i is aperiodic. If fi ¼ 1 and fij ¼ 1, we can deﬁne a mean recurrence time and a mean ﬁrst-passage time from state i to state j as follows:

mi ¼

1 X

mfi

mij ¼

m¼1

1 X

mfij

(20:9:26)

m¼1

With this knowledge we can classify the states even further. If the mean recurrence time mi is ﬁnite, then the state is called positive recurrent; if it is inﬁnite, it is called null recurrent. Finally, a state i is ergodic if it is aperiodic and positive recurrent, or a ¼ 1, fi ¼ 1 and mi , 1. A ﬁnite-state aperiodic irreducible Markov chain is always ergodic, and the occupancy probability of the states as n ! 1 will tend to an equilibrium distribution independent of the initial probabilities as in Examples 20.9.1 – 20.9.3. In transient states the equilibrium distribution will be 0.

Example 20.9.4

The transition matrix P of a four-state Markov chain is given below: 1 8 2 1 1 > > > 3 > 4 0 > : 4 0:3

2 0 0:4

3 0 0:6

0:8 0

0:2 0:2

4 3 0 0 7 7 7 0 5 0:5

and the state transition ﬂow diagram is shown in Fig. 20.9.5. We can obtain the classiﬁcation of the states by inspection of the transition ﬂow diagram of Fig. 20.9.5. From the ﬂow diagram state 1 is absorbing because p11 ¼ 1, and once the transition occurs to state 1, it never gets out of it. It is also a recurrent state. States 2 and 3 are recurring states because the transition occurs only between these two states. If the transition occurs out of state 4, it can never get back in. Hence, state 4 is a transient state. We will determine the equilibrium probabilities P from

546

Classiﬁcation of Random Processes

FIGURE 20.9.5

Eq. (20.9.18). The diagonal eigenvalue matrix L is given by 2

0:5 60 L¼6 40 0

0 1 0 0

0 0 0:4 0

3 0 07 7 05 1

2

0 60 n lim L ¼ 6 40 n!1 0

0 1 0 0

3 0 07 7 05 1

0 0 0 0

The presence of two 1s for eigenvalues indicates the presence of sub-Markov chains. The modal matrix and its inverse are 2

0

0

0

0:8575

3 7 7 7 5

60 6 M¼6 40

0:6804 0:5907 0:6804 0:7877

0 0

1

0:2722 0:1750

0:5145

2

0:6

6 0 6 M1 ¼ 6 4 0 1:1622

0:3556

0:0444

0:8398

0:6299

0:7255 0

0:7255 0

1

3

07 7 7 05 0

and the equilibrium probability P is given by

P ¼ M lim Ln M1 n!1

1 2 8 2 0 1 1 > > > 6 < 2 6 0 0:5714 ¼ 6 > > 3 4 0 0:5714 > : 4 0:6 0:2286

3

4

0

0

3

0:4286 0 7 7 7 0:4286 0 5 0:1714 0

State 4 is transient since p4j ¼ 0, j ¼ 1, . . . , 4. State 1 is absorbing since p11 ¼ 1. States 2 and 3 are recurrent with p23 ¼ 0.4286 and p32 ¼ 0.5714.

20.9

Markov Chains

547

Example 20.9.5 We will slightly change the transition probabilities by introducing a path between states 2 and 1 by inserting p21 ¼ 0.02 and modifying p23 ¼ 0.58: 1 2 8 2 1 0 1 > > > 6 < 2 6 0:02 0:4 P¼ 6 > 0:8 340 > > : 0 4 0:3

3 0

4 3 0 0:58 0 7 7 7 0:2 0 5 0:2 0:5

The state transition ﬂow diagram is shown in Fig. 20.9.6. This small change makes a big difference in the properties of the Markov chain. From the ﬂow diagram, the only state that is recurrent is 1. The other states (2 –4) are transient because everything ﬂows into state 1 directly or indirectly. This will be conﬁrmed from the equilibrium probabilities P of the states. The diagonal eigenvalue matrix L and the limn!1 are given by 2

0:5 60 L¼6 40 0

0 0:9885 0 0

0 0 0:3885 0

2

3 0 07 7 05 1

0 60 n lim L ¼ 6 40 n!1 0

0 0 0 0

0 0 0 0

3 0 07 7 05 1

Since there is only one eigenvalue of 1, there are no sub-Markov chains. The modal matrix and its inverse are 2

0 60 M¼6 40 1

3 0 0 0:5 0:6739 0:5830 0:5 7 7 0:6837 0:7926 0:5 5 0:28 0:1784 0:5

2

3 0:5853 0:3687 0:0461 1 6 1:4748 0:8497 0:6251 0 7 7 M1 ¼ 6 4 0:0106 0:7330 0:7225 0 5 2 0 0 0

FIGURE 20.9.6

548

Classiﬁcation of Random Processes

and the equilibrium probability P is given by 1 2 1 1 26 61 P ¼ M lim Ln M1 ¼ 6 n!1 341

2 3 4 3 0 0 0 0 0 07 7 7 0 0 05

4 1 0 0 0 The equilibrium probability matrix shows that states 2 –4 are transient because the probabilities are zero. State 1 is the absorbing recurrent state. Example 20.9.6

A state transition matrix P for a seven-state Markov chain is given by

1 8 2 1 > > > 6 0:6 > > 2 6 0:7 > > > 6 > > 6 > 6 > > 6 > >560 > > 6 > > 6 4 0:2 > > : 7 0

2

3

4

5

6

7

0:4

0

0

0

0

0

0:3 0

0 0:7

0 0 0:3 0

0 0

0 0

0 0 0 0

0:4 0:3 0:5 0:4 0:1 0:1 0:3 0:2

0:3 0:1 0:1 0:4

3

7 7 7 7 7 7 0 0 7 7 0 0 7 7 0:2 0:3 7 5 0:1

The corresponding state transition ﬂow diagram is shown in Fig. 20.9.7.

FIGURE 20.9.7

20.9

Markov Chains

549

We will classify the states by inspection from Fig. 20.9.7. The ﬂow diagram shows that states 1 and 2 form a sub-Markov chain since there are no paths out of that pair. Hence they form recurring states. States 3– 5 form another sub-Markov chain, and there are no paths leading out of these three states. Hence they form another set of recurring states. All paths lead out of states 6 and 7 and hence are transient states. We can conﬁrm these ﬁndings by computing the equilibrium probability P. The diagonal eigenvalue matrix L and the limn!1 Ln are given by 2

0:2

60 6 6 60 6 6 L ¼ 60 6 60 6 6 40 0

0

0

0

0

0

0:1 0

0 1

0 0

0 0

0 0

0 0

0 0

0:2303 0 0 0:1303

0 0

0

0

0

0

1

0

0

0 2

0

0

0 60 6 6 60 6 6 n lim L ¼ 6 0 n!1 6 60 6 6 40

0

3

7 7 7 7 7 7 0 7 7 0 7 7 7 0 5 0:1 3 0 07 7 7 07 7 7 07 7 07 7 7 05 0 0

0

0

0

0 0

0 0

0 1

0 0

0 0 0 0

0

0

0

0 0

0 0

0 0

0 0

0 0 0 1

0 0

0

0

0 0

0

Since there are two eigenvalues of 1, we conclude that there is a sub-Markov chain. The modal matrix M and its inverse M 21 are given by 2

0

60 6 6 60 6 6 M ¼ 60 6 60 6 6 41 0 2

1

M

0

0

0

0

0:6963 0:4710

0

0

0

0

0:6963

0 0

0:4682 0:028 0:4682 0:0438

0:1665 0:4607

0 0

0 0:9487

0:4682 0:3511

0:0271 0:9946

0:4388 0:4821

0 0:1741

0:3162

0:4682

0:0861

0:5789

0:0833 6 0 6 6 6 0 6 6 ¼6 0 6 6 0 6 6 4 0:9139 0:7720

0

18:25 2:1082

8:75 4:2164

13:25 1:0541

0

1:2521

0:6629

0:2210

0 0

16:6314 0:3084

5:0356 1:0184

11:5958 1:3268

0:3333 0

0:5222 0:7720

0 0

0 0

0 0

3

0:8243 7 7 7 7 0 7 7 0 7 7 7 0 7 7 0:3140 5 0 3 1 3 0 3:1623 7 7 7 7 0 0 7 7 0 0 7 7 7 0 0 7 7 0 0 5 0 0

550

Classiﬁcation of Random Processes

The equilibrium probability P is given by

P ¼ M lim Ln M1 n!1

1 2 3 4 5 8 2 1 0:6364 0:3636 0 0 0 > > > 6 > > 2 6 0:6364 0:3636 0 0 0 > > > > 6 > 6 0 0:5862 0:3103 0:1035 > 6 > >560 0 0:5862 0:3103 0:1035 > > 6 > > 6 > > 6 4 0:1591 0:0909 0:4396 0:2328 0:0776 > > : 0 0:5862 0:3103 0:1035 7 0

6

7

0

0

3

07 7 7 07 7 7 0 07 7 0 07 7 7 0 05 0 0 0 0

The matrix P shows that there are two closed sets of recurring states f1,2g and f3,4,5g with equilibrium probabilities. States 6 and 7 have zero probabilities, thus conﬁrming that they are transient states. An interesting point to note is that there is no direct transition from state 6 to state 2 in the state transition matrix but there is a direct transition from state 6 to state 2 in the equilibrium probability distribution.

Properties of Markov Chains 1. Let X ¼ fXn: n ¼ 0, 1, . . . , Ng be a Markov chain with state space S ¼ fi, j, k, . . .g. The m-step transition probability p(m) ij (n) is the probability of Xnþ1 ¼ j given that Xn¼i, or P{Xnþm ¼ jjXn ¼ i} ¼ P{Xnþm ¼ jjXn ¼ i} ¼ pij(m) (n) The Markov chain is homogeneous when P{Xnþm ¼ jjXn ¼ i} ¼ P{Xm ¼ jjX0 ¼ i} ¼ p(m) ij The transition probabilities are independent of time n. 2. If P is the matrix of one-step transition probabilities, then pij is the (i, j)th element in P as shown below: 0 1 ... 8 2 p00 p01 > > >06 > > p p11 10 > > 16 > 6 . > .. > < 6 6 .. . 6 P¼ 6 > i 6 pi0 pi1 > > > 6 > .. > 6 .. > > 4 . . > > :N pN0 pN1

j

... N

p0j p1j .. .

pij .. . pNj

p0N p1N .. .

3

7 7 7 7 7 7 piN 7 7 .. 7 7 . 5 pNN

P is called the stochastic matrix or Markov matrix with row sum equal to 1, or N X pij ¼ 1, i ¼ 0, 1, . . . , N j¼0

3. The Markov chain satisﬁes the Chapman –Kolmogorov equation X X (m1) P{Xm ¼ jjXm1 ¼ k}P{Xm1 ¼ kjX0 ¼ i} ¼ pkj pik pij(m) ¼ k

k

20.10 Martingale Process

551

4. Let the row of the initial probability of the states of a Markov chain at time 0 vector (0) (0) . Then the row vector of state occupancy probabilities , p , . . . , p be p(0) ¼ p(0) N 0 1 (n) (n) at time n, p(n) ¼ p(n) , p , . . . , p N , is given by 0 1 p(n) ¼ p(n1) P ¼ p(0) Pn 5. A Markov chain with Markov matrix P is called irreducible if every state can be reached from every other state in a ﬁnite number of steps. The long-term statistical equilibrium occupancy probabilities p of the states can be obtained from solving p ¼ pP

or p(I P) ¼ 0

6. Only one eigenvalue of an irreducible Markov chain is 1, and the absolute values of all other eigenvalues are less than 1. 7. Starting from a state i at time n, the probability of ﬁrst return to i at time n þ m as fii(m) is deﬁned by fii(m) ¼ P{Xnþ1 = i, . . . , Xnþm1 = i, Xnþm ¼ ijXn ¼ i} and starting from a state i at time n the probability of ﬁrst passage to j at time n þ m as fij(m) is deﬁned by fij(m) ¼ P{X1 = j, . . . , Xm1 = j, Xm ¼ jjX0 ¼ i} The probabilities of eventual ﬁrst return to state i or ﬁrst passage to state j are fi ¼

1 X

fii(m)

m¼1

fij ¼

1 X

fij(m)

m¼1

If fi or fij is equal to 1, then the state i or j is certain, and both are called recurrent. If they are less than 1, then the states are called transient. 8. The mean recurrence time and the mean ﬁrst passage time can be deﬁned as mi ¼

1 X

mfi

m¼1

mij ¼

1 X

mfij

m¼1

If mi , 1, then the state i is positive recurrent, and if mi ¼ 1, then i is null recurrent and similar deﬁnitions apply to state j if mij , 1, or mij ¼ 1 9. In a Markov chain with transitionP matrix P, if E is a proper subset of the state space S, or if E , S, then E is closed if j[E pij ¼ 1 for all i [ E. The states in E form a sub-Markov chain in which case there will be two eigenvalues of 1. 10. A closed set of states E , S containing no proper subsets that are closed is called irreducible. If the number of states within an irreducible set is ﬁnite, then each state in E is recurrent. For any irreducible set, every state communicates with every other state. 11. If all the states of a Markov chain are irreducible, then a steady-state row vector deﬁned by p can be determined from p ¼ pP, or p(I P) ¼ 0.

B 20.10

MARTINGALE PROCESS

The meaning of a martingale in both British and American dictionaries refers to a strap preventing a horse from rearing. This has no relevance in the mathematical sense,

552

Classiﬁcation of Random Processes

where it conveys the sense of a fair game. It is a stochastic process where the best estimate of the future value conditioned on the past, including the present, is the present value. It is a process without trend and hence unpredictable. Martingale concepts are widely used in engineering and stochastic ﬁnance. Many ﬁltering problems in engineering such as the Kalman ﬁlter [27] can be cast in a martingale framework. The option pricing model developed by Black and Scholes [7] in 1973 for pricing stock options can be formulated within the framework of martingale theory. Just like a Markov process, conditional expectations play a key role in martingale processes. For detailed study of martingales, see Refs. 16, 26, 35, 37, and 40. A random sequence fYn, n 0g is a discrete martingale with respect to the sequence fXm, m ng if the following conditions are satisﬁed: (1) EjYn j , 1 (2) E ½Yn jXm , Xm1 , . . . , X0 ¼ Ym ; m n

(20:10:1)

If the second condition in Eq. (20.10.1) is modiﬁed as (2a) E ½Yn jXm , Xm1 , . . . , X0 Ym , m n

(20:10:2)

then the random sequence fYn, n 0g is a submartingale with respect to the sequence fXm, m 0g and if modiﬁed as (2b) E ½Yn jXm , Xm1 , . . . , X0 Ym , m n

(20:10:3)

then the random sequence fYn, n 0g is a supermartingale with respect to the sequence fXm, m 0g. A random process fY(t), t 0g is a continuous martingale with respect to the process fX(s), 0 s , tg if the following conditions are satisﬁed: (3) EjY(t)j , 1 (4) E½Y(t)jX(s), 0 s , t ¼ Y(s)

(20:10:4)

If the second condition in Eq. (20.10.1) is modiﬁed as (4a) E ½Y(t)jX(s), 0 s , t Y(s)

(20:10:5)

then the random process fY(t), t 0g is a continuous submartingale with respect to the process fX(s), 0 s , tg and if modiﬁed as (4b) E ½Y(t)jX(s), 0 s , t Y(s)

(20:10:6)

then the random process fY(t), t 0g is a continuous supermartingale with respect to the underlying process fX(s), 0 s , tg. We can explain the concept of a martingale by a simple example. A player tosses a fair coin p ¼ q ¼ 12 and wins a dollar if heads come up and loses a dollar if tails come up. This is the modiﬁed Bernoulli process described by Eq. (20.4.11). Let Wn ¼ Z1 þ Z2 þ . . . þ Zn be his fortune at the end of n tosses. The average fortune of the player at the (n þ 1)st toss is the same as his current fortune Wn, and this is not affected by how he arrived at his current fortune, that is, E ½wnþ1 jZn , Zn1 , . . . , Z1 ¼ Wn , thus capturing the notion that the game is fair. In the same spirit we can think of the game being biased in favor of the player (submartingale) or biased against the player (supermartingale). Unless otherwise stated, we will assume that the condition EjYn j , 1 or EjY(t)j , 1 is always satisﬁed in the following examples.

20.10 Martingale Process

553

Example 20.10.1 Random-Walk Process Let fZig be the modiﬁed Bernoulli process described by Eq. (20.4.11) with mZ ¼ p 2 q and s2Z ¼ 4pq. The corresponding random-walk process is a modiﬁed form of Eq. (20.4.15): Wn ¼

n X

Zi ¼

n1 X

i¼1

Zi þ Zn ¼ Wn1 þ Zn

(20:10:7)

i¼1

The mean value of Wn from Eq. (20.4.16) is mW ¼ n( p 2 q), and the variance of Wn from Eq. (20.4.17) is s2W ¼ 4npq. The conditional expectation E ½Wnþ1 jZn , Zn1 , . . . , Z1 is given by E ½Wnþ1 jZn , Zn1 , . . . , Z1 ¼ E ½Znþ1 þ Wn jZn , Zn1 , . . . , Z ¼ ( p q) þ Wn

(20:10:8)

If we now deﬁne Yn ¼ Wn 2 n( p 2 q) in Eq. (20.10.5), we can write E ½Ynþ1 jZn , Zn1 , . . . , Z1 ¼ E ½Znþ1 ( p q) þ Yn jZn , Zn1 , . . . , Z ¼ Yn

(20:10:9)

Hence Yn is a discrete martingale with respect to the sequence fZk, 1 k ng. Example 20.10.2 The characteristic function of Wn in Eq. (20.10.4) is FWn (v) ¼ E ½ejvWn , and we deﬁne Yn ¼

e jvWn FWn (v)

and

E ½Yn ¼ 1

For m n, we can write for the following conditional expectation: jv(Wn Wm ) jvWm e e Wm , . . . ,W1 E ½Yn jWm , . . . ,W1 ¼ E FWn (v) Because of the independent increment property, (Wn 2 Wm) is independent of Wm and hence E ½e jv (Wn Wm ) jWm , . . . , W1 ¼ E ½e jv (Wn Wm ) and the preceding equation can be written as E ½Yn jWm , . . . ,W1 ¼

E ½e jv(Wn Wm ) e jvWm e jvWm ¼ ¼ Ym , FWnm (v) FWm (v) FWm (v)

mn

Hence Yn is a martingale with respect to the sequence fWm, . . . , W1g. Example 20.10.3 The sequence fZig of Example 20.10.2 is now a set of i.i.d. random variables with p ¼ q ¼ 12 with variance s2Z. Here the mean value mZ ¼ 0. We form the sequence !2 n X Zi ns2Z Yn ¼ i¼1

Since E(

Pn

i¼1

2

ns2Z ,

we have E[Yn] ¼ 0. For m n, we can write 82 3 !2 nm < X Z s2Z (n m)5 E ½Yn jZm , . . . , Z1 ¼ E 4 : i¼m i

Zi ) ¼

2 þ4

m X i¼1

!2 Zi

9 3 = ms2Z 5Zm , . . . , Z1 ;

554

Classiﬁcation of Random Processes

Pnm

2

s2Z (n m) is independent of the sequence fZm, . . . , Z1g we have 2 3 2 3 !2 !2 nm nm X X Zi s2Z (n m)Zm , . . . , Z1 5 ¼ E4 Zi s2Z (n m)5 ¼ 0 E4 i¼m i¼m

Since E

i¼m

Zi

and hence 2 E ½Yn jZm , . . . , Z1 ¼ E4

m X i¼1

!2 Zi

3 !2 m X 2 msZ Zm , . . . , Z1 5 ¼ Zi ms2Z ¼ Ym i¼1

and Yn is a discrete martingale with respect to the sequence fZk, 1 k ng. Example 20.10.4 (Poisson Martingale) The Poisson process fN(t), t 0g with mean mN(t) ¼ lt is given by the probability function P{N(t) ¼ k} ¼ elt

(lt)k k!

Let Y(t) ¼ N(t) 2 lt. We will show that Y(t) is a martingale. For s t we can write E ½Y(t) j Y(s), s t ¼ E ½Y(t) Y(s) þ yðsÞ j Y(s), s t Since Y(t) is an independent increment process, Y(t) 2 Y(s) is independent of Y(s). Hence E ½Y(t) Y(s) j Y(s), s t ¼ E ½Y(t) Y(s), s t ¼ 0 Hence we have E ½Y(t)jY(s) ¼ Y(s),

st

showing that Y(t) is a martingale with respect to the process fY(s), s tg. Example 20.10.5 (Wiener Martingale) The Wiener process fW(t), t 0g has been deﬁned in Section 20.7 as a zero mean independent increment process with density function given by Eq. (20.7.1): 1 2 2 fW (w, t) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ eð1=2Þ(w =s t) 2pst Using the same procedure as in previous examples, we can write E ½W(t)jW(s), s t ¼ E ½W(t) W(s) þ W(s)jW(s), s t Since W(t) is an independent increment process, W(t) 2 W(s) is independent of W(s). Hence E ½W(t) W(s)jW(s), s t ¼ E ½W(t) W(s), s t ¼ 0 Hence we have E ½W(t)jW(s) ¼ W(s),

st

showing that W(t) is a martingale with respect to the process fW(s), s tg. In all the ﬁve examples given above the underlying random process was an independent increment process. The question arises as to whether all independent increment processes with zero mean are martingales. We will show that this is indeed the case. If

20.10 Martingale Process

555

X(t) is an independent increment process, then from Eq. (20.4.1), we have E{½X(ti ) X(ti1 )½X(tiþ1 ) X(ti )} ¼ E ½X(ti ) X(ti1 )E ½X(tiþ1 Þ X(ti )

(20:4:1)

If the mean of X(t) is mX(t), then we can deﬁne a zero mean independent increment process Y(t) ¼ X(t) 2 mX(t) and write the conditional expectation E ½Y(t)jY(s) for 0 s t as follows: E ½Y(t) j Y(s) ¼ E ½Y(t) Y(s) j Y(s) þ E ½Y(s) j Y(s),

0st

(20:10:10)

Using the independent increment property, we obtain E ½Y(t) Y(s) j Y(s) ¼ E ½Y(t) Y(s) ¼ 0,

0st

(20:10:11)

0st

(20:10:12)

Substituting Eq. (20.10.11) in Eq. (20.10.10), we obtain E ½Y(t) Y(s) j Y(s) ¼ E ½Y(t) Y(s) ¼ 0,

thus showing that a zero mean independent increment process is a martingale with respect to the process fY(s), 0 s tg. However, not all martingales are independent increment processes, as shown in the following example. Example 20.10.6 (Likelihood Ratio Martingale) The likelihood ratio under the two hypotheses H0 and H1 in Eq. (18.6.17) is modiﬁed as LX (n) ¼

fX (Xn , . . . , X1 j H1 ) fX (Xn , . . . , X1 j H0 )

We will show that under the hypothesis H0, the likelihood ratio LX(n) is a martingale with respect to the sequence fXn, . . . , X1g. The conditional expectation E[LX(n þ 1)jXn, . . . , X1jH0] can be written as E ½LX (n þ 1) j Xn , . . . , X1 j H0 ð1 fX (xnþ1 , xn , . . . , x1 j H1 ) fX (xnþ1 , xn , . . . , x1 j H0 )dxnþ1 ¼ f 1 X (xnþ1 , xn , . . . , x1 j H0 ) From the deﬁnition of conditional probabilities, we can substitute the following in the equation above fX (xnþ1 , xn , . . . , x1 j H0 ) ¼ fX (xnþ1 j xn , . . . , x1 j H0 ) fX (xn , . . . , x1 j H0 ) and write E ½LX (n þ 1) j Xn , . . . , X1 j H0 ð1 fX (xnþ1 , xn , . . . , x1 j H1 ) fX (xnþ1 j xn , . . . , x1 j H0 ) dxnþ1 ¼ fX (xn , . . . , x1 j H0 ) fX (xnþ1 j xn , . . . , x1 j H0 ) 1 ð1 1 ¼ fX (xnþ1 , xn , . . . , x1 j H1 )dxnþ1 ¼ LX (n) fX (xn , . . . , x1 j H0 ) 1

556

Classiﬁcation of Random Processes

thus showing that LX (n) is a martingale with respect to the sequence fXn, . . . , X1g under the hypothesis H0. Example 20.10.7 The Wiener process fW(t), t 0g has mean 0 and variance s2t. We form the function Y(t) with W(t) as Y(t) ¼ ekW(t)k

2

(s2 t=2)

t.0

,

where k is an arbitrary constant. We will show that Y(t) is a martingale with respect to the Wiener process fW(t), t 0g. We form the conditional expectation with s t: E ½Y(t) j W(s), 0 s , t ¼ ek

2

(s2 t=2)

E½ekW(t) j W(s)

¼ ek

2

(s2 t=2)

E ½ek½W(t)W(s) eW(s) j W(s)

Using the independent increment property E ½ek½W(t)W(s) jW(s) ¼ E ½ek½W(t)W(s) in the preceding equation, we ﬁnd that E ½Y(t) j W(s), 0 s , t ¼ ekW(s) ek

2

(s2 t=2)

E½ek½W(t)W(s)

But E ½ek½W(t)W(s) is the moment generating function (MGF) of the process W(t) 2 W(s), which has zero mean and variance s2(t 2 s). From Example 11.5.2 the MGF is given by 2 2 E ½ek½W(t)W(s) ¼ ek ½s (ts)=2 , and substituting this equation, we obtain E ½Y(t) j W(s), 0 s , t ¼ ekW(s) ek

2

ðs2 t=2) k2 ½s2 ðtsÞ=2

e

¼ e kWðsÞ ek

2

ðs2 s=2Þ

¼ Y(s)

thus showing the required result. Example 20.10.8 The random process is given by Y(t) ¼ W 2(t), where W(t) is a Wiener process with variance s2t. We will show that Y(t) is a submartingale. We form the conditional expectation with s , t: E ½Y(t) j W(s) ¼ E ½W 2 (t) W 2 (s) W 2 (s) j W(s),

0s,t

With 0 s , t and, using the independent increment property, this equation can be written as E ½Y(t) j W(s) ¼ E ½W 2 (t) W 2 (s) W 2 (s) ¼ s2 (t s) W 2 (s) . Y(s) showing that Y(t) is indeed a submartingale. However, we can show that V(t) ¼ W 2(t) 2 s2t is a martingale as shown below: E ½V(t) j W(s) ¼ E ½W 2 (t) s2 t W 2 (s) s2 s þ V(s) ¼ V(s),

0 s , t

Thus, V(t) is a martingale with respect to the Wiener process fW(t), t 0g. Example 20.10.9 If fYn, n 0g is a martingale with respect to the sequence fXm, m 0g and f (.) is some convex function, then f(Yn) is submartingale with respect to fXm, m 0g as shown below: From Jensen’s inequality [Eq. (14.6.3), E ½h(X) h(E ½X), we can write E ½f(Yn )jXm f{E ½Yn jXm } ¼ f(Ym ) Showing that f(Yn) is a submartingale.

20.11 Periodic Random Process

557

As particular cases the following convex functions of a martingale Yn are all submartingales: (1) jYnj, (2) Y2n, (3) jYnja. Similarly, convex functions of a continuous martingale Y(t) are submartingales as shown below: 1. Poisson martingale N(t) 2 lt. [N(t) 2 lt]2 is a submartingale. 2. Wiener martingale W(t). We have already seen in Example 20.10.8 that W 2(t) is a submartingale. 3. W 2(t) 2 s2t is a martingale. [W 2(t) 2 s2t]2 is a submartingale. A martingale process is based on expected values and a Markov process on conditional probabilities, or E ½X(t) j X(k), 0 k s ¼ X(s) P ½X(t) j X(k), 0 k s ¼ P ½X(t) j X(s)

martingale Markov

Hence a martingale process need not be a Markov process or vice versa. However, all independent increment processes are both martingales and Markov processes. Decomposition of Submartingales

Doob [16] and Meyer [41] have shown that under certain regularity conditions any submartingale can be uniquely decomposed into a martingale and an increasing predictable process. Heuristically, a process A(t) or An is predictable if it can be completely determined from a knowledge of either fX(s), 0 s , t)g or fXk, 0 , k ng. All deterministic functions are predictable. The Doob–Meyer decomposition states that if Yt is a submartingale with respect to an underlying random process fXt, t 0g, then the unique decomposition is given by Yt ¼ M t þ A t

(20:10:13)

where Mt is a martingale with respect to the same underlying random process fXt, t 0g and At is an increasing predictable process. It is the decomposition aspect of the submartingales that gave rise to an elegant proof [19] for the nonlinear ﬁltering problem of which the Kalman linear ﬁltering problem is a special case. Complete details can be found in Refs. 26, 35, and 37.

B 20.11

PERIODIC RANDOM PROCESS

The continuity of the sample paths of a periodic random processes is of concern to us for expanding them in Fourier series. Among the different types of continuity the mean-square continuity is of importance. A random process X(t) is continuous in the mean-square sense if the following condition is satisﬁed: lim EjX(t) X(s)j2 ! 0 s!t

(20:11:1)

The mean-square continuity of the random process is tied to the autocorrelation RX(t,s) of the process. We will show that a random process X(t) is mean-square continuous if and only if RX(t,s) is continuous at the diagonal point s ¼ t ¼ t: EjX(t) X(s)j2 ¼ RX (t, t) 2RX (t, s) þ RX (s, s)

(20:11:2)

558

Classiﬁcation of Random Processes

Since RX(t,s) is continuous at s ¼ t, by assumption the righthand side is zero at s ¼ t, thus showing the necessity. Conversely RX (t, t) RX (s, s) ¼ E ½X(t)X(t) E ½Xs2 ¼ E{½X(t) X(s)X(t)} þ E{½X(t) X(s)X(s)} From the preceding equation, using Schwarz’s inequality of Eq. (14.5.13), we can form RX (t, t) RX (s,s) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E ½jX(t) X(s)j2 E ½X 2 (t) þ E ½jX(t) X(s)j2 E ½X 2 (s)

(20:11:3)

From the assumption of mean-square continuity of X(t) the righthand side goes to zero as t, s ! t and hence RX(t,s) is continuous at s ¼ t. A random process X(t) is periodic in the wide sense with period Tc if mX (t þ kTc ) ¼ mX (t) for all t and integer k RX (t, s) ¼ RX (t þ kTc , s) ¼ RX (t, s þ kTc ) for all t, s, and integer k

(20:11:4)

The autocorrelation function RX(t,s) is periodic in both arguments t and s and is shown in Fig. 20.11.1 in terms of constant-value contours. If X(t) is a stationary process, then with t ¼ t 2 s, Eq. (20.11.4) simpliﬁes to RX (t) ¼ RX (t þ kTc )

for all t and integer k

(20:11:5)

and the mean-square continuity of X(t) guarantees that RX(t) is uniformly continuous in t. Such wide-sense periodic stationary processes are shown in Examples 19.2.1 –19.2.3. As a consequence of the continuity, the Fourier series of RX(t) given by RX (t) ¼

1 X

Rk e jkvc (st) ,

t¼st

(20:11:6)

n¼1

converges uniformly to RX(t). As a consequence of the uniform continuity of RX(t), the zero mean stationary periodic random process X(t) with fundamental frequency vc ¼ 2p/Tc can be represented in

FIGURE 20.11.1

20.11 Periodic Random Process

559

the mean-square sense by a Fourier series 1 X

X(t) ¼

Xn e jnvc t , X0 ¼ 0 and

Xn ¼

n¼1

1 Tc

ð Tc

X(t)ejnvc t dt

(20:11:7)

0

The Fourier coefﬁcients fXng are in general complex random variables, and we will investigate their properties. Since X(t) is a zero mean process, E[Xn] ¼ 0. The crosscorrelation between two of the coefﬁcients can be calculated using Eq. (20.11.7): E ½Xm Xn

1 ¼ 2 Tc 1 ¼ 2 Tc

ð Tc ð Tc 0

E ½X(s)X(t)ejvc (msnt) ds dt

0

ð Tc ð Tc 0

RX (s t)ejvc (msmt) dt ds

(20:11:8)

0

Substituting for RX(s 2 t) from Eq. (20.11.6) in Eq. (20.11.8), we obtain E ½Xm Xn

1 ¼ 2 Tc ¼

ð Tc ð Tc X 1 0

Rk e jkvc (st) ejvc (msnt) dt ds

0 n¼1

ð Tc ð Tc 1 1 X jvs(km) R e ds e jvt(nk) dt k T 2 n¼1 0 0

(20:11:9)

The complex exponentials {ejnvc s } and {ejnvc t } are orthogonal sets and hence 1 Tc

ð Tc e

jvs(km)

ds ¼ dkm ¼

0

1, 0,

¼ dnk ¼

k¼m

1 : T k=m c 1, n ¼ k 0,

ð Tc

e jvt(nk) ds

0

(20:11:10)

n=k

Substituting Eq. (20.11.10) into Eq. (20.11.9), we obtain the result E ½Xm Xn

1 1 X Rm , ¼ 2 Rk dkm dnk ¼ Rm dmn ¼ 0, Tc n¼1

n¼m n=k

or E ½Xm2 ¼ Rm

(20:11:11)

Thus, the Fourier coefﬁcients of a zero mean periodic stationary random process X(t) are orthogonal and uncorrelated. If it is a Gaussian process, then these coefﬁcients are also independent. Example 20.11.1 given by

The autocorrelation of a zero mean white Gaussian noise W(t) is

E ½W(s)W(t) ¼ RW (t s) ¼ s2 d(t s)

560

Classiﬁcation of Random Processes

where s2 is the variance parameter. The process may be considered periodic with period Tc ! 1. Substituting for RW(t 2 s) in Eq. (20.11.8), we obtain s2 Tc !1 Tc2

E ½Wm Wn ¼ lim

ð Tc ð Tc 0

s2 dmn Tc !1 Tc2

d(s t)ejvc (msnt) dt ds ¼ lim

0

or s2 ¼ s2W Tc !1 Tc2

E ½Wm2 ¼ lim

and the Fourier coefﬁcients Wm and Wn are independent for any m and n provided m = n. Cyclostationary Process

A random process X(t) is called strictly cyclostationary [21] if the joint distribution function for any set of samples at times t1,t2, . . . ,tn is the same for any integer multiples of some period T, that is, if the cdf and pdf satisfy the equations FX (x1 , . . . , xn ; t1 , . . . , tn ) ¼ FX (x1 , . . . , xn ; t1 þ kT, . . . , tn þ kT) fX (x1 , . . . , xn ; t1 , . . . , tn ) ¼ fX (x1 , . . . , xn ; t1 þ kT, . . . , tn þ kT)

(20:11:12)

for any n and k. This is somewhat restrictive, and hence we deﬁne a wide-sense cyclostationary process if the mean mX(t) and autocorrelation RX(t1,t2) are periodic: mX (t þ kT) ¼ mX (t) RX (t1 þ kT, t2 þ kT) ¼ RX (t1 , t2 )

(20:11:13)

The autocorrelation function of a wide-sense cyclostationary process is shown in Fig. 10.11.2. The difference between Figs. 20.11.2 and 20.11.1 should be noted. Example 20.11.2 A random process X(t) is deﬁned by X(t) ¼ A cos(vc t) þ B sin(vc t), where A and B are two zero mean independent Gaussian random variables with variances

FIGURE 20.11.2

20.11 Periodic Random Process

561

s2A and s2B respectively. The mean value of this process is E ½X(t) ¼ mX (t) ¼ E ½A cos(vc t) þ E ½B sin(vc t) ¼ 0,

E ½A ¼ E ½B ¼ 0

The autocorrelation function RX(t,s) is given by RX (t, s) ¼ E ½X(t)X(s) ¼ E{½A cos(vc t) þ B sin(vc t)½A cos(vc s) þ B sin(vc s)} ¼ E ½A2 cos(vc t) cos(vc s) þ E ½B2 sin(vc t) sin(vc s) þ E ½AB½cos(vc t) sin(vc s) þ sin(vc tÞ cos(vc s) Since A and B are zero mean and independent, E ½A2 ¼ s2A , E ½B2 ¼ s2B , and E ½AB ¼ E ½AE ½B ¼ 0 and hence 1 (s t, s þ t) ¼ s2A ½cos (vc (s t)) þ cos(vc (s þ t)) 2 1 þ s2B ½cos(vc (s t)) cos(vc (s þ t)) 2 ¼

s2A þ s2B s2 s2B cos(vc (s t)) þ A cos(vc (s þ t)) 2 2

Since the autocorrelation function RX(s2t, s þ t) is dependent on both t and s, the process is not stationary. However, with mX(t) ¼ 0 and Tc ¼ 2p/vc , we have RX (s t, s þ t) ¼ RX (s t þ kTc , s þ t þ kTc ) in the preceding equation. Hence the process is wide-sense cyclostationary. Example 20.11.3

A pulse-amplitude-modulated random process is of the form 1 X

X(t) ¼

Ak p(t kTc )

k¼1

where fAkg is a stationary digital sequence of random variables with mean mA and autocorrelation function RA (h) ¼ E ½Ak Akþh . p(t) is a deterministic pulse of duration Tc. We want to ﬁnd the mean and the autocorrelation function of X(t): Mean: mX (t):

1 X

E ½X(t) ¼ mX (t) ¼

1 X

E ½Ak p(t kTc ) ¼ mA

k¼1

p(t kTc )

k¼1

and the mean is a periodic function with period Tc. Autocorrelation: E ½X(t)X(s) ¼ RX (t, s) ¼

1 1 X X

E ½Ak Aj p(t kTc )p(s jTc )

k¼1 j¼1

¼

1 1 X X

RA (k, j)p(t kTc )p(s jTc )

k¼1 j¼1

¼ RX (t kTc , s kTc ), k ¼ +1, +2, . . .

562

Classiﬁcation of Random Processes

FIGURE 20.11.3

Since both conditions of Eq. (20.11.13) are satisﬁed, we conclude that X(t) is a cyclostationary process. We will make the result more concrete by assuming that p(t) is a rectangular pulse of unit height and width Tc. The random variable sequence fAkg is an independent set of equiprobable þ1 or 21 representing the symbols 1 or 0 respectively. Since Ak is equiprobable, mA ¼ 0, and we obtain mX(t) ¼ 0. A sample waveform of X(t) corresponding to sending symbols f0100110010100g is shown in Fig. 20.11.3. Since the Ak values are independent with zero mean, we have

E ½X(t)X(s) ¼ Rx (t, s) ¼

1 X

E ½A2k p(t kTc )p(s kTc )

k¼1

1 1 With E ½A2k ¼ 1:1 þ (1) (1) ¼ 1, we obtain 2 2 2 3 kTc t , (k þ 1)Tc 1, kTc s , (k þ 1)Tc 5 RX (t, s) ¼ 4 0, otherwise The function RX (t, s) is shown in Fig. 20.11.4, which corresponds to Fig. 20.11.2, showing that X(t) is a cyclostationary process.

FIGURE 20.11.4

20.12 Aperiodic Random Process (Karhunen– Loeve Expansion)

563

B 20.12 APERIODIC RANDOM PROCESS (KARHUNEN – LOEVE EXPANSION) In the last section the coefﬁcients of the Fourier expansion of a mean-square continuous periodic random process were found to be orthogonal random variables. We can attempt a similar Fourier expansion in the interval [a,b] for a mean-square continuous aperiodic random process X(t) as X(t) ¼

1 X

Xn fn (t),

atb

(20:12:1)

n¼1

where the sequence ffn(t)g is an orthonormal set in the interval [a,b] deﬁned by ðb a

fn (t)fm (t)dt ¼

1, 0,

m¼n m=n

(20:12:2)

and the generalized Fourier coefﬁcients (FCs) Xn are given by ðb Xn ¼ a

X(t)fn (t)dt,

n ¼ 0, 1, . . .

(20:12:3)

If the sequence ffn(t)g is any orthonormal set, then the coefﬁcients fXng will not be orthogonal, unlike the periodic random processes as shown in the following example. Example 20.12.1 Let X(t), 0 t T, be a zero mean stationary Gaussian random process with continuous ACF RX(t). A Fourier series expansion of X(t) is given by X(t) ¼ lim

N!1

N X

Xn e jnv0 t :

v0 ¼

n¼N

2p T

The Fourier coefﬁcient Xn is given by 1 Xn ¼ T

ð T=2

X(t)ejnv0 t dt

with

E ½Xn ¼ 0,

E ½X(t) ¼ 0

T=2

We will show that the random variables fXng are not orthogonal. E½Xn Xm can be written as E ½Xn Xm

1 ¼ 2 T

ð T=2 ð T=2

E½X(t)X(s)e jv0 (ntms) dt ds

T=2 T=2

Substituting m ¼ n 2 k in this equation, we obtain E ½Xn Xm

1 ¼ 2 T

ð T=2 ð T=2

RX (t s)e jv0 n(ts) ejv0 ks dt ds

T=2 T=2

Substituting t ¼ t 2 s in this equation, yields E ½Xn Xm

1 ¼ 2 T

ð T=2 ð T=2s RX (t)e T=2

T=2s

jv0 nt

dt ejv0 ks ds

(A)

564

Classiﬁcation of Random Processes

We will deﬁne the quantity in Eq. (A): ð T=2s RX (t)ejv0 nt dt ¼ SX (n, s) and

lim SX (n, s) ¼ SX (nv0 )

T!1

T=2s

Substituting the preceding equation in Eq. (A), we ﬁnd that ð 1 T=2 SX (n, s)e jv0 ks ds = 0 E½Xn Xm ¼ 2 T T=2 Hence we have shown that if X(t) is expanded in any orthonormal set, the resulting Fourier coefﬁcients fXng are not orthogonal. However as T ! 1, we see that SX(n, s) tends to SX(nv0) and hence ð 1 T=2 lim 2 SX (nv0 )e jv0 ks ds ! 0 T!1 T T=2 Thus, the Fourier coefﬁcients are orthogonal as T ! 1. We are interested in ﬁnding the necessary and sufﬁcient conditions for that orthonormal set that will yield an orthogonal Fourier coefﬁcient set. We ﬁrst multiply Eq. (20.12.1) by Xm and take expectations E½X(t)Xm ¼

1 X

E½Xn Xm fn (t),

atb

(20:12:4)

n¼1

Under the orthogonality condition of the FCs we can substitute E½Xn Xm ¼ 0, n = m in Eq. (20.12.4) and obtain E½X(t)Xm ¼ EjXm j2 fm (t),

atb

(20:12:5)

We can obtain a second equation by multiplying the following equation for Xm ðb Xm ¼ X (s)fm (s)ds a

by X(t) and taking expectations. Or, for a t b ðb ðb E½X(t)Xm ¼ E½X(t)X (s)fm (s)ds ¼ RX (t, s)fm (s)ds

(20:12:6)

Comparing Eqs. (20.12.5) and (20.12.6), we can write ðb E½X(t)Xm ¼ RX (t, s)fm (s)ds ¼ EjXm2 jfm (t),

(20:12:7)

a

a

atb

a

Deﬁning the signal energy EjXn2 j ¼ ln , Eq. (20.12.7) can be rewritten as ðb RX (t, s)fn (s)ds ¼ ln fn (t), a t b a

or ðb RX (t, s)f(s)ds ¼ lf(t), a

atb

(20:12:8)

20.12 Aperiodic Random Process (Karhunen– Loeve Expansion)

565

This equation which is called the Karhunen – Loeve (K-L) integral equation, which gives the necessary and sufﬁcient conditions for the random process X(t) to be expanded as a series in terms of the orthonormal set ffm(t)g such that the generalized Fourier coefﬁcients Xm, called Karhunen –Loeve (K-L) coefﬁcients, are orthogonal. The expansion of Eq. (20.12.1) is called the Karhunen –Loeve (K-L) expansion. The orthonormal functions ffm(t)g are called the eigenfunctions and flmg are called the eigenvalues of the K-L integral equation. It is to be particularly noted that the process X(t) can be nonstationary. Summary: K-L equation satisfied ) Fourier coefﬁcients{Xn } are orthogonal Fourier coefﬁcients{Xn } are orthogonal ) K-L equation is satisfied Example 20.12.2 We are given a white-noise process with autocorrelation function RV(t, s) ¼ s2d(t 2 s). Even though this process violates the second-order condition in the sense that the autocorrelation function is not ﬁnite, we can formally ﬁnd the orthonormal set such that the Fourier coefﬁcients are orthogonal in the interval [2a, a]. Substituting for RX(t, s) in Eq. (20.12.8), we obtain ðb

s2 d(t s)fn (s)ds ¼ ln fn (t)

a

or s2 fn (t) ¼ ln fn (t) and

s2 ¼ ln ,

at a

This result conveys that any orthonormal set ffn(t)g with eigenvalues s2 will generate K-L coefﬁcients Vn for the white-noise process in the interval [2a, a] with E[V2n] ¼ ln ¼ s2. Example 20.12.3

A random process Y(t) not necessarily stationary is given by Y(t) ¼ X(t) þ V(t)

where V(t) is a zero mean white Gaussian noise uncorrelated with X(t). We want to ﬁnd the eigenfunctions ffn(t)g in the interval [2a, a] that will give K-L coefﬁcients. The autocorrelation function of Y(t) is RY (t, s) ¼ RX (t, s) þ s2 d(t s) The eigenfunctions ffX(t)g and ffY (t)g for representing X(t) and Y(t) can be found by applying the K-L integral equation Eq. (20.12.8) to RX(t,s) and RY(t,s) as given below: ða RX (t, s)fX (s)ds ¼ lX fX (t),

a,ta

RY (t, s)fY (s)ds ¼ lY fY (t),

a,t a

a ða a

566

Classiﬁcation of Random Processes

Since X(t) and V(t) are uncorrelated, we can substitute RY (t, s) ¼ RX (t, s) þ s2 d(t s) in the preceding integral equation for RY(t, s) and obtain ða ½RX (t, s) þ s2 d(t s)fY (s)ds ¼ lY fY (t), a , t a a

or ða

RX (t, s)fY (s)ds ¼ ½lY s2 fY (t),

a,t a

a

From the uniqueness of the K-L equation, fX (t) ¼ fY (t) and lY ¼ lX þ s2 . Hence, we can use the same eigenfunction set ffX (t)g for the series expansion of both X(t) and Y(t), and from Example 20.12.1 the same can be used for expanding V(t). Thus, all three signals X(t), Y(t), and V(t) can be represented for 2a , t a in the same function space as X(t) ¼

1 X

Xn fn (t),

Y(t) ¼

n¼1

1 X

Yn fn (t),

V(t) ¼

n¼1

1 X

Vn fn (t)

n¼1

The previous two examples are simple ones. However, solving the general K-L integral equation can be difﬁcult. In particular cases, closed-form solutions can be obtained as shown in the following examples. Example 20.12.4 (Nonstationary Process) Wiener process W(t) is

The autocorrelation function RW(t, s) of the

RW (t, s) ¼ s2 min(t, s)

(20:12:9)

The series expansion for W(t) for the time of observation [0, T ] can be obtained from the K-L integral equation [Eq. (20.12.8)] as follows: ðT s2 min(t, s)f(s)ds ¼ lf(t) 0

or s2

ð t

ðT sf(s)ds þ t

0

f(s)ds ¼ lf(t),

0 , t, s T

(20:12:10)

t

The usual procedure to solve integral equations of this type is to differentiate them as many times as possible so that they are converted into differential equations. The resulting differential equations are then solved from the boundary conditions obtained from their form. Differentiating Eq. (20.12.10) with respect to t, there results ðT 2 s tf(t) þ f(s)ds tf(t) ¼ lf(t) t

or s2

ðT f(s)ds ¼ lf(t),

0 , t, s T

(20:12:11)

t

Differentiating again with respect to t [Eq. (20.12.11)], we obtain a second-order homogeneous differential equation given by f(t) þ

s2 ¼ 0, l

0 , t T, l = 0

(20:12:12)

20.12 Aperiodic Random Process (Karhunen– Loeve Expansion)

567

pﬃﬃﬃ The characteristic equation is s 2 þ (s2/l) ¼ 0, and the roots are s1 , s2 ¼ +(s= l). The solution to the differential equation is st st f(t) ¼ A cos pﬃﬃﬃ þ B sin pﬃﬃﬃ (20:12:13) l l There are three unknowns—A, B, and l—to be resolved. The boundary conditions are obtained as follows. Substituting t ¼ 0 þ in Eq. (20.12.10), we can write Ð 0þ s2 0 min (t, s)f(s)ds ¼ lf(0 þ), or lf(0 þ) ¼ 0 and f(0 þ) ¼ 0. Substituting this initial condition in Eq. (20.12.13), we have A ¼ 0 and the solution becomes st f(t) ¼ B sin pﬃﬃﬃ (20:12:14) l ÐT Substituting t ¼ T2 in Eq. (20.12.11), we obtain s2 T f(s)ds ¼ lf(T ), or pﬃﬃﬃ pﬃﬃﬃ f(T ) ¼ 0. Differentiating Eq. (20.12.14), f(t) ¼ B cos (st= l)s= l and we have 2 sT sT ,n 1 (20:12:15) B cos pﬃﬃﬃ ¼ 0 from which ln ¼ (2n 1)p=2 l The constant B is obtained from the normalization integral given by ðT B2 sin2 (kt)dt ¼ 1

(20:12:16)

0

where, for convenience, we have deﬁned s s(2n 1)p (2n 1)p ¼ k ¼ pﬃﬃﬃﬃﬃ ¼ 2sT 2T ln The integral in Eq. (20.12.16) yields ðT ðT 2 B B2 T sin(2kT) 2 2 ½1 cos(2kt)dt ¼ ¼1 B sin (kt)dt ¼ 2 2k 0 0 2

(20:12:17)

Withpkﬃﬃﬃﬃﬃﬃﬃﬃ ¼ (2n 1)p=2T, the term sin (2kT)=2k in Eq. (20.12.17) goes to zero and B ¼ 2=T . Hence the eigenfunctions fn(t) are rﬃﬃﬃﬃ 2 (2n 1)pt n1 sin fn (t) ¼ , (20:12:18) 0,tT T 2T The K-L expansion of the Wiener process W(t) is given by rﬃﬃﬃﬃ 1 X 2 (2n 1)pt Wn sin , 0,tT W(t) ¼ T 2T n¼1 where the coefﬁcients Wn are given by an equation similar to Eq. (20.12.3) rﬃﬃﬃﬃ ðT 2 (2n 1)pt n1 sin dt, Wn ¼ W(t) 0,tT T 2T 0

(20:12:19)

(20:12:20)

with the property E ½Wn2 ¼ ln , n 1. Example 20.12.5 A sinusoid with random phase is given by X(t) ¼ A cos(vt þ F), where A and v are constants and F is uniformly distributed in (2p, p). The

568

Classiﬁcation of Random Processes

autocorrelation function as derived in Example 19.2.1 is RX(t, s) ¼ (A 2/2) cos [v(t 2 s)]. We have to ﬁnd the K-L expansion for X(t) in the period [2T/2, T/2] and show that the K-L coefﬁcients Xn are orthogonal. Substituting for the autocorrelation function in the K-L equation [Eq. (20.12.8)], we obtain ð A2 T=2 T T cos½v(t s)f(s)ds ¼ lf(t), t, s 2 T=2 2 2 Taking derivative with respect to t once, we get ð T=2 A2 v sin½v(t s)f(s)ds ¼ lf(t), 2 T=2

T T t, s 2 2

Taking the derivative with respect to t again, we have ð A2 2 T=2 v cos½v(t s)f(s)ds ¼ lf(t) 2 T=2 or v2 lf(t) ¼ lf(t) or

f(t) þ v2 f(t) ¼ 0,

T T t, s 2 2

The solution to the second-order homogeneous differential equation above is given by f(t) ¼ C cos(vt) þ D sin(vt) From the form of f(t) we conclude that the eigenfunctions are rﬃﬃﬃﬃ rﬃﬃﬃﬃ 2 2 cos(vt) f2 (t) ¼ sin(vt) f1 (t) ¼ T T There are only two terms in the ﬁnite K-L expansion given by rﬃﬃﬃﬃ rﬃﬃﬃﬃ 2 2 cos(vt) þ X2 cos(vt) X(t) ¼ X1 T T We can ﬁnd these two K-L coefﬁcients by equating like coefﬁcients in the expansion of X(t) given by X(t) ¼ A cos(vt) cos(F) A sin(vt) sin(F) Thus rﬃﬃﬃﬃ T X1 ¼ A cos(F) 2

and

rﬃﬃﬃﬃ T X2 ¼ A sin(F) 2

With F uniformly distributed in (2p, p), the joint moment E[X1X2] is E ½X1 X2 ¼

A2 T A2 T E ½cos (F) sin (F) ¼ E ½sin (2F) ¼ 0 2 2

showing that the K-L coefﬁcients are orthogonal. We will now ﬁnd the eigenvalue l1 by substituting f1(t) in the K-L equation: rﬃﬃﬃﬃ rﬃﬃﬃﬃ ð A2 T=2 2 2 cos½v(t s) cos(vs)ds ¼ cos(vt) 2 T=2 T T

20.12 Aperiodic Random Process (Karhunen– Loeve Expansion)

569

or A2 2

ð T=2

1 {cos(vt) þ cos½v(t 2s)}ds ¼ l1 cos(vt) T=2 2

or A2 T cos(vt) ¼ l cos(vt) and 4

l1 ¼

A2 T 4

In a similar manner, substituting f2(t) in the K-L equation, l2 ¼ A 2T/4. The energy values in the K-L coefﬁcients are E ½X12 ¼

A2 T A2 T E ½cos2 (F) ¼ ¼ l1 2 4

E ½X12 ¼

A2 T A2 T E ½sin2 (F) ¼ ¼ l2 2 4

Example 20.12.6 The autocorrelation function of a random process X(t) is RX (t, s) ¼ e2jtsj . The signal is observed in the interval [22, 2]. We want to ﬁnd the eigenvalues and eigenfunctions. The function RX (t, s) ¼ e2jtsj is shown in Fig. 20.12.1. This example a little more involved than the previous ones, and since X(t) is not speciﬁed, we cannot obtain the K-L coefﬁcients directly. The K-L equation is obtained by substituting RX(t,s) in Eq. (20.12.8). The result is ðt

e2(ts) f (s)ds þ s,t

2

ð2 t

e2(ts) f (s)ds ¼ lf(t), 2 t 2 st

Differentiating this equation with respect to t, we obtain ðt e

2 2

2(ts)

ð2 f(s)ds þ 2

e2(ts) f(s)ds ¼ lf(t)

t

FIGURE 20.12.1

570

Classiﬁcation of Random Processes

Differentiating again yields ð t ð2 e2(ts) f(s)ds þ e2(ts) f(s)ds ¼ lf(t) 4f(t) þ 4 t

2

The quantity within braces is the K-L equation and equals lf(t). Hence we obtain lf(t) þ 4(1 l)f(t) ¼ 0,

2t 2

a second-order homogeneous differential equation. The characteristic equation is s2 þ 4(1 l)=l ¼ 0, and the oscillatory roots are given by v2 ¼ 4(1 l)=l. We have an implicit relationship between v and l with l ¼ 4=(4 þ v2 ). The two eigenfunctions are of the form f1 (t) ¼ A cos(vt) and f2 (t) ¼ B sin(vt), where the constants A and B are obtained from normalization. Substituting f1(t) in the K-L equation, we obtain ðt ð2 e2t e2s cos(vs)ds þ e2t e2s cos(vs)ds ¼ l cos(vt) ðBÞ 2 t ðt 1 e2s cos(vs)ds ¼ {2 cos(vt) þ v sin(vt) e4 e2t ½2 cos(2v) þ v sin(2v)} e2t 4 þ v2 2 ð2 1 e2t e2s cos(vs)ds ¼ {2 cos(vt) v sin(vt) e4 e2t ½2 cos(2v) þ v sin(2v)} 4 þ v2 t Substituting these equations in Eq. (B) we obtain 1 {4 cos(vt) þ e4 (e2t þ e2t )½v sin(2v) 2 cos(2v)} ¼ l cos(vt) 4 þ v2 Equating like coefﬁcients on both sides of this equation, yields l¼

4 4 þ v2

and

v sin(2v) 2 cos(2v) ¼ 0

The ﬁrst equation has been obtained earlier. The second equation can be simpliﬁed as tan(2v) ¼ 2/v. This is a transcendental equation that has to be solved graphically for v. It is simpler to solve graphically g(v) ¼ (v=2) tan(2v) ¼ 1 for v as shown in Fig. 20.12.2.

FIGURE 20.12.2

20.12 Aperiodic Random Process (Karhunen– Loeve Expansion)

571

The solution to g(v) ¼ 1 is v ¼ 0.6323. Having found v, we can ﬁnd A and B from ð2 sin(4v) 2 2 2 A cos (vt)dt ¼ 1; or A 2 þ ¼ 1, hence A ¼ 0:6383 2v 2 ð2 sin(4v) 2 2 2 B sin (vt)dt ¼ 1, or B 2 ¼ 1, hence B ¼ 0:8044 2v 2 The eigenvalue is l ¼ 4=(4 þ v2 ) ¼ 0:9091. The eigenfunctions are f1 (t) ¼ 0:6383 cos(0:6323t) and f2 (t) ¼ 0:8044 sin(0:6323t). The K-L expansion is X(t) ¼ X1 f1 (t) þ X2 f2 (t), 2 t 2 and E[X21] ¼ E[X22] ¼ l ¼ 0.9091. Discrete Karhunen– Loeve Transform

In the last section we discussed the continuous Karhunen – Loeve expansion. The same type of an expansion is applicable to a discrete aperiodic random sequence fX (n)g. We shall assume that fX (n)g is a second-order random sequence with autocorrelation RX(n, m). Similar to the continuous-time case, the Fourier expansion for X(n) can be given by X(n) ¼

N 1 X

Xi fi (n),

n ¼ 0, . . . , N 1

(20:12:21)

i¼0

where Xi is the discrete generalized Fourier transform and ffi (n)g is an orthonormal set in the interval 0 n N 2 1. We want to ﬁnd the discrete equivalent of the integral equation (20.12.8) satisﬁed by the orthonormal sequence ffi(n)g. The generalized Fourier coefﬁcients Xi are given by Xi ¼

N 1 X

X(n)fi (n),

0iN1

(20:12:22)

n¼0

If the sequence ffi(n)g is any orthonormal set, then the coefﬁcients fXig in general will not be orthogonal. We are interested in ﬁnding the necessary and sufﬁcient conditions for that orthonormal set ffi(n)g that will yield an orthogonal set of Fourier coefﬁcient fXig. Multiplying Eq. (20.12.21) by Xk and taking expectations, we can write E ½X(n)Xk ¼

N 1 X

E ½Xi Xk fi (n),

0nN1

(20:12:23)

i¼0

Substituting the desired orthogonality criterion E ½Xi Xk ¼ 0, i = k of the Fourier coefﬁcients in Eq. (20.12.23) yields E ½X(n)Xk ¼ EjXk j2 fk (n),

0 n, k N 1

(20:12:24)

We can obtain a second equation by multiplying the equation for X k Xk ¼

N 1 X

X (m)fk (m),

0k N 1

m¼0

by X(n) and taking expectations. Thus, for 0 n, k N 2 1, we obtain E ½X(n)Xk ¼

N 1 X m¼0

E ½X(n)X (m)fk (m) ¼

N 1 X m¼0

RX (n, m)fk (m)

(20:12:25)

572

Classiﬁcation of Random Processes

Comparing Eqs. (20.12.24) and (20.12.25), we can write E ½X(n)Xk ¼

N1 X

RX (n, m)fk (m) ¼ EjXk j2 fk (n),

0nN1

(20:12:26)

m¼0

As in the continuous case, we can deﬁne the signal energy EjXk j2 ¼ lk and rewrite Eq. (20.12.26) as N 1 X

RX (n, m)fk (m) ¼ lk fk (n),

0nN 1

(20:12:27a)

m¼0

or N 1 X

RX (n, m)f(m) ¼ lf(n),

0nN1

(20:12:27b)

m¼0

Equation (20.12.27) is a necessary and sufﬁcient condition for the existence of an orthonormal set that will give an orthogonal set of coefﬁcients fXkg in the Fourier series expansion for random sequence X(n). The convergence of Eq. (20.12.21) is in the mean-square sense, or 2 N 1 X Xk fk (n) ¼ 0, E X(n) k¼0

n ¼ 0, . . . , N 1

(20:12:28)

Matrix Formulation of K-L Equation

The K-L equation [Eq. (20.12.27a)] can be compactly expressed in matrix form. The complex random sequence fX(n)g can be expressed as an N-vector X as X ¼ X(0)

X(1)

X(N 1)

T

(20:12:29)

with the autocorrelation matrix RX ¼ E[XX H] given by 2

RX (0, 0) 6 RX (1, 0) 6 RX ¼ 6 .. 4 . RX (N 1, 0)

3 RX (0, 1) RX (0, N 1) 7 RX (1, 1) RX (1, N 1) 7 7 .. .. 5 . . RX (N 1, 1) RX (N 1, N 1)

(20:12:30)

where the superscript H stands for conjugate transpose. We have to ﬁnd the condition under which F an N N transformation matrix given by 2 3 f00 f01 f0N1 f f11 f1N1 7 6 6 10 7 F ¼ f0 f 1 fN1 ¼ 6 . (20:12:31) 7 .. . .. 4 .. 5 . fN10 fN11 FN1N1

20.12 Aperiodic Random Process (Karhunen– Loeve Expansion)

573

transforms the random sequence vector X in Eq. (12.20.29) into an orthogonal random sequence vector X given by T X ¼ FH X ¼ X0 X1 XN1 (20:12:32) with E ½XXH ¼ L, a diagonal matrix with components flkg. In this case the transformation matrix F has to satisfy the matrix K-L equation given by RX F ¼ LF

(20:12:33)

This equation is similar to the matrix eigenequation (16.2.5). Hence the transforming matrix F is the modal matrix of RX, and since RX is symmetric and positive deﬁnite, F is unitary, that is, F 21 ¼ F H. The eigenvalues of RX constitute the diagonal matrix L with elements lk ¼ E[X2k ], n ¼ 0, 1, . . . , N 2 1.

CHAPTER

21

Random Processes and Linear Systems

B 21.1

REVIEW OF LINEAR SYSTEMS

Consider the deterministic linear time-invariant system shown in Fig. 21.1.1 with input x(t) and output y(t). The system function is characterized in the time domain as the output to a Dirac delta function d(t) with x(0) ¼ 0 and is called the impulse response h(t). A causal system is one that does not anticipate the input meaning that the impulse response h(t) ¼ 0, t , 0. The output y(t) can be given as a convolution of the input x(t) and the impulse response h(t). Depending on the causality of x(t) and h(t), the solution y(t) in continuous time is given by a convolution integral and may contain an initial condition term x(0) as shown below: Ðt Ðt 2 x(0)h(t) þ 0 x(t)h(t t)dt ¼ x(0)h(t) þ 0 x(t t)h(t)dt, x(t), h(t) causal Ð1 6Ðt y(t) ¼ 4 1 x(t)h(t t)dt ¼ 0 x(t t)h(t)dt, x(t) noncausal, h(t) causal Ð1 Ð1 x(t), h(t) noncausal 1 x(t)h(t t)dt ¼ 1 x(t t)h(t)dt, (21:1:1) and similar equations in discrete time for the solution y(n) with initial condition x(0) 2 n P 6x(0)h(n) þ m¼1 x(m)h(n m) 6 6 n 6 ¼ P x(n m)h(m), x(n), h(n) causal 6 6 m¼0 6 P 6 n 6 x(m)h(n m) 6m¼1 y(n) ¼ 6 (21:1:2) 1 6 6 ¼ P x(n m)h(m), x(n) noncausal, h(n) causal 6 m¼0 6 6 P 6 1 x(m)h(n m) 6 6m¼1 6 1 4 P ¼ x(n m)h(m), x(n), h(n) noncausal m¼1

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

574

21.1

Review of Linear Systems

575

FIGURE 21.1.1

The Fourier transform (FT) of the impulse response h(t) is called the transfer function H(v) and is shown below: ð1 h(t)ejvt dt (21:1:3) H(v) ¼ 0

The input FT X(v) and the output FT Y(v) are related by Y(v) ¼ H(v)X(v)

(21:1:4)

The time-invariant system can also be characterized in continuous time by state equations of the form d x(t) ¼ Ax(t) þ Bu(t), x(0) d

(21:1:5)

y(t) ¼ Cx(t) and in discrete time by x(n þ 1) ¼ Ax(n) þ Bu(n), x(0) y(n) ¼ Cx(n)

(21:1:6)

In Eqs. (21.1.5) and (21.1.6) x is an n-vector, A is an n n matrix, B is an n m matrix, u is an m-vector, y is a p-vector, C is a p n matrix, and D is a p m matrix. If these equations are time-varying, then the matrices A, B, and C will become A(t), B(t), and C(t). The corresponding matrix solutions for y for a causal system with initial condition x(0) are Continuous Time: ðt F(t t)Bu(t)dt, time-invariant

y(t) ¼ CF(t)x(0) þ 0

(21:1:7)

ðt F(t, t)B(t)u(t)dt, time-varying

y(t) ¼ C(t)F(t, 0)x(0) þ 0

Discrete Time: y(n) ¼ CF(n)x(0) þ

n X

F(n m)Bu(m), time-invariant

m¼0

y(n) ¼ C(n)F(n, 0)x(0) þ

n X

(21:1:8) F(n, m)B(m)u(m), time-varying

m¼0

In Eqs. (21.1.7) and (21.1.8) the term F (.) is called the state transition matrix. The time-invariant system in Eq. (21.1.5) can be solved in the frequency domain by Laplace transform (LT) techniques. Taking unilateral LT of the ﬁrst equation in

576

Random Processes and Linear Systems

Eq. (21.1.5) and using the initial condition x(0), we obtain sX(s) x(0) ¼ AX(s) þ BU(s) or (sI A)X(s) ¼ x(0) þ BU(s) Hence X(s) ¼ (sI A)1 x(0) þ (sI A)1 BU(s)

(21:1:9)

The inverse LT of the matrix (sI 2 A)21 term by term yields the state transition matrix F(t). The solution for Y(s) is given by substituting Eq. (21.1.9) in the LT of the second equation in Eq. (21.1.5): Y(s) ¼ C(sI A)1 x(0) þ C(sI A)1 BU(s)

(21:1:10)

In general, deterministic systems deal with causal input x(t) and causal impulse response h(t), and the resulting output y(t) will also be causal. However, for random inputs to a system, x(t) need not be causal whereas h(t) will be causal. The resulting output y(t) also will not be causal, as shown in the following example. Example 21.1.1 In the system represented by Fig. 21.1.1, x(t), is a noncausal signal given by x(t) ¼ e 22jtj and the impulse response is causal, given by h(t) ¼ et u(t). We will ﬁnd the output using Laplace transforms (LTs). The LTs of x(t) and h(t) are given by X(s) ¼

4 4 s2

and

H(s) ¼

1 sþ1

The LT Y(s) of the output y(t) is given by Y(s) ¼ H(s)X(s) ¼

4 1 4 s2 s þ 1

The partial fraction expansion of Y(s) yields Y(S) ¼

4 1 4 1 1 ¼ þ 2 4 s s þ 1 3(s þ 1) s þ 2 3(2 s)

and the inverse LT gives the output as 4 1 y(t) ¼ et u(t) e2t u(t) þ e2t u(t) 3 3 x(t), h(t), and y(t) are shown in Fig. 21.1.2. The results can also be obtained in the time domain directly from the convolution integral ð1 x(t t)h(t)dt y(t) ¼ 0

and h(t) and x(t 2 t) are shown in Fig. 21.1.3. From the ﬁgure we can evaluate the convolution. Since x(t 2 t) is given by 2(tt) , tt e x(t t) ¼ e2jttj ¼ e2(tt) , t . t

21.1

Review of Linear Systems

577

FIGURE 21.1.2

we can calculate y(t) in the two regions t 0 and t . 0. For t 0, we have ð1 y(t) ¼

e2(tt) et dt ¼ e2t

0

ð1 0

1 e3t dt ¼ e2t , 3

t 0, t . t

The case for t . 0 is shown in Fig. 21.1.3. We calculate the convolution integral in the two regions t t and t . t as shown below: ðt e

y(t) ¼ 0

2(tt) t

e

tt

ð1 dt þ

e t

2(tt)

t

e dt ¼ e

2t

ðt

t

e dt þ e

2t

ð1

0

t.t

1 ¼ et e2t þ et , 3 The ﬁnal result for y(t) is as obtained by the transform method.

FIGURE 21.1.3

e3t dt

t

t.0

578

Random Processes and Linear Systems

B 21.2 RANDOM PROCESSES THROUGH LINEAR SYSTEMS The input to the system of Fig. 21.1.1 is a random process fX(t), 21 , t , 1g. The system function h(t) is causal. The sample output Y(t, j) of the system for each sample function X(t, j) is given by ð1 X(t t, j)h(t)dt (21:2:1) Y(t, j) ¼ 0

This is a more convenient form for a noncausal X(t) and causal h(t) than the form ðt X(t, j)h(t t)dt (21:2:2) Y(t, j) ¼ 1

Under the following mild conditions on h(t) ð1 ð1 2 jh(t)j dt , 1 and h(t t)CX (t, s)h(t s)dt , 1 0

(21:2:3)

0

the output fY(t), 21 , t , 1g is also a random process. We can now obtain the statistics of the random process Y(t) under the condition that the input random process X(t) is stationary. (It has already been mentioned that stationary means wide-sense stationary.) The mean value of Y(t) is obtained by taking expectations of both sides of the equation ð1 X(t t)h(t)dt Y(t) ¼ 0

or ð1

ð1 E½X(t t)h(t)dt ¼ mX

E½Y(t) ¼ mY (t) ¼ 0

and on substituting H(0) ¼

Ð1 0

h(t)dt ¼ mY

(21:2:4)

0

h(t)dt from Eq. (21.1.3), we obtain mY ¼ mX H(0)

(21:2:5)

Thus the mean value of Y(t) is a constant. The ﬁrst step in obtaining the autocorrelation function RY (t, t) of the output Y(t) is to premultiply Eq. (21.2.4) by X(t – t) and take expectations. Thus ð1 E½X(t t)X(t j)h(j)dj E½X(t t)Y(t) ¼ 0

or ð1 RX (t j)h(j)dj

RXY (t) ¼ 0

or RXY (t) ¼ RX (t) h(t)

(21:2:6)

Thus the cross-correlation function RXY(t) between the input X(t) and the output Y(t) is the convolution between the input autocorrelation function RX(t) and the impulse

21.2

Random Processes through Linear Systems

579

FIGURE 21.2.1

response h(t). The second step is to postmultiply Eq. (21.2.4) by Y(t þ t) and take expectations. Thus ð1 E½X(t j)Y(t þ t)h(j)dj E½Y(t)Y(t þ t) ¼ 0

or ð1 RXY (t þ j)h(j)dj

RY (t) ¼ 0

or RY (t) ¼ RXY (t) h(t)

(21:2:7)

Combining Eqs. (21.2.6) and (21.2.7), we obtain RY (t) ¼ RX (t) h(t) h(t)

(21:2:8)

In many cases it is easier to solve Eq. (21.2.8) in the frequency domain. Taking the Fourier transform of Eq. (21.2.8), the power spectral densities of the input and the output are related by SXY (v) ¼ SX (v)H(v) SY (v) ¼ SXY (v)H (v)

(21:2:9)

SY (v) ¼ SX (v)jH(v)j2 The schematic of the input –output transformation is shown in Fig. 21.2.1. In conclusion, we can state that if a stationary random process X(t) is passed through a linear system whose impulse response h(t) satisﬁes the regularity conditions of Eq. (21.2.3), then the output Y(t) is also a stationary process. In most problems involving random processes through linear systems, the output autocorrelation RY(t) can be more easily found by ﬁrst ﬁnding the psd SY(v) from Eq. (21.1.9) and taking the inverse Fourier transform. Example 21.2.1 A white-noise process X(t) with autocorrelation function RX(t) ¼ s2d(t) is passed through a linear system whose impulse response h(t) is given by h(t) ¼ e 2atu(t). We want to ﬁnd the mean value mY of the output, the cross-correlation function RXY(t), the output autocorrelation function RY (t), and the variance of the output s2Y. The transfer function H(v) and its absolute value squared jH(v)j2 are given by H(v) ¼

1 1 , jH(v)j2 ¼ 2 a þ jv a þ v2

and the input power spectral density is given by SX(v) ¼ s2. The mean value mY is m mY ¼ mX H(0) ¼ X ¼ 0 since mX ¼ 0 a

580

Random Processes and Linear Systems

The cross-spectral density SXY(v) and the output psd SY(v) from Eq. (21.2.9) are SXY (v) ¼ s2 H(v) ¼

s2 ; a þ jv

SY (v) ¼ SX (v)jH(v)j2 ¼

a2

s2 þ v2

Taking inverse Fourier transforms of SXY(v) and SY(v), we obtain the cross-correlation function RXY(t) and the autocorrelation function RY(t): RXY (t) ¼ s2 eat u(t);

RY (t) ¼

s2 ajtj e 2a

Since the mean mY ¼ 0, the variance of the output is s2Y ¼ RY (0) ¼ E½Y 2 ¼ s2 =2a. This ﬁlter is called a shaping ﬁlter since it shapes a white noise to any desired psd by means of the factor a. Example 21.2.2 The autocorrelation function RX(t) of the input random process X(t) in the previous example is modiﬁed as RX(t) ¼ a 2 þ s2d(t) with the impulse response h(t) ¼ e – atu(t). The mean value of X(t) from Eq. (19.2.26) is mX ¼ a. Hence the mean value of Y(t) is m a mY ¼ mX H(0) ¼ X ¼ a a The input psd is SX(v) ¼ a 22pd(v) þ s2. The cross-spectral density SXY(v) and the output psd SY(v) from Eq. (21.2.9) are SXY (v) ¼ s2 þ 2pa2 d(v) H(v) ¼ SY (v) ¼

s2 2pa2 þ d(v) a þ jv a

s2 a2 þ 2pd(v) a2 þ v2 a2

The corresponding cross-correlation and autocorrelation functions of Y(t) are RXY (t) ¼ s2 eat u(t) þ

a2 ; a

RY (t)¼

s2 ajtj a2 e þ 2 2a a

Example 21.2.3 In this example the input X(t) is a random telegraph wave as in Example 19.2.7 with h(t) ¼ e – atu(t) and autocorrelation function RX (t) ¼ 14 {1 þ e2ljtj }. We have to ﬁnd the statistics of the output process Y(t). The mean value, the autocovariance, and the variance X(t) are rﬃﬃﬃ 1 1 1 1 ¼ CX (t) ¼ e2ljtj s2X ¼ CX (0) ¼ mX ¼ 4 2 4 4 The input psd is SX (v) ¼

1 4l 2pd(v) þ 2 4 4l þ v2

The mean value of the output is mY ¼ mX H(0) ¼

mX 1 ¼ a 2a

21.2

Random Processes through Linear Systems

581

The cross-spectral density SXY(v) and the output psd SY(v) from Eq. (21.2.9) are 1 4l 1 2pd(v) l SXY (v) ¼ 2pd(v) þ 2 ¼ þ 4 4a 4l þ v2 a þ jv (4l2 þ v2 )(a þ jv) 1 4l 1 2pd(v) l ¼ þ 2pd(v) þ 2 SY (v) ¼ 2 2 2 2 2 2 4 4a 4l þ v a þ v (4l þ v )(a2 þ v2 ) The partial fraction expansions of SXY(v) and SY(v) are 1 1 l 2pd(v) 4(a 2l) 4(a þ 2l) a2 4l2 þ þ SXY (v) ¼ a þ jv 2l þ jv 2l jv 4a SY (v) ¼

2pd(v) 1 4l l 2a þ 4a2 4(a2 4l2 ) (4l2 þ v2 ) 2a(a2 4l2 ) (a2 þ v2 )

The corresponding cross-correlation and autocorrelation functions of Y(t) are 2 2lt 1 e 1 þ 6 4 a þ 2l a , t 0 6 RXY (t) ¼ 6 2lt 41 e 1 leat þ , t .0 2 4 a 2l a a 4l2 1 e2ljtj 1 leajtj þ RY (t) ¼ 4 a2 4l2 a2 2a(a2 4l2 ) The cross-spectral density SXY (v), the cross-correlation function RXY (t), the power spectral density SY(v), and the autocorrelation function RY (t) are shown for l¼1 and a ¼ 0.5 in Figs. 21.2.2 –21.2.5. The variance of the output process Y(t) is given by s2Y ¼ RY (0) m2Y ¼

4(a2

1 l ¼ 0:2 2 2 4l ) 2a(a 4l2 )

FIGURE 21.2.2

582

Random Processes and Linear Systems

FIGURE 21.2.3

FIGURE 21.2.4

FIGURE 21.2.5

21.2

Random Processes through Linear Systems

583

Example 21.2.4 The input to a linear system is a zero mean stationary random process X(t), and the output is Y(t). Their psds are given by SX (v) ¼

1 4 : ; SY (v) ¼ 2 2 4þv (4 þ v )(64 þ v2 )

The autocorrelation of the input and the output are given by their respective inverse Fourier transforms (IFT): 1 4 e2jtj RX (t) ¼ IFT ¼ 4 4 ð4 þ v2 Þ 4 1 4 1 16 ¼ IFT RY (t) ¼ IFT (4 þ v2 )(64 þ v2 ) 60 (4 þ v2 ) 240 (64 þ v2 ) ¼

1 2jtj e8jtj 4e 240

Since the mean mX ¼ 0, the mean mY ¼ 0. The variances are 1 s2X ¼ , 4

s2Y ¼

1 1 1 ¼ 60 240 80

Transfer function H(v) and the impulse response h(t) can be found as follows. From Eq. (21.2.9) we can write jH(v)j2 ¼

SY (v) 4 2 2 ¼ ¼ H(v)H(v) ¼ 2 SX (v) 64 þ v (8 þ jv) (8 jv)

Hence the transfer function and the impulse responses are given by H(v) ¼

2 ; h(t) ¼ 2e8t u(t) (8 þ jv)

The cross-spectral density is given by SXY (v) ¼

1 2 2 (4 þ v ) (8 þ jv)

The partial fraction expansion of SXY(v) yields SXY (v) ¼

1 1 1 þ 12(2 þ jv) 20(2 jv) 30(8 þ jv)

and the cross-correlation function RXY(t) is given by the IFT of SXY(v): RXY (t) ¼

1 2t 1 1 e u(t) þ e2t u(t) e8t u(t) 20 12 30

The Cross-correlation function RXY(t) is shown in Fig. 21.2.6. Example 21.2.5 A random process X(t) with autocorrelation function RX (t) ¼ 1 þ 4e2jtj excites a linear system with impulse response h(t). The autocorrelation function of the output Y(t) is given by RY (t) ¼ 1 þ 83 ejtj 43 e2jtj . We will ﬁnd the various statistics of this system. Since RX(1) ¼ 1, the mean value mX ¼ 1. The variance s2X ¼ 4.

584

Random Processes and Linear Systems

FIGURE 21.2.6

The power spectral densities of X(t) and Y(t) are the FTs of the corresponding autocorrelations: 16 4 þ v2 16 16 SY (v) ¼ 2pd(v) þ 3(1 þ v2 ) 3(4 þ v2 )

SX (v) ¼ 2pd(v) þ

¼ 2pd(v) þ

16 (1 þ

v2 )(4

þ v2 )

The presence of the delta function complicates the analysis somewhat. If H(v) is the required transfer function, then from Eq. (21.2.9) we have 16 16 jH(v)j2 ¼ 2pd(v) þ 2pd(v) þ 4 þ v2 (1 þ v2 )(4 þ v2 ) 2pd(v)jH(0)j2 þ

16 16 jH(v)j2 ¼ 2pd(v) þ 2 2 4þv (1 þ v )(4 þ v2 )

Since the singularity term on the lefthand side can be matched only by the singularity term on the righthand side, we have jH(0)j2¼1 and 16 16 jH(v)j2 ¼ 4 þ v2 (1 þ v2 )(4 þ v2 ) from which jH(v)j2 ¼

1 1 1 ¼ (1 þ v2 ) 1 þ jv 1 jv

Hence, the transfer function H(v) ¼ 1=(1 þ jv) and the impulse response h(t) ¼ et u(t). The mean value of the output mY ¼ mX H(0) ¼ 1. The variance of the output is s2Y ¼ RY (0) m2Y ¼

8 4 4 ¼ 3 3 3

21.2

Random Processes through Linear Systems

585

FIGURE 21.2.7

The cross-spectral density SXY(v) is given by SXY (v) ¼ 2pd(v) þ

16 4 þ v2

1 16 1 ¼ 2pd(v) þ 1 þ jv 4 þ v2 1 þ jv

and the cross-correlation function RXY(t) by 4 16 RXY (t) ¼ 1 þ e2t u(t) 4e2t u(t) þ et u(t) 3 3 which is shown in Fig. 21.2.7. Example 21.2.6 A zero mean stationary Gaussian stochastic process X(t) with autocorrelation function RX(t) ¼ 100e 2100jtj is applied to a system shown in Fig. 21.2.8. From the diagram the output Y(t) is another random process given by Y(t) ¼ X(t)– X(t– 0.1p). We will determine the various statistics of the random processes. Since mX ¼ 0, it follows that RX(t) ¼ CX(t) and hence the variance of X(t), s2X ¼ CX (0) ¼ 100. The joint density of fX1 ¼ X(t – 0.1p), X2 ¼ X(t)g can be found by ﬁrst obtaining the correlation coefﬁcient rX1X2: E½X1 X2 ¼ E½X(t 0:1p)X(t) ¼ RX (0:1p) ¼ CX (0:1p) ¼ 100e10p

FIGURE 21.2.8

586

Random Processes and Linear Systems

Hence rX1 X2 ¼

CX (0:1p) ¼ e10p CX (0)

The joint Gaussian density between X(t1) and X(t2) can be obtained from Eq. (9.3.1) as follows: 2 3 1 6 2(1 r2 ) 7 1 X1 X2 6 7 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp6 2 fX1 X2 (x1 , x2 ) ¼ 2 7 x 2r x x þ x 4 2 2 X1 X2 1 2 2 5 2psX 1 rX1 X2 1 2 sX 2 3 1 6 2(1 e20p ) 7 1 2 7 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp6 ¼ 10p 2 4 x1 2e x1 x2 þ x2 5 200p 1 e20p 100 The power spectral density SX(v) of the input random process X(t) is SX (v) ¼

20,000 10,000 þ v2

The transfer function of the system is H(v) ¼

Y(v) (1 ej0:1pv )X(v) ¼ ¼ 1 ej0:1pv X(v) X(v)

Hence the mean value mY ¼ mXH(0) ¼ 0. The cross-spectral density SXY (v) and the power spectral density SY (v) are obtained from Eqs. (21.2.9) as follows: SXY (v) ¼ SX (v)H(v) ¼

20,000 (1 ej0:1pv ) 10,000 þ v2

SY (v) ¼ SX (v)H(v)H(v) ¼

20,000 (1 ej0:1pv )(1 e j0:1pv ) 10,000 þ v2

¼

20,000 (2 ej0:1pv e j0:1pv ) 10,000 þ v2

¼

40,000½1 cos (0:1pv) 10,000 þ v2

The graphs of jSXY (v)j and SY (v) are shown in Figs. 21.2.9 and 21.2.10. Taking the IFTs of SXY (v) and SY (v) from the FT tables yields the correlation functions RXY (t) and RY (t), given by

RXY (t) ¼ IFT½SXY (v) ¼ 100 e100jtj e100jt0:1pj

RY (t) ¼ FT½SY (v) ¼ 100 2e100jtj e100jt0:1pj e100jtþ0:1pj The graphs of RXY(t), and RY(t) are shown in Figs. 21.2.11 and 21.2.12. The variance of the process Y(t) is given by s2Y ¼ CY (0) ¼ RY (0) ¼ 200. With 2 sY ¼ 200, the pdf of Y(t) is given by fY (y) ¼

1 2 pﬃﬃﬃﬃ e(1=2){y =200} 20 p

21.2

Random Processes through Linear Systems

FIGURE 21.2.9

FIGURE 21.2.10

FIGURE 21.2.11

587

588

Random Processes and Linear Systems

FIGURE 21.2.12

FIGURE 21.2.13

from which the probability of Y(t) . y can be calculated from ðy 2 1 P{Y(t) . y} ¼ 1 FY (y) ¼ 1 pﬃﬃﬃﬃ e(1=2){z =200} dz 20 p 1 and P(Y(t) . 10) can be evaluated. The function 1 2 FY ( y) is shown in Fig. 21.2.13. However, this probability can be evaluated from the Gaussian tables by ﬁrst substituting pﬃﬃﬃ z ¼ y=s ¼ y=10 2 in the integral above and obtaining ð pﬃ ð 10 1 1 1= z (z2 =2) (1=2){y2 =200} P{Y(t) . 10} ¼ 1 pﬃﬃﬃﬃ e dy ¼ 1 pﬃﬃﬃﬃﬃﬃ e dz 20 p 1 2p 1 ﬃﬃ p pﬃﬃﬃﬃﬃﬃ Ð 1= 2 2 and from the tables P{Y 10} ¼ 1= 2p 1 e(z =2) dz ¼ 0:7603. Hence P{Y(t) . 10} ¼ 1 0:7603 ¼ 0:2397 Example 21.2.7 (Threshold Filter) This example is a computer simulation of using the threshold ﬁlter to retrieve from the output signal corrupted with white noise, the input

21.2

Random Processes through Linear Systems

589

FIGURE 21.2.14

sinusoidal signal in an environment of high signal-to-noise ratio (SNR). The input sinusoidal signal is given by 5 s(t) ¼ 2 sin (4v0 t) þ cos (7v0 t) þ 3 sin (7v0 t) 2 where v0 ¼ (2p=T0 ) ¼ (2p=64) and the period T0 ¼ 64. The input s(t) is discretized with Dt ¼ 1, and the discretized signal s(iDt) ¼ si, given by 2pi 5 2pi 2pi si ¼ 2 sin 8 þ cos 14 þ 3 sin 18 , 128 2 128 128

i ¼ 0, . . . ,127

is shown in Fig. 21.2.14. The number of points N is chosen to be N ¼ 27 ¼ 128 for ease in applying the fast Fourier algorithm (FFT) [36] to transform si into the frequency domain. The zero mean additive discrete white noise wi with variance 4 is shown in Fig. 21.2.15. The input power Psi is N 1 127 1X 1 X 2pi 5 2pi s2i ¼ 2 sin 8 þ cos 14 N i¼0 128 i¼0 128 2 128 2pi 2 ¼ 9:625 þ 3 sin 18 128

Ps i ¼

FIGURE 21.2.15

590

Random Processes and Linear Systems

FIGURE 21.2.16

and the noise power Pni is N1 127 1X 1 X w2i ¼ w2 ¼ 4 N i¼0 128 i¼0 i

Pn i ¼

and SNR ¼ 9:625=4 ¼ 2:406. The output signal yi, given by yi ¼ si þ w i ,

i ¼ 0, . . . , N 1

is shown in Fig. 21.2.16. The output power Pyi is given by P yi ¼

N 1 127 1X 1 X y2i ¼ y2 ¼ 14:811 N i¼0 128 i¼0 i

The input and the noise signals are transformed into the frequency domain by using the discrete Fourier transform (DFT) [36], deﬁned by Sk ¼

N 1 X

si ejk(2p=N)i ¼

i¼0

Wk ¼

N 1 X

si ejk(2p=128)i

i¼0

wi ejk(2p=N)i ¼

i¼0

127 X

127 X

wi ejk(2p=128)i

i¼0

For the purely sinusoidal si the DFT Sk is given by Kro¨necker delta functions as N 5 2jdk8 þ dk14 3jdk18 Sk ¼ 2 2 where dm is the Kro¨necker delta equaling 1 when m ¼ 0 and 0 when m = 0. The DFT of the output yi is given by Yk ¼ S k þ W k ,

k ¼ 0, . . . , N 1

The input and the output spectra jSkj and jYkj are shown in Fig. 12.2.17. At frequncies of 8, 14, and 18, the peak values of the spectra are given in Table 21.2.1. In Fig. 21.2.17, if we choose those data points Yk that are above a threshold value a, then we can isolate all the peaks. A simple threshold ﬁlter can be designed by multiplying

21.2

Random Processes through Linear Systems

591

FIGURE 21.2.17

TABLE 21.2.1 f 8 14 18

Input Spectrum

Output Spectrum

jS8j ¼ 128 jS14j ¼ 160 jS18j ¼ 192

jY8j ¼ 145.47 jY14j ¼ 167.11 jY18j ¼ 200.04

fYk, k ¼ 0, . . . , N 2 1g by a unit step function u(jYkj 2 a), deﬁned by 1, jYk j a u(jYk j a) ¼ 0, jYk j , a The output of this ﬁlter will be only those values of Yk that are above the threshold. If 55 a 145, then the output will be all three peaks. If 146 a 167, then the output will be only two peaks, and if 168 a 200, the output will be only one peak. Finally, if a . 200, the output will be zero. The output Zk of this ﬁlter will be given by Zk ¼ Yk u(jYk j a),

k ¼ 0, . . . , N 1

TABLE 21.2.2 k

jSkj

jYkj

u(jYkj 2 a)

jZkj

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0 0 0 128 0 0 0 0 0 160 0 0 0 192 0 0

19.709 5.858 11.158 145.473 11.091 23.615 28.109 23.032 15.388 167.111 15.412 10.366 22.076 200.036 17.753 32.344

0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0

0 0 0 145.473 0 0 0 0 0 167.111 0 0 0 200.036 0 0

592

Random Processes and Linear Systems

FIGURE 21.2.18

To clarify the operation of the threshold ﬁlter, the quantities, jSkj, jYkj, u(jYkj 2 a), and jZkj are tabulated in Table 21.2.2. The inverse DFT of Zk yields the signal zi that will approximate the input signal si: IDFT{Zk } ¼ zi si With a ¼ 65, this ﬁlter as applied and the IDFT of Zk yields the retrieved signal zi shown in Fig. 21.2.18 along with the superposed original signal si. We see from the ﬁgure that the retrieved signal is almost the same as the original signal. The power Pzi of the retrieved signal is Pzi ¼

B 21.3

N 1 127 1X 1 X z2i ¼ z2 ¼ 10:877 N i¼0 128 i¼0 i

LINEAR FILTERS

The previous section dealt with only lowpass ﬁlters. Among ideal ﬁlters we can deﬁne lowpass, highpass, and bandpass ﬁlters. The transfer functions and impulse responses of these ﬁlters are deﬁned as follows: Ideal Lowpass: Hlow (v) ¼

sin (vB t) jvj vB ; hlow (t) ¼ otherwise pt

(21:3:1)

sin (vB t) jvj . vB ; hhigh (t) ¼ d(t) otherwise pt

(21:3:2)

1, 0,

Ideal Highpass: Hhigh (v) ¼

1, 0,

Ideal Bandpass: Hband (v) ¼

1,

jv vc j vB

0,

otherwise

sin (vB t) cos (vc t), hband (t) ¼ pt

; (21:3:3)

vc . 2vB

In these equations vB is the cutoff frequency and vc is the carrier frequency. The transfer functions and the impulse responses are shown in Fig. 21.3.1.

21.3

Linear Filters

593

FIGURE 21.3.1

As seen in the ﬁgures, the impulse responses are noncausal and hence the ﬁlters are not physically realizable. All physically realizable transfer functions must satisfy the Paley – Wiener criterion [45], given by ð1 ln jH(v)j dv , 1 (21:3:4) 2 1 1 þ v Nevertheless, these transfer functions can be approximated by physically realizable transfer functions. Example 21.3.1 A random telegraph wave X(t) with mean mX ¼ 0 and autocorrelation function RX(t) ¼ Ke 2ajtj is multiplied with a carrier signal cos(vct þ F), where F is a random phase independent of X(t) and K ¼ 2, a ¼ 3, and vc ¼ 5p. We will calculate the statistics of the output Y(t) ¼ X(t)cos(vct þ F) and devise a scheme to retrieve the input signal. Using the independence of X(t) and F, we have mY ¼ mX E[cos(vct þ F)] ¼ 0 since mX and E[cos(vct þ F)] ¼ 0. Again, from the independence of X(t) and F, the autocorrelation function RY (t) can be calculated as follows: RY (t) ¼ E½Y(t)Y(t þ t) ¼ E½X(t)X(t þ t)E½cos (vc t þ F) cos½vc (t þ t) þ F 1 ¼ E½X(t)X(t þ t) E½cos (vc t) þ cos½v0 (2t þ t) þ 2F 2 1 K ¼ RX (t) cos (vc t) ¼ eajtj cos (vc t) 2 2

594

Random Processes and Linear Systems

FIGURE 21.3.2

Hence the random process Y(t) is also stationary. The two autocorrelation functions RX(t) and RY (t) are shown in Fig. 21.3.2. The psd SX(v) of the input process X(t) is given by

SX (v) ¼

2Ka a2 þ v2

From the form of RY (t), we conclude that Y(t) is a bandpass signal, and the psd SY (v) of the output process Y(t) is given by 1 SX (v) p½d(v þ vc ) þ d(v vc ) 2p 2 1 ¼ ½SX (v þ vc ) þ SX (v vc ) 4 2Ka 1 1 ¼ þ 4 a2 þ (v þ vc )2 a2 þ (v vc )2

SY (v) ¼

The ﬁrst step in retrieving the original signal X(t) is to multiply Y(t) by a carrier cos(vct þ Q) where Q is another random phase angle independent of Y(t). Thus, Z(t) ¼ Y(t)cos(vct þ Q), and the autocorrelation RZ(t) is given by RZ (t) ¼ E½Z(t)Z(t þ t) ¼ E½Y(t)Y(t þ t)E½cos (vc t þ Q) cos½vc (t þ t) þ Q 1 ¼ E½Y(t)Y(t þ t) E½cos (vc t) þ cos½vc (2t þ t) þ 2Q 2 1 K Keajtj ¼ RY (t) cos (vc t) ¼ eajtj cos2 (vc t) ¼ ½1 þ cos (2vc t) 8 2 4 The two autocorrelation functions RY(t) and RZ(t) are shown in Fig. 21.3.3.

21.3

Linear Filters

595

FIGURE 21.3.3

Clearly, Z(t) is also a stationary process. Its psd is given by 1 SY (v) p½d(v þ vc ) þ d(v vc ) 2p 2 1 ¼ ½SY (v þ vc ) þ SY (v vc ) 4 Ka 1 1 2 ¼ þ þ 8 a2 þ (v þ 2vc )2 a2 þ (v 2vc )2 a2 þ v2

SZ (v) ¼

The power spectral densities SX(v), SY(v), and SZ(v) are shown in Fig. 21.3.4. The maximum value of SX(v) occurs at zero frequency and is equal to SX;max ¼ SX (v)jv¼0 ¼

FIGURE 21.3.4

4 3

596

Random Processes and Linear Systems

The maximum value of SY (v) occurs at frequencies vc ¼ +5p and is equal to, SY, max ¼ SY (v)jv¼+5p ¼

2 9 þ 50p2 ¼ 0:3364 3 9 þ 100p2

The value of SZ(v) occurring at zero frequency is equal to SZ (v)jv¼0 ¼

1 9 þ 50p2 ¼ 0:1682 3 9 þ 100p2

and the values of SZ(v) occurring at frequencies of +10p are equal to SZ (v)jv¼+10p ¼

1 81 þ 3150p2 þ 10,000p4 ¼ 0:085 3 (9 þ 100p2 )(9 þ 400p2 )

As shown in Fig. 21.3.4, passing the random process Z(t) through a lowpass ﬁlter of bandwidth 2vc and amplitude equal to 1 9 þ 50p2 ¼ 0:1682 3 9 þ 100p2 multiplied by a suitable gain factor will reconstruct the original signal X(t). Matched Filters

In the previous section we discussed basic ﬁltering methods where a causal transfer function H(v) was used to shape an input random process X(t) to any desired output random process Y(t). We will take this process one step further where we want to design an optimum ﬁlter for any given signal corrupted by noise with known statistics. There are three main factors in designing optimum systems: 1. Speciﬁcations of the input or a minimum knowledge of the characteristics of the input. We will assume that the input to the ﬁlter consists of a known deterministic signal s(t) with an additive stationary Gaussian noise N(t). The autocorrelation of the noise is RN (t) with psd FN(v): X(t) ¼ s(t) þ N(t),

E½N(t)N(t þ t) ¼ RN (t)

(21:3:5)

2. The nature of the impulse response h(t) such as linear time-invariant, linear time-varying, nonlinear, or causal, must be speciﬁed. We will consider linear time-invariant impulse responses, not necessarily causal. In this case the transfer function will be given by H(v). 3. The criterion of optimality has to speciﬁed. Some of the criteria of optimality similar the criteria of estimation discussed in Section 18.1 are (a) Maximum SNR (b) Minimum mean-square error (c) Bayesian optimality criteria such as maximum a posteriori and maximum likelihood The criterion of optimality chosen will be maximizing the signal-to-noise ratio (SNR). In the schematic shown in Fig. 21.3.5, the signal s(t) is corrupted with an additive stationary Gaussian noise N(t) with autocorrelation function RN(t). The input to the system is X(t), given by Eq. (21.3.5). We have to ﬁnd the optimum impulse response h(t), not necessarily causal, such that the output SNR go(t) at the time of observation T is maximized.

21.3

Linear Filters

597

FIGURE 21.3.5

Since the system is linear, the output signal Y(t) will be given by Y(t) ¼ so (t) þ No (t)

(21:3:6)

where the output signal so(t) and the output noise No(t) will be convolutions of the noncausal impulse response h(t) with the input signal s(t) and the input noise N(t). Or ð1 ð1 so (t) ¼ s(t)h(t t)dt : No (t) ¼ N(t)h(t t)dt (21:3:7) 1

1

The instantaneous input and output SNRs gi(t) and go(t) can be deﬁned as follows: gi (t) ¼

js2 (t)j js2 (t)j js2o (t)j js2o (t)j ¼ ; g ¼ (t) ¼ o E½N 2 (t) RN (0) E½No2 (t) RNo (0)

(21:3:8)

Since So (v) ¼ S(v)H(v), we can express so(t) as ð 1 1 so (t) ¼ S(v)H(v)e jvt dv 2p 1

(21:3:9)

From Eq. (21.3.9) we can relate the output noise psd FNo (v) to the input noise psd FN(v) as FNo (v) ¼ jH(v)j2 FN (v) We can express RNo (0) in terms of Eq. (21.3.10) as ð ð 1 1 1 1 2 jvt jH(v)j FN (v)e dv ¼ jH(v)j2 FN (v)dv RN (0) ¼ 2p 1 2p 1 t¼0

(21:3:10)

(21:3:11)

Substituting Eqs. (21.3.10) and (21.3.11) in the output SNR go(t) in Eq. (21.3.8), we have 1 Ð1 Ð1 2 2 jvt p 1 1 S(v)H(v)e jvt dv 1 S(v)H(v)e dv 2 Ð1 ¼ (21:3:12) go (t) ¼ 1 Ð 1 2 2p 1 jH(v)j2 FN (v)dv 2 p 1 jH(v)j FN (v)dv The problem now is to ﬁnd the transfer function H(v) in Eq. (21.3.12) that maximizes the output SNR go(t) without regard to causality. We will address the question of causality later. The Schwarz inequality [Eq. (14.5.2)] modiﬁed in the frequency form given by Eq. (21.3.13) will be used to solve for H(v): 2 ð 1 ð1 ð1 2 A(v)B (v)dv jA(v)j dv jB(v)j2 dv (21:3:13) 1

1

1

with equality holding when A(v) ¼ KB(v). We deﬁne the following quantities: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A(v) ¼ H(v) FN (v)

and

S(v)e jvt B (v) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ FN (v)

(21:3:14)

598

Random Processes and Linear Systems

Substituting Eq. (21.3.14) in Eq. (21.3.13), Schwartz’ inequality becomes 2 2 ð 1 ð 1 ð 1 S(v) jvt 2 dv H(v)S(v)e dv H (v)FN (v)FN (v)dv 1 1 1 FN (v) or Ð1 2 ð 1 jvt S(v) 2 1 H(v)S(v)e dv Ð1 dv 2 1 FN (v) 1 H (v)FN (v)dv

(21:3:15)

Substituting Eq. (21.3.15) in Eq. (21.3.12), we obtain Ð1 2 2 ð 1 1 S(v)H(v)e jvt dv 1 1 S(v) go (t) ¼ dv Ð 2p 1 H(v) 2 FN (v)dv 2p 1 FN (v)

(21:3:16)

1

The equality holds when A(v) ¼ KB(v), or pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ S (v)ejvt H(v) FN (v) ¼ K pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ FN (v)

and

S (v)ejvt Hopt (v) ¼ K FN (v)

(21:3:17)

If T is the time of observation, then the optimum transfer function will be given by S (v)ejvT Hopt (v) ¼ K FN (v)

(21:3:18)

where K is an arbitrary real gain constant. Since Hopt(v) is matched to the input signal, it is called a matched ﬁlter. Under these optimum conditions the maximum output SNR go;max for the time of observation T becomes go, max

1 ¼ 2p

2 ð 1 S(v) 1

FN (v)

dv

(21:3:19)

Simpliﬁcations in Eq. (21.3.18) can result if the noise N(t) is white with psd FN(v) ¼ N0/2 and the gain constant is adjusted to be equal to K ¼ N0/2. In this case the optimum transfer function is given by Hopt (v) ¼ S (v)ejvT

(21:3:20)

The optimum impulse response is the IFT of Hopt(v) and is given by hopt (t) ¼ s(T t)

(21:3:21)

This equation shows clearly that the optimum output impulse response hopt(t) is matched to the input signal s(t). The autocorrelation of the signal s(t) is deﬁned by ð1 s(t t)s(t)dt (12:3:22) Rs (t) ¼ s(t) s( t) ¼ 1

Taking FT on both sides of Eq. (12.3.22), we obtain FT½Rs (t) ¼ Fs (v) ¼ S(v)S (v) ¼ jS(v)j2

(12:3:23)

21.3

Linear Filters

599

jS(v)j2 is called the energy spectral density. The autocorrelation function Rs(t) can be given as the IFT of jS(v)j2: ð 2 1 1 S(v) e jvt dv Rs (t) ¼ (12:3:24) 2p 1 The optimum output signal spectrum So(v) under the condition K ¼ N0/2 is given by the product So (v) ¼ S(v)Hopt (v) ¼ S(v)S (v)ejvT ¼ jS(v)j2 ejvT Taking the IFT of So(v), the optimum output signal so(t) is given by ð 2 1 1 S(v) ejvT e jvt dv ¼ Rs (t T), t 0 so (t) ¼ 2p 1

(21:3:25)

(21:3:26)

Thus, the optimum output signal so(t) under the conditions of additive white noise is the autocorrelation of the input signal Rs(t) shifted by the time of observation T. The corresponding maximum SNR is given by ð 1 2 1 jS(v)j2 dv (21:3:27) go, max ¼ 2p N0 1 Ð1 Ð1 Using Parseval’s theorem (1/2p) 1 jS(v)j2 dv ¼ 1 s2 (t)dt in Eq. (21.3.27), we obtain ð 2 1 2 2Es s (t)dt ¼ (21:3:28) go, max ¼ N0 N0 1 where Es is the input energy. Summary of Matched-Filters Properties

Some of the properties of matched ﬁlters under the assumptions that (1) the additive noise N(t) is white with psd SN(v) ¼ N0 =2 and (2) the gain constant K ¼ N0/2 can be summarized as follows: 1. The Fourier spectrum So(v) of the output signal so(t) is equal to the energy spectral density jS(v)j2 multiplied by a delay factor e 2jvT of the input signal s(t). This can be shown from Eq. (21.3.20) as follows: So (v) ¼ S(v)Hopt (v) ¼ S(v)S (v)ejvT ¼ jS(v)j2 ejvT

(21:3:29)

2. The output response so(t) is equal to the autocorrelation Rs(t) of the input signal shifted in time by T as follow: so (t) ¼ Rs (t T),

t0

3. From Eq. (21.3.28), the optimum SNR go,max is ð ð 1 1 Fs (v) 1 2 1 1 2Es dv ¼ go, max ¼ Fs (v)dv ¼ N0 2p 1 FN (v) 2p No 2p 1

(21:3:26)

(21:3:30)

4. Causality: If the original signal s(t) is time limited to (0,T ), then the matched ﬁlter given by s(T 2 t) is causal. In the following three examples (Examples 21.3.2 –21.3.4) the additive noise N(t) is white with psd SN(v) ¼ N0/2 and the gain constant K ¼ N0/2.

600

Random Processes and Linear Systems

The signal s(t) is a rectangular pulse deﬁned by

Example 21.3.2 (Rectangular Pulse) s(t) ¼

A, 0,

0,t,T otherwise

From the tables, the FT of the signal s(t) is given by S(v) ¼ AT

sin (vT=2) jvT=2 e (vT=2)

Hence, from Eq. (21.3.20) the optimum transfer function is Hopt (v) ¼ AT

sin (vT=2) jvT=2 jvT sin (vT=2) jvT=2 e ¼ AT e e (vT=2) (vT=2)

and the impulse response hopt(t) ¼ s(T 2 t) ¼ s(t) and the matched ﬁlter are the same as the input signal. The autocorrelation function of the input from Eq. (12.3.22) is 2 A (T þ t), T , t 0 Rs (t) ¼ 0,tT A2 (T t), The output so(t) is obtained from Eq. (12.3.26) as follows: 2 0,tT A t, Rs (t T) ¼ A2 (T t), T , t 2T From Eq. (21.3.26), the maximum output SNR is go, max

2 ¼ N0

ð1

2 s (t)dt ¼ N0 1 2

ðT 0

A2 dt ¼

2A2 T N0

Example 21.3.3 (Sinusoidal Pulse) The input signal s(t) is a time-limited sinusoidal pulse with energy Es ¼ 1 as given below: 2 rﬃﬃﬃﬃ 2 2pn sin (v0 t), 0 t T, v0 ¼ 6 s(t) ¼ 4 T T 0, otherwise The signal s(t) is shown in Fig.(21.3.6a). It is simpler to approach the problem from the time domain. From Eq. (21.3.21) the optimum impulse response is given by 2 rﬃﬃﬃﬃ 2 2pn v0 ¼ sin½v0 (T t), 0 t T, 6 hopt (t) ¼ s(T t) ¼ 4 T T 0, otherwise The optimum output signal so(t) is given by the convolution integral so (t) ¼ s(t) s(T t) ¼

ð1 s(t)s(T t þ t)dt 1

21.3

Linear Filters

FIGURE 21.3.6

The convolution integral can be written as follows: 2 Ð 2 t 6 T 0 sin (v0 t) sin½v0 (T t þ t)dt, 6 6 so (t) ¼ 6 2 Ð T sin (v0 t) sin½v0 (T t þ t)dt, 6 4 T tT 0, The integrals are evaluated, and the results are 2 1 ½2v0 t cos (v0 t) sin (v0 t), 6 v0 T 6 6 so (t) ¼ 6 1 ½v (2T t) cos (v0 t) þ sin (v0 t), 6 4 v0 T 0 0, The output signal so(t) is shown in Fig. (21.3.6b).

0tT T t 2T otherwise

0tT T t 2T otherwise

601

602

Random Processes and Linear Systems

It is also instructive to evaluate the output so(t) as the shifted version of the autocorrelation of the input signal s(t) as given in Eq. (21.3.26). The autocorrelation Rs(t) of the input s(t) is given by the following integrals: 2 Ð 2 tþT 6 T 0 sin½(v0 (t t) sin (v0 t)dt, T t 0 6 6 Rs (t) ¼ 6 2 Ð T 0tT 6 t sin½v0 (t t) sin (v0 t)dt, 4T 0, otherwise In this example the integrand in the integral shown above is the same as those in so(t) except that the limits are different. Evaluation of these integrals results in 2 1 ½v (T þ t) cos (v0 t) sin (v0 t), T t 0 6 v0 T 0 6 6 Rs (t) ¼ 6 1 ½v (T t) cos (v0 t) þ sin (v0 t), 0tT 6 4 v0 T 0 0,

otherwise

and indeed Rs(t 2 T ) ¼ so(t) and its graph also superpose the graph of so(t) exactly in Fig. 21.3.6, thus validating property 2 of Eq. (21.3.26).

Example 21.3.4 signal, given by

In this example, the input s(t) is a more complicated time-limited " s(t) ¼

et sin 0,

pt 2

,

0t4 otherwise

The psd of noise N(t) is given by N0/2 ¼ 2. The time of observation T ¼ 4 The FT of s(t) is given by 2p S(v) ¼ ½1 e4(1þjv) 4(1 þ jv)2 þ p2 From Eq. (21.3.20) the optimum ﬁlter Hopt(v) is given by 2p Hopt (v) ¼ S (v)ejvT ¼ ½1 e4(1jv) e4jv 4(1 jv)2 þ p2 From Eq. (21.3.25) the output spectrum So(v) is given by So (v) ¼ jS(v)j2 ejvT ¼

4p2 ½1 2e4 cos (4v) þ e 8 4jv e 16(1 þ v2 )2 þ 8p2 (1 v2 ) þ p4

The amplitude spectra jS(v)j and jHopt(v)j are the same and are shown in Fig. 21.3.7 along with the amplitude spectrum of jSo(v)j. The impulse response hopt(t) corresponding to the transfer function Hopt(v) can be obtained from Eq. (21.3.21) as ð 1 1 S (v)ejv(4t) dv hopt (t) ¼ 2p 1 ð 1 1 p(4 t) S (v)e jv(4t) dv ¼ s(4 t) ¼ e(4t) sin , 0t4 ¼ 2p 1 2

21.3

Linear Filters

603

FIGURE 21.3.7

It is easier to obtain the output signal so(t) from the shifted version of the autocorrelation Rs(t) of the input s(t) rather than from the IFT of So(v). Rs(t) can be obtained from the following integrals: 2 Ð Tþt

pt p(t t) sin sin dt, 6 0 e 2 2 6 6Ð

T Rs (t) ¼ 6 6 e2tþt sin pt sin p(t t) dt, 6 t 2 2 4 2tþt

0,

T t 0 0tT otherwise

After some involved algebra, these integrals can be evaluated as

pt

pt ½e(8þt) þ et þ p cos ½e(8þt) et , 2 2 6 p 6

pt

pt 6 Rs (t) ¼ (8t) t 2 sin ½e ½e(8t) et , þ e p cos 4(4 þ p2 ) 6 4 2 2 0, 2

2 sin

4 t 0 0 t 4 otherwise

and the output signal so(t) can be evaluated from the convolution integral or by shifting the autocorrelation Rs(t) by T ¼ 4: so (t) ¼ s(t) s(4 t) ¼ Rs (t 4) The signal s(t), the optimum impulse response hopt(t), the autocorrelation Rs(t), and the output signal so(t) are shown in Fig. 21.3.8. The maximum SNR from Eq. (21.3.19) is go, max ¼

1 2p

ð1

jS(v)j2 1 2 dv ¼ 2p N0 1 SN (v)

The input energy Es ¼ go,max ¼ 0.0889.

ÐT 0

ð1 1

jS(v)j2 dv ¼

2 N0

ðT 0

s2 (t)dt ¼

2Es N0

s2 (t)dt ¼ 0:17784, and with N0/2 ¼ 2, the maximum SNR is

604

Random Processes and Linear Systems

FIGURE 21.3.8

Example 21.3.5 (Nonwhite Gaussian Noise) In this example the signal s(t) and the psd FN(v) of the nonwhite stationary noise N(t) are given by 2 3 t 3t

e 2 e 2 , 4 4 s(t) ¼ 0,

t0

FN (v) ¼

t,0

4 1 þ 4v2

The time of observation T is equal to 5, and the dc value of the optimum transfer function Hopt(0) ¼ 1. The Fourier spectrum of s(t) can be obtained from the tables as S(v) ¼

3 3 ¼ (1 þ 2jv)(3 þ 2jv) 3 4v2 þ 8jv

The transfer function Hopt(v) of the matched ﬁlter is found from Eq. (21.3.18) as Hopt (v) ¼ K

S (v)ejvT 3ejv5 4 þ v2 ¼K 2 jFN (v)j 3 4v 8jv 4

Since H(0) ¼ 1, we obtain K ¼ 1, and Hopt(v) is Hopt (v) ¼

3(4 þ v2 )ejv5 4(3 4v2 8jv)

The Fourier spectrum So(v) of the optimum output signal so(t) is given by So (v) ¼ S(v)Hopt (v) ¼

jS(v)j2 ejv5 9(4 þ v2 )e5jv ¼ FN (v) 4(16v4 þ 40v2 þ 9)

The amplitude spectra jS(v)j, jHopt(v)j, and jSo(v)j are shown in Fig. 21.3.9.

21.3

Linear Filters

605

FIGURE 21.3.9

The optimum impulse response hopt(t) and the output time function so(t) are obtained from the IFTs of their respective Fourier spectra Hopt(v) and So(v) and are given by 45 (t5)=2 21 3(t5)=2 3 e e u(5 t) d(5 t) 64 64 16 135 (t5)=2 21 3(t5)=2 so (t) ¼ u(5 t) e e 512 512 135 (5t)=2 21 3(5t)=2 þ e e u(t 5) 512 512

hopt (t) ¼

The input signal s(t), the impulse response hopt(t), and the output time function so(t) are shown in Fig. 21.3.10.

FIGURE 21.3.10

606

Random Processes and Linear Systems

The optimum SNR is obtained from Eq. (21.3.19) and is given by ð ð 1 1 jS(v)j2 1 1 9(4 þ v2 ) dv ¼ dv go, max ¼ 2p 1 FN (v) 2p 1 4(16v4 þ 40v2 þ 9) 1 135 1 21 57 ¼ tan (1) tan1 (1) ¼ 2p 128 128 256

B 21.4.

BANDPASS STATIONARY RANDOM PROCESSES

Review of Bandpass Signals

Bandpass signals form the backbone of signal transmission. The channel through which the signal is transmitted is bandwidth-limited to a band of frequencies centered about the carrier. If the bandwidth 2vB is much smaller than the carrier frequency vc, then such signals are called narrowband signals. They can be represented in three ways as 1. Natural Envelope Representation. This is the usual representation of a narrowband process and is given by s(t) ¼ a(t) cos½vc t þ u(t)

(21:4:1)

where a(t) (called the natural envelope) is a lowpass signal bandlimited to 2vB, vc is the carrier frequency, and u(t) is the phase angle. If the Fourier spectrum of a(t) is A(v), then the spectrum of the signal S(v) is given by 1 A(v) p½d(v þ vc ) þ d(v vc ) 2p 1 ¼ ½A(v þ vc ) þ A(v vc ) 2

S(v) ¼

(21:4:2)

The amplitude spectra of A(v) and S(v) are shown in Fig. 21.4.1. In Fig. 21.4.1 2vB , vc, and hence the signal s(t) is bandpass. The band of frequencies centered about vc is called the upper sideband, and the band centered about 2vc is called

FIGURE 21.4.1

21.4. Bandpass Stationary Random Processes

607

the lower sideband. Each of the sidebands carries the same information as the original lowpass signal s(t). 2. Quadrature Representation. The signal s(t) in Eq. (21.4.1) can be expanded and written as s(t) ¼ a(t) cos½vc t þ u(t) ¼ a(t)½cos (vc t) cos u(t) sin (vc t) sin u(t)

(21:4:3)

In Eq. (21.4.3) we can deﬁne the quadrature components sc(t) and ss(t) as sc (t) ¼ a(t) cos u(t), ss (t) ¼ a(t) sin u(t),

in phase quadrature

(21:4:4)

s(t) ¼ sc (t) cos u(t) ss (t) sin u(t)

(21:4:5)

and write Eq. (21.4.3) as

The envelope a(t) and the phase u(t) are obtained from the quadrature components as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 ss (t) 2 2 a(t) ¼ sc (t) þ ss (t); u(t) ¼ tan (21:4:6) sc (t) Since the signal is narrowband a(t) varies slowly compared to tc ¼ 2p=vc . 3. Analytic Signal Representation. Since both sidebands in a bandpass signal carry the same information, we should be able to transform the spectrum into positive frequencies. This is accomplished by using Hilbert transforms (HTs), [36]. The Hilbert transform s^ (t) of any signal s(t) is a time-domain transformation, deﬁned by ð1 s(t) 1 ¼ dt (21:4:7) s^ (t) ¼ s(t) pt p(t t) 1 under the assumptions, that (a) s(t) is square integrable and (b) the FT S(v) is minimum phase. The improper integral in Eq. (21.4.7) is evaluated in the sense of Cauchy principal value, deﬁned by (ð ) ð1 ð1 t1 s(t) s(t) s(t) dt ¼ lim dt þ dt 1!0 1 p(t t) tþ1 p(t t) 1 p(t t) The inverse Hilbert transform (IHT) is deﬁned by ð1 1 s^ (t) s(t) ¼ ^s(t) ¼ dt pt p(t t) 1

(21:4:8)

Since (1=pt) , jsgn(v), the FT of s^ (t) is obtained by using the convolution property as ^ FT{^s(t)} ¼ S(v) ¼ S(v)j sgn(v) We can also deﬁne the HT of the Fourier transform of S(v) as, ð1 1 S(p) ^^ ^ ¼ dp FT{S(v)} ¼ S(v) ¼ S(v) pv p(v p) 1

(21:4:9)

(21:4:10)

608

Random Processes and Linear Systems

^^ ^ In general, S(v) = S(v), but if s(t) is a real causal time function, then it can be shown that ^^ ^ S(v) ¼ S(v)sgn(v)

(21:4:11)

The analytic signal zs(t) corresponding to s(t) is deﬁned as zs ðtÞ ¼ sðtÞ þ j^sðtÞ

(21:4:12)

The Fourier spectrum of the analytic signal zs(t) can be obtained by using Eq. (21.4.9): Zs ðvÞ ¼ Sðv þ jðjÞSðvÞsgnðvÞ ¼ 2SðvÞuðvÞ

(21:4:13)

Thus the spectrum Zs(v) is causal in the frequency domain. In most problems it may be easier to ﬁnd the HT from the frequency representation of Eq. (21.4.9) rather than the convolution integral of Eq. (21.4.7). Example 21.4.1

We will ﬁnd the Hilbert transforms for the following functions:

(a) xc(t) ¼ a(t)cos(vct): The FT of xc(t) is Xc (v) ¼

1 1 A(v) p½d(v þ vc ) þ d(v vc ) ¼ ½A(v þ vc ) þ A(v vc ) 2p 2

and from Eq. (21.4.9) 1 j X^ c (v) ¼ ½A(v þ vc ) þ A(v vc )½jsgn(v) ¼ ½A(v þ vc ) A(v vc ) 2 2 Hence x^ c (t) ¼ a(t) sin (vc t) (b) xs(t) ¼ a(t) sin(vct): The FT of xs(t) is Xs (v) ¼

1 j A(v) jp½d(v þ vc ) d(v vc ) ¼ ½A(v þ vc ) A(v vc ) 2p 2

and from Eq. (21.4.9) j 1 X^ s (v) ¼ ½A(v þ vc ) A(v vc )½jsgn(v) ¼ ½A(v þ vc ) þ A(v vc ) 2 2 Hence x^ s (t) ¼ a(t) cos (vc t) (c) y(t) ¼ sin(vBt) sin(vct), vB vc: We will solve this problem from basics and show that the HT y^ (t) follows directly from the previous example (b). The

21.4. Bandpass Stationary Random Processes

Fourier spectrum Y(v) is obtained from the tables as 1 { jp(v þ vB ) d½v vB )} {jp½d(v þ vc ) d(v vc )} 2p p ¼ {d½v þ vB þ vc d½v þ vB vc 2

Y(v) ¼

d½v vB þ vc þ d½v vB vc } p ¼ {d½v þ (vc þ vB ) þ d½v (vc þ vB )} 2 p þ {d½v þ (vc vB ) þ d½v (vc vB )} 2 ^ From Eq. (21.4.9) the FT Y(v) is given by ^ Y(v) ¼ Y(v)( j sgn(v)) jp {d½v þ (vc þ vB ) d½v (vB þ vc )} 2 jp {d½v þ (vc vB ) d½v (vc vB )} ¼ 2 ¼

The functions Y(v), 2jsgn(v), and Y(v) are shown in Fig. 21.4.2

FIGURE 21.4.2

609

610

Random Processes and Linear Systems

TABLE 21.4.1 No.

Table of Hilbert transforms Time Function s(t)

Hilbert Transform s^ (t)

1. 2. 3.

cos(vct) sin(vct) a(t)cos(vct)

sin(vct) 2cos(vct) a(t)sin(vct)

4.

a(t)sin(vct)

2a(t)cos(vct)

sin (vc t) vc t 1 1þ t2 1 1 u t u tþ 2 2

1 cos (vc t) vc t t 1 þ t2 1 t þ 1=2 ln p t 1=2 1 pt 2pd(t)

5. 6. 7. 8. 9.

d(t) 1 t

Comments

a(t) bandlimited to 2vB, ,vc a(t) bandlimited to 2vB, ,vc

Hence the HT of y(t) is 1 y^ (t) ¼ { sin½(vc vB )t sin½(vc þ vB )t} 2 ¼ sin (vB t) cos (vc t) A result that can be obtained by substituting a(t) ¼ sin(vBt) in example (b) above. A summary of some Hilbert transforms is given in Table 21.4.1. Substituting s(t) ¼ a(t) cos½vc t þ u(t) from Eq. (21.4.1) and s^ (t) ¼ a(t) sin½vc t þ u(t) from Example 21.4.1 in Eq. (21.4.12) for the analytic signal, we can write zs (t) ¼ a(t){ cos½vc t þ u(t) j sin½vc t þ u(t)} ¼ a(t)e j½vc tþu(t) ¼ ½a(t)e ju(t) e jvc (t) ¼ s~ (t)e jvc t

(21:4:14)

where s~ (t) ¼ a(t)eju(t) is deﬁned as the complex envelope of the analytic signal zs(t). The bandpass signal s(t) can be written in terms of the analytic signal as s(t) ¼ Re½zs (t) ¼ Re½~s(t)e jvc t ¼ a(t) cos½vc t þ u(t)

(21:4:15)

Using the relationship Re[z] ¼ 12[z þ z ] we can rewrite Eq. (21.4.15) as s(t) ¼

1 s(t)e jvc t þ s (t)ejvc t 2

(21:4:16)

and its Fourier spectrum as 1 S(v) ¼ ½S(v vc ) þ S (v þ vc ) 2

(21:4:17)

The absolute value js˜(t)j of the complex envelope s˜(t) is the natural envelope a(t), and s˜(t) can be written in terms of the quadrature components of Eq. (21.4.4) as s~ (t) ¼ sc (t) þ jss (t)

(21:4:18)

21.4. Bandpass Stationary Random Processes

611

It should be noted that a(t), s˜(t), sc(t), and ss(t) are all lowpass signals, whereas s(t) and zs(t) are bandpass signals. Example 21.4.2 A narrowband bandpass signal X(t) with bandwidth 2vB is passed through a bandpass ﬁlter with bandwidth 2vB and impulse response h(t). We will show that the output of this system Y(t) can be represented by 1 Y(t) ¼ Re½zh (t) zX (t) 2 where zh(t) and zX(t) are the analytic signals corresponding to h(t) and X(t). We will now evaluate ð1 ð1 ^ ^ zh (t)zX (t)dt ¼ Re ½h(t) þ jh(t)½X(t) jX(t)dt Re 1

ð1 ¼

1

^ X(t)dt ^ ½h(t)X(t) þ h(t)

1

By Parseval’s theorem, we obtain ð1 ð 1 1 ^ ^h(t)X(t)dt ^ ¼ H(v)X^ (v)dv 2p 1 1 ð 1 1 j sgn(v)H(v) j sgn(v)X (v)dv ¼ 2p 1 ð 1 1 ¼ sgn2 (v)H(v) X (v)dv 2p 1 ð1 ð 1 1 ¼ H(v) X (v)dv ¼ h(t)X(t)dt 2p 1 1 Hence ð 1 Re

zh (t)zX (t)dt

ð1

ð1

¼2

1

h(t)X(t)dt ¼ 2 1

Re½zh (t)Re½zX (t)dt

(A)

1

We will evaluate the output Y(t) given by the convolution integral ð1 h(t)X(t t)dt Y(t) ¼ 1

^ and zX (t) ¼ X(t) þ jX(t) ^ we have and with zh (t) ¼ h(t) þ jh(t) ð1 Re½zh (t)Re½zX (t t)dt

Y(t) ¼ 1

Substituting the result (A) derived earlier in this equation, we obtain ð1 1 1 zh (t)zX (t t)dt ¼ Re½zh (t) zX (t) Y(t) ¼ Re 2 2 1 Example 21.4.3 A pulsed radiofrequency (RF) wave x(t) ¼ ApT=2 (t) cos(vc t), where pT=2 is a pulse of width T deﬁned by pT=2 ¼ u(t þ T=2) u(t T=2), is applied to an ideal bandpass ﬁlter shown as in Fig. 21.4.3.

612

Random Processes and Linear Systems

FIGURE 21.4.3

We want to ﬁnd the output y(t) using Hilbert transform techniques. The HT of the input can be obtained form the HT table (Table 21.4.1) as x^ (t) ¼ ApT=2 (t) sin (vc t), the analytic signal and the complex envelope corresponding to x(t) are zx (t) ¼ ApT=2 (t)e jvc t ;

x~ (t) ¼ ApT=2 (t)e jt 0

and the FT of zx (t) is Zx (v) ¼ AT

sin½(v vc )T=2 (v vc )T=2

The equivalent lowpass ﬁlter for H(v) can be obtained by retaining the positive half of ~ H(v) and backshifting it to the origin. Thus H(v) and the corresponding complex ~ impulse response h(t) are given by ~ H(v) ¼ u(v þ vB ) u(v vB )e jv0 ;

~ ¼ vB sin (vB t) h(t) p vB t

The complex envelope y~ (t) of the output y(t) is the convolution given by ð T=2 vB sin (vB t) ~ ¼ ApT=2 (t) vB sin (vB t) ¼ A PT=2 ðt tÞdt y~ (t) ¼ x~ (t) h(t) p vB t vB t T=2 p Ðx Using the sine integral Si(x) ¼ 0 ðsin(t)=tÞdt, the preceding equation can be written as A T T Si vB t þ y(t) ¼ Si vB t p 2 2 The RF pulse x(t) is shown in Fig. 21.4.4a, and y(t) is shown in Fig. 21.4.4b. Bandpass Stationary Random Processes

A stationary random process N(t) is called a narrowband bandpass process if the power spectral density (psd) SN(v) satisﬁes two conditions: 1. SN(v) is zero outside a band of frequencies centered around the carrier frequency +vc. 2. The width 2vB of the band of frequencies is very much smaller than the carrier frequency vc. These conditions are similar to those of deterministic narrowband signals except that the Fourier spectrum has been replaced by the power spectral density. Similar to the deterministic process, these narrowband stationary processes can also be represented in three ways.

21.4. Bandpass Stationary Random Processes

613

FIGURE 21.4.4

The natural envelope representation is N(t) ¼ W(t) cos½vc t þ Q(t)

(21:4:19)

where W(t) is a lowpass stationary random process with psd SW(v) and bandwidth 2vB and Q(t) is a random phase. The quadrature representation can be obtained by expanding Eq. (21.4.19) as N(t) ¼ W(t)½cos (vc t) cos½Q(t) sin (vc t) sin½Q(t) and rewriting it as N(t) ¼ Nc (t) cos (vc t) Ns (t) sin (vc t)

(21:4:20a)

where Nc (t) ¼ W(t) cos½Q(t);

Ns (t) ¼ W(t) sin½Q(t)

(21:4:20b)

are the lowpass stationary quadrature components. The analytic signal representation is N(t) ¼ Re½zN (t) ¼ Re½N(t)e jvc t

(21:4:21)

614

Random Processes and Linear Systems

~ where zN (t), the analytic signal corresponding to N(t), and N(t), the complex envelope in terms of the quadrature components Nc(t) and Ns(t), are given by ^ zN (t) ¼ N(t) þ jN(t) ~ ¼ Nc (t) þ jNs (t) N(t)

(21:4:22)

^ is Hilbert transform of N(t). where N(t) Quadrature Representation

We will now investigate the quadrature representation of Eq. (21.4.20a). Since N(t) is stationary, the correlation functions of any two stationary random processes N(t) and M(t) are governed by the symmetry conditions E½N(t)N(t þ t) ¼ RN (t) ¼ E½N(t þ t)N(t) ¼ RN ( t) E½N(t)M(t þ t) ¼ RNM (t) ¼ E½M(t þ t)N(t) ¼ RMN ( t)

(21:4:23)

We will determine the autocorrelation functions RN(t), RNc (t), and RNs (t) and the crosscorrelation functions RNc Ns (t) and RNs Nc (t): RN (t) ¼ E{½Nc (t) cos (vc t) Ns (t) sin (vc t) ½Nc (t þ t) cos½vc (t þ t) Ns (t þ t) sin½vc (t þ t)} ¼ RNc (t) cos (vc t) cos½vc (t þ t) þ RNs (t) sin (vc t) sin½vc (t þ t) RNc Ns (t) cos(vc t) sin½vc (t þ t) RNs Nc (t) sin(vc t) cos½vc (t þ t)

(21:4:24)

Using trigonometric identities, we obtain 1 RN (t) ¼ ½RNc (t) þ RNs (t) cos (vc t) 2 1 þ ½RNc (t) RNs (t) cos½vc (2t þ t) 2 1 ½RNc Ns (t) RNs Nc (t) sin (vc t) 2 1 ½RNc Ns (t) þ RNs Nc (t) sin½vc (2t þ t) 2

(21:4:25)

Since there can be no dependence on t for a stationary process, the following conditions must be satisﬁed by N(t) in Eq. (21.4.25): RNc (t) ¼ RNs (t) RNc Ns (t) ¼ RNs Nc (t)

(21:4:26)

Substituting Eqs. (21.4.6) in Eq. (21.4.25), we obtain the autocorrelation RN(t) also in quadrature form as RN (t) ¼ RNc(t) cos(vc t) RNc Ns (t) sin (vc t)

(21:4:27)

^ of the HT of N(t), but it is In a similar manner we can also derive the autocorrelation N(t) easier to use the HT table (Table 21.4.1) and write RN^ (t) ¼ RNc (t) sin(vc t) þ RNc Ns (t) cos (vc t)

(21:4:28)

21.4. Bandpass Stationary Random Processes

615

Equations (21.4.27) and (21.4.28) can be combined as a matrix and written as RN (t) RNc (t) cos (vc t) sin (vc t) (21:4:29) ¼ RN^ (t) RNc Ns (t) sin (vc t) cos (vc t) and the correlation functions of quadrature components Nc(t) and Ns(t) can be given in ^ as terms of the correlation functions of N (t) and N(t) RNc (t) cos (vc t) sin (vc t) RN (t) (21:4:30) ¼ RNc Ns (t) sin (vc t) cos (vc t) RN^ (t) Equations (21.4.29) and (21.4.30) are orthogonal transformations.

Analytic Signal Representation

We will investigate the analytic signal representation of the narrowband bandpass stationary random process N(t). The analytic signal corresponding to N(t) is jvc t ^ ¼ N(t)e ~ zN (t) ¼ N(t) þ jN(t)

(21:4:31)

~ will be given by and the complex envelope N(t) jvc t ^ ~ ¼ Nc (t) þ jNs (t) ¼ zN (t)ejvc t ¼ ½N(t) þ jN(t)e N(t)

(21:4:32)

where Nc(t) and Ns(t) are the lowpass quadrature components given by Eq. (21.4.20b). The quadrature components Nc(t), Ns(t) can be expressed in terms of the components of ^ by expanding Eq. (21.4.32) as the analytic signal N(t), N(t) ^ (vc t) j sin (vc t) Nc (t) þ jNs (t) ¼ ½N(t) þ jN(t)½cos ^ sin (vc t) þ j½N(t) ^ cos (vc t) N(t) sin (vc t) ¼ N(t) cos (vc t) þ N(t) and equating real and imaginary parts to obtain, in matrix form Nc (t) sin (vc t) Nc (t) cos (vc t) ¼ Ns (t) sin (vc t) cos (vc t) N^ s (t)

(21:4:33)

Inverting the matrix equation [Eq. (21.4.33)] the components of the analytic signal, N(t), ^ are expressed in terms of the quadrature components N(t) N(t) cos (vc t) sin (vc t) Nc (t) ¼ (21:4:34) ^ N(t) cos (vc t) Ns (t) sin (vc t) Equations (21.4.33) and (21.4.34) are again orthogonal transformations.

Properties of Quadrature Components

Without loss in generality we shall assume that N(t) is a zero mean stationary narrowband bandpass Gaussian process. ^ 1. Since N(t) is Gaussian, N(t), which is linear transformation, is also a zero mean stationary narrowband bandpass Gaussian process. ^ and 2. From Eq. (21.4.33), Nc(t) and Ns(t) are linear transformations of N(t) and N(t) hence they are also stationary Gaussian processes.

616

Random Processes and Linear Systems

3. Because of the stationarity of N(t), Eq. (21.4.26) must be satisﬁed. RNc (t) ¼ RNs (t) RNc Ns (t) ¼ RNs Nc (t)

(21:4:26)

4. Since RNc (t) ¼ RNs (t), Nc (t) and Ns (t) have the same psd SNc (v) ¼ SNs (v). 5. Substituting t ¼ 0 in Eq. (21.4.27), we obtain RN (0) ¼ RNc (0) and from Eq. (21.4.26) RN (0) ¼ RNc (0) ¼ RNs (0) ¼ s2N

(21:4:35)

Thus, the original random process N(t) and its quadrature components Nc(t) and Ns(t) have the same variance s2N . 6. The autocorrelation function RN (t) will be computed: RN (t) ¼ E{½Nc (t) þ jNs (t)½Nc (t þ t) jNs (t þ t)} ¼ RNc (t) þ RNs (t) j½RNc Ns (t) RNs Nc (t) ¼ 2½RNc (t) jRNc Ns (t) ½from Eq. (21:4:26)

(21:4:36)

Multiplying Eq. (21.4.36) by ejvc t , we obtain RN (t)ejvc t ¼ 2½RNc (t) jRNc Ns (t)½cos (vc t) j sin (vc t) ¼ 2½RNc (t) cos (vc t) RNc Ns sin (vc t) 2j½RNc (t) sin (vc t) þ RNc Ns (t) cos (vc t)þ ¼ 2½RN (t) jRN^ (t)

½ from Eq. (21:4:29)

(21:4:37)

Taking the real parts of both sides of Eq. (21.4.37) results in 1 RN (t) ¼ Re½RN (t)ejvc t 2

(21:4:38)

7. The symmetry condition of any cross-correlation function is RNc Ns (t) ¼ RNs Nc ( t), and together with the stationarity condition of Eq. (21.4.26), RNc Ns (t) ¼ RNs Nc (t) implies that RNc Ns (t) ¼ RNs Nc ( t) ¼ RNs Nc (t)

(21:4:39)

Hence the cross-correlation function RNc Ns (t) is odd. 8. The autocorrelation function RN (t) has Hermitian symmetry. From Eq. (21.4.36) and the fact that RNc Ns (t) is odd, from Eq. (21.4.39), we obtain RN ( t) ¼ 2½RNc (t) þ jRNc Ns (t) ¼ 2½RNc (t) jRNc Ns (t), ¼ RN (t) showing Hermitian symmetry.

RNc Ns (t) odd function (21:4:40)

21.4. Bandpass Stationary Random Processes

617

9. The cross-spectral density SNc Ns (v) is purely imaginary, as shown below: ð1 ð1 jvc t SNc Ns (v) ¼ RNc Ns (t)e dt ¼ RNc Ns ( t)ejvc t dt 1 ð1

1

¼ 1

RNc Ns (t)e jvc t dt ¼ SNc Ns (v)

(21:4:41)

The only way that Eq. (21.4.41) can be satisﬁed is when Re½SNc Ns (v) ¼ 0. 10. Since RNc Ns (t) is odd from, Eq. (21.4.39), we have RNc Ns (0) ¼ 0. Hence E½Nc (t)Ns (t) ¼ 0; showing that the quadrature components Nc(t) and Ns(t) are orthogonal. Since they are zero mean, they are uncorrelated, and because they are Gaussian, they are also independent. 11. Structure of SN (v) and SN (v). Since RN (t) has Hermitian symmetry, from Eq. (21.4.40), its FT SN (v) is a real-valued function of the frequency v. Using the relationship Re½z ¼ (z þ z )=2 in Eq. (21.4.38), we can write 1 RN (t) ¼ ½RN (t)ejvc t þ RN (t)e jvc t 4

(21:4:42)

Taking FT on both sides of Eq. (21.4.42), we have 1 SN (t) ¼ ½SN (v vc ) þ SN (v þ vc ) 4

(21:4:43)

12. Structure of SNc (v) ¼ SNc (v). We will assume that the bandwidth 2vB of SNc ðvÞ is governed by –vB vc. vB. From Eq. (21.4.30) we have RNc (t) ¼ RNs (t) ¼ RN (t) cos(vc t) þ RN^ (t) sin(vc t)

(21:4:44)

Taking FT on both sides of Eq. (21.4.14), we obtain 1 SN (v) p½d(v þ vc ) þ d(v vc ) 2p j SN (v)sgn(v) jp½d(v þ vc ) (v vc ) 2p 1 ¼ SN (v þ vc )½1 þ sgn(v þ vc ) 2 1 þ SN (v vc )½1 sgn(v vc ), vB vc vB 2 (21:4:45)

SNc (v) ¼ SNs (v) ¼

In Eq. (21.4.45) the terms containing the signum functions are given by 2, v . vc 0, v . vc ; ½1 sgn(v vc ) ¼ (21:4:46) ½1 þ sgn(v þ vc ) ¼ 0, v vc 2, v vc Substituting Eq. (21.4.46) in Eq. (21.4.45), we obtain SNc (v) ¼ SNs (v) ¼ SN (v þ vc )u(v þ vc ) þ SN (v vc )u½(v vc ), vB vc vB

(21:4:47)

The development of the psd SNc (v) from the psd SN(v) is shown in Fig. 21.4.5.

618

Random Processes and Linear Systems

FIGURE 21.4.5 Natural Envelope Representation

The natural envelope representation of a narrowband bandpass stationary random process is given by N(t) ¼ W(t) cos½vc t þ Q(t)

(21:4:19)

where W(t) in terms of the quadrature components Nc(t) and Ns(t) is given by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ W(t) ¼ Nc2 (t) þ Ns2 (t) (21:4:48) and the phase angle Q(t), by Q(t) ¼ tan1

Ns (t) Nc (t)

(21:4:49)

If N(t) is Gaussian-distributed, then from property 2 above, the quadrature components Nc(t) and Ns(t) are also Gaussian-distributed. We will assume that they are zero mean. Under these conditions we want to ﬁnd the distributions of the natural envelope W(t) and the phase angle Q(t). By property 10, Nc(t) and Ns(t) are independent, and by property 5, their variance s2N is the same as the variance of N(t). Hence their joint

21.4. Bandpass Stationary Random Processes

619

density can be given as follows: fNc Ns (nc , ns ) ¼

1 2 2 2 e(1=2)½(nc þns )=sN 2 2psN

(21:4:50)

The transformation between Nc(t) and Ns(t) to W(t) is given by Nc (t) ¼ W(t) cos½Q(t) Ns (t) ¼ W(t) sin½Q(t)

(21:4:51)

The pdf fW(w) can be found by the techniques described in Section 13.3. The Jacobian of the transformation is 2 3 @Nc (w, u) @Nc (w, u) 6 7 cos (u) w sin(u) @u (21:4:52) ¼ kJ(w, u)k ¼ 4 @w 5 @Ns (w, u) @Ns (w, u) sin (u) w cos(u) @w @u or kJ(w, u)k ¼ w½cos2 (u) þ sin2 (u) ¼ w The solutions for w and u are qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ w ¼ n2c þ n2s ;

ns u ¼ tan nc 1

(21:4:53)

Hence, from Eq. (13.3.5), we obtain fNc Ns (nc , ns ) ¼

1 w (1=2)(w2 =s2N ) 2 2 2 e(1=2)½(nc þns )=sN ¼ e ¼ fW, Q (w, u) 2 2psN 2ps2N

(21:4:54)

which is the desired joint density fW, Q (w;u). Integrating Eq. (21.4.54) with respect to u, we obtain ð 2p w (1=2)(w2 =s2N ) w 2 2 fW (w) ¼ e du ¼ 2 e(1=2)(w =sN ) , w . 0 (21:4:55) 2 2ps s 0 N N which is a Rayleigh distribution shown in Fig. 7.8.2. Similarly, integrating with respect to w, we obtain ð 1 2p w (1=2)(w2 =s2N ) 1 , 0 u 2p (21:4:56) e du ¼ fQ (u) ¼ 2p 0 s2N 2p which is a uniform distribution in (0, 2p). Thus we ﬁnd that W(t) and Q(t) are also independent. This derivation holds good only if the mean values of the quadrature components Nc(t) and Ns(t) are zero. We will now discuss the case where the means are not zero. Narrowband Bandpass Stationary Process with Additive Sinusoid

A stationary bandpass random process X(t) is given by X(t) ¼ N(t) þ a cos½vc t þ C

(21:4:57)

where N(t) is a zero mean Gaussian stationary narrowband bandpass random process deﬁned by Eq. (21.4.19) with quadrature components Nc(t) and Ns(t), deﬁned by Eq. (21.4.51). C is a random phase uniformly distributed in (0, 2p), and a is a constant.

620

Random Processes and Linear Systems

The variances of N(t), Nc(t), and Ns(t) are all equal to s2N . The natural envelope of X(t) will be given by X(t) ¼ R(t) cos½vc t þ F(t)

(21:4:58)

where F(t) is some function of both C and Q(t). Expanding Eq. (21.4.58), we obtain X(t) ¼ R(t)½cos½F(t) cos(vc t) sin½F(t) sin(vc t)

(21:4:59)

Substituting Eq. (21.4.20a) for N(t) in Eq. (21.4.57), we obtain X(t) ¼ ½Nc (t) þ a cos(C) cos(vc t) ½Ns (t) þ a sin(C) sin(vc t)

(21:4:60)

and the quadrature components of X(t) can be deﬁned as Xc (t) ¼ Nc (t) þ a cos(C), in phase Xs (t) ¼ Ns (t) þ a sin(C), quadrature

(21:4:61)

Unlike Nc(t) and Ns(t), which are zero mean Gaussian lowpass random processes, Xc(t) and Xs(t) are Gaussian but not zero mean processes. Their mean values are given by mXc ¼ a cos(C);

mXs ¼ a sin(C) and

a¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m2Xc þ m2Xs

(21:4:62)

Without loss of generality, we can assume that C ¼ c is a constant and the quadrature components Xc(t) and Xs(t) are independent. Under these assumptions their joint Gaussian density is fXc Xs (xc , xs ) ¼

2 2 1 2 e(1=2sN )½xc a cos(c) þ½xs a sin(c) 2ps2N

1 2 2 2 2 ¼ e(1=2sN ){xc þxs þa 2a½xc cos(c)þxs sin(c)} 2 2psN

(21:4:63)

Comparing terms in Eqs. (21.4.59) and (21.4.60), we can write Xc (t) ¼ R(t) cos½F(t) ¼ Nc (t) þ a cos(c) Xs (t) ¼ R(t) sin½F(t) ¼ Ns (t) þ a sin(c)

(21:4:64)

To obtain the joint density of R(t) and F(t), we use the following transformation: xc ¼ r cos(f);

xs ¼ r sin(f)

(21:4:65)

The Jacobian of the transformation of Eq. (21.4.65) kJk ¼ r. In Eq. (21.4.63) we note that the term xc cos(c) þ xs sin(c) ¼ r cos(f) cos(c) þ r sin(f) sin(c) ¼ r cos(f c)

(21:4:66)

Substituting Eqs. (21.4.65) and (21.4.66) in Eq. (21.4.63), the joint pdf fRF(r, f) can be written as follows: r 0r ,1 (1=2sN2 ){r 2 þa2 2ar cos(fc)} e , fR, F (r, f) ¼ (21:4:67) 0 f 2p 2ps2N

21.4. Bandpass Stationary Random Processes

621

The marginal density of the envelope R(t) can be found by integrating Eq. (21.4.67) with respect to f. Substituting u ¼ f 2 c in Eq. (21.4.67), we obtain ð ar cosðuÞ r ½(r2 þa2 )=2s2N 1 2p r ½(r2 þa2 )=2s2N ar s2N e du ¼ 2 e I0 2 (21:4:68) fR (r) ¼ 2 e 2p 0 sN sN sN where we have utilized the fact that ear cosðuÞ is periodic with period 2p, that is, ear cosðuÞ ¼ ear cos(2pþu) . The integral in Eq. (21.4.68) is the zero-order modiﬁed Bessel function of the ﬁrst kind given in Eq. (7.8.13) and shown in Fig. 7.8.4. ð ar 1 2p ½ar cos(u)=s2N I0 2 ¼ e du (21:4:69) 2p 0 sN Equation (21.4.68) can be normalized by substituting r a z¼ ; k¼ sN sN and written as 2

fZ (z) ¼ ze½(z þk

2

)=2

I0 (kz)

(21:4:70)

This is called Rice’s density and is shown in Fig. 7.8.5 and reproduced in Fig. 21.4.6 for k ¼ 0, . . . , 5. If a ¼ 0, the mean values are 0, and we obtain the Rayleigh density shown in Fig. 21.4.6. To obtain the phase density function, we integrate Eq. (21.4.67) over r, yielding ð 2p 1 2 2 2 re(1=2sN ){r þa 2ar cos(fc)} dr (21:4:71) fF (f) ¼ 2ps2N 0 Completing the squares in the expression r2 þ a2 2ar cos(f c), we can rewrite Eq. (21.4.71) as follows: 2 ð 1 (1=2s2N )½1cos2 (fc) 1 r 1 r 2 a cos(f c) dr exp (21:4:72) fF (f) ¼ e 2p s 2 s s N N N 0

FIGURE 21.4.6

622

Random Processes and Linear Systems

Making the following substitutions for convenience in Eq. (21.4.72) z¼

r a ;k¼ ; u ¼ f c; b ¼ k cos(u) sN sN

as in the previous case, we can write fF (f) ¼ e

(1=2)k2 sin2 (u)

1 2p

ð1

2

ze(1=2)(zb) dz

(21:4:73)

0

Again substituting w ¼ z b in Eq. (21.4.73) and rearranging terms, we obtain ð 2 1 1 2 2 (w þ b)e(1=2)w dw fF (f) ¼ e(1=2)k sin (u) 2p b (21:4:74) ð ð1 b 1 1 (1=2)w2 (1=2)k2 sin2 (u) 1 (1=2)w2 fF (f) ¼ e we dw þ pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ e dw 2p b 2p 2p b Evaluating the integrals, we obtain fF (f) ¼ e(1=2)k

2

sin2 (u)

1 (b2 =2) b þ pﬃﬃﬃﬃﬃﬃ F(b) e 2p 2p

(21:4:75)

where F(b) is the Gaussian integral 1 F(b) ¼ pﬃﬃﬃﬃﬃﬃ 2p

ð1

1 2 e(1=2)w dw ¼ pﬃﬃﬃﬃﬃﬃ 2p b

ðb

2

e(1=2)w dw

1

Substituting back for b ¼ k cos(f 2 c) in Eq. (21.4.75), we have fF (f) ¼

1 (k2 =2) k cos (f c) (1=2)k2 sin2 (fc) pﬃﬃﬃﬃﬃﬃ e e þ F(b) 2p 2p

(21:4:76)

which is the desired result for the phase density. When k ¼ 0, the phase f is uniformly distributed in (0, 2p). However, if k = 0, then the quadrature components are not independent, unlike the case when k ¼ 0. The phase density is shown in Fig. 21.4.7 for c ¼ p.

FIGURE 21.4.7

21.4. Bandpass Stationary Random Processes

623

FIGURE 21.4.8

Example 21.4.4

A received signal is of the form X(t) ¼ A cos½(vc þ vd )t þ N(t)

(21:4:77)

where N(t) is a bandpass white-noise signal with psd as shown in Fig. 21.4.8. We want to express N(t) in the form N(t) ¼ Nc0 (t) cos½(vc þ vd )t Ns0 sin½(vc þ vd )t

(21:4:78)

with the quadrature components of N(t) given by [Eq. (21.4.20a)] N(t) ¼ Nc (t) cos(vc t) Ns (t) sin(vc t)

(21:4:79)

We will express Nc0 (t) and Ns0 (t) in terms of Nc(t) and Ns(t). Expanding the righthand side (RHS) of Eq. (21.4.78), we obtain Nc0 (t) cos½(vc þ vd )t Ns0 (t) sin½(vc þ vd )t ¼ ½Nc0 (t) cos(vd t) Ns0 (t) sin(vd t) cos(vc t) ½Nc0 (t) sin(vd t) þ Ns0 (t) cos(vd t) sin(vc t)

(21:4:80)

Equating like coefﬁcients in Eqs. (21.4.80) and (21.4.79), we obtain, in matrix form Nc (t) cos(vd t) sin(vd t) Nc0 (t) (21:4:81) ¼ Ns0 (t) Ns (t) sin(vd t) cos(vd t) The multiplying matrix in Eq. (21.4.81) is called a rotation matrix and is orthogonal. The inverse is the transpose, and hence we can write 0 Nc (t) cos(vd t) sin(vd t) Nc (t) (21:4:82) ¼ Ns0 (t) sin(vd t) cos(vd t) Ns (t) or Nc0 (t) ¼ Nc (t) cos(vd t) þ Ns (t) sin(vd t) Ns0 (t) ¼ Nc (t) sin(vd t) þ Ns (t) cos(vd t)

(21:4:83)

We can now ﬁnd the autocorrelation functions, RNc0 (t) and RNs0 (t), in terms of RNc (t) and RNs (t) as follows: RNc0 (t) ¼ E½Nc0 (t)Nc0 (t þ t) ¼ E ½Nc (t) cos(vd t) þ Ns (t) sin(vd t)

½Nc (t þ t) cos½vd (t þ t) þ Ns (t þ t) sin½vd (t þ t) ¼ RNc (t) cos(vd t) cos½vd (t þ t) þ RNs (t) sin(vd t) sin½vd (t þ t) þ RNc Ns cos(vd t) sin½vd (t þ t) þ RNs Nc sin(vd t) cos½vd (t þ t)

(21:4:84)

624

Random Processes and Linear Systems

FIGURE 21.4.9

FIGURE 21.4.10

Substituting RNc ðtÞ ¼ RNs (t); RNc Ns (t) ¼ RNs Nc (t) from Eq. (21.4.26) in Eq. (21.4.84), we obtain RNc0 (t) ¼ RNc (t) cos(vd t) þ RNc Ns sin(vd t)

(21:4:85)

This equation [Eq. (21.4.85)] applies for any stationary random process N(t). For a whitenoise process, we have RNc Ns (t) ¼ 0,

t=0

(21:4:86)

and substituting Eq. (21.4.86) in Eq. (21.4.85), we obtain RNs0 (t) ¼ RNc (t) cos(vd t)

(21:4:87)

In a similar manner, we can obtain RNs0 (t) ¼ RNc (t) cos(vd t)

(21:4:88)

From Eqs. (21.4.87) and (21.4.88), we ﬁnd that RNc0 (t) ¼ RNs0 (t). The psd SNc (v) is obtained from Eq. (21.4.47) as SNc (v) ¼ SNs (v) ¼ N0 ½u(v þ vB ) u(v vB )

(21:4:89)

and is shown in Fig. 21.4.9. The psd SNc0 (v) is obtained by using the modulation property of FT and is given by 1 SN (v) p½d(v þ vd ) þ d(v vd ) 2p c 1 ¼ ½SNc (v þ vd ) þ SNc (v vd ) 2

SNc0 (v) ¼

SNc0 (v) is shown in Fig. 21.4.10 for the two cases of vd . vB and vd , vB.

(21:4:90)

CHAPTER

22

Weiner and Kalman Filters

B 22.1

REVIEW OF ORTHOGONALITY PRINCIPLE

Random Variables

The estimate Y^ of a random variable Y by a function of another random variable g(X ) using the mean-square error criterion J ¼ E[Y 2 g(X )]2 is found in Chapter 18 as the conditional expectation given by Eq. (18.2.7): g(X) ¼ Y^ ¼ E½Y j X

(18:2:7)

and this estimator is also the orthogonal projection of Y in the space of X in the sense ^ ¼ E ½Y g(X)g(X) ¼ 0 E½1Y

(22:1:1)

In the case of linear estimation given by g(X) ¼ Y^ ¼ aT X

(22:1:2)

the coefﬁcient vector a is determined from Eq. (18.2.23) as a ¼ R1 X RXY

(22:1:3)

where RX is the correlation matrix E[XX T] and RXY is the cross-correlation vector E[XY ]. We will now consider the one-dimensional case where Y^ ¼ aX. The orthogonality condition of Eq. (22.1.1) yields E ½Y aXaX ¼ E½XY aE½X 2 or

a¼

E½XY E½X 2

(22:1:4)

and using the orthogonality principle, the minimum mean-square error is Jmin ¼ E½12 ¼ E½1(Y aX) ¼ E½1Y ¼ E½Y 2

{E½XY}2 E½X 2

(22:1:5)

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

625

626

Weiner and Kalman Filters

and if X and Y have zero mean, then we can deﬁne rXY as the correlation coefﬁcient between X and Y and write Eq. (22.1.5) as Jmin ¼ E½Y 2 ½1 r2XY

(22:1:6)

Random Processes

We can extend the preceding analysis to random processes. The input to the system shown in Fig. 22.1.1 is a zero mean random process X(t) consisting of a zero mean signal process S(t) and a zero mean additive noise N(t) that is independent of S(t), both deﬁned in the estimation time interval T: X(t) ¼ S(t) þ N(t),

t [ T; E½S(t)N(t) ¼ E½S(t)E½N(t) ¼ 0,

t[T

(22:1:7)

The output of the system is Y(t). The system is observed during the observation time interval U. T and U may be different or the same. We want to design a linear system with impulse response h(t,u), u [ U such that the output Y(t) approximates the desired signal S(t) according to the minimum mean-square error criterion given by J ¼ min E ½S(t) Y(t)2 , h(t,u)

t [ T, u [ U

(22:1:8)

where ð Y(t) ¼

h(t,u)X(u)du,

t[T

(22:1:9)

U

For this case the orthogonality condition E{½S(t) Y(t)Y(t)} ¼ 0 becomes ð ð h(t,v)X(v)dv ¼ 0, t [ T E S(t) h(t,u)X(u)du U

(22:1:10)

U

This equation can be rearranged as follows: ð h(t,v) E½S(t)X(v) h(t,u)E½X(u)X(v)du dv,

ð U

t[T

(22:1:11)

U

If Eq. (22.1.11) has to be satisﬁed, then the quantity within braces must be equal to 0: ð hopt (t,u)E½X(u)X(v)du ¼ E½S(t)X(v), U

FIGURE 22.1.1

t [ T, v [ T

(22:1:12)

22.2 Wiener Filtering

We can now express Eq. (22.1.12) in terms of correlation functions as ð hopt (t,u)RX (u,v)du ¼ RXS (t,v), t [ T, v [ T

627

(21:1:13)

U

where hopt(t) is the optimum impulse response. The corresponding minimum mean-square error Jmin is given by Jmin ¼ E ½S(t) Yo (t)S(t) ð 2 ¼ E½S (t) hopt (t,u)E½X(u)S(t)du ð ¼ RS (t,t)

U

hopt (t,u)RXS (t,u)du,

t[T

(22:1:14)

U

If S(t) is replaced by S(t þ a) in Eqs. (22.1.12), then we have ð hopt (t,u)E½X(v)X(u)du ¼ E½X(v)S(t þ a), t [ T, v [ T

(22:1:15)

U

and Eq. (21.1.13) becomes ð hopt (t,u)RX (u,v)du ¼ RXS (t þ a,v),

t [ T, v [ T

(22:1:16)

U

In Eq. (22.1.16), if a ¼ 0, we have a ﬁltering problem; if a . 0, we have a prediction or extrapolation problem; and if a , 0, we have a smoothing or interpolation problem. The corresponding equation for the minimum mean square error is ð Jmin ¼ E½S2 (t þ a) hopt (t,u)E½X(u)S(t þ a)du U

ð

hopt (t,u)RXS (t þ a,u)du,

¼ RS (t,t)

t[T

(22:1:17)

U

B 22.2

WIENER FILTERING

Noncausal Weiner Filtering (Unrealizable Filter)

We will now assume that the estimation time interval T and the observation time interval U are the real line, or, T ¼ U ¼ (21, 1). The processes S(t),N(t) are stationary, and the system is time-invariant but not causal. Under these assumptions Eq. (22.1.16) can be rewritten as ð1 hopt (t u)RX (u v)du ¼ RXS (t þ a v), t [ T, v [ T (22:2:1) 1

We make a change of variables t 2 u ¼ j and t 2 v ¼ t in Eq. (22.2.1) and we have ð1 hopt (j)RX (t j)dj ¼ RXS (t þ a) 1

Or hopt (t) RX (t) ¼ RXS (t þ a),

t[T

(22:2:2)

628

Weiner and Kalman Filters

The optimum transfer function Hopt(v) is obtained by taking FT of Eq. (22.2.2), yielding Hopt (v) ¼

SXS (v)e jva SX (v)

(22:2:3)

Since the signal and noise are assumed independent, the relevant correlation functions and the spectral densities are RX (t) ¼ RS (t) þ RN (t) RXS (t) ¼ E ½S(t) þ N(t)S(t þ t) ¼ RS (t) SX (v) ¼ SS (v) þ SN (v);

(22:2:4)

SXS (v) ¼ SS (v)

Substituting Eq. (22.2.4) into Eq. (22.2.3), we obtain Hopt (v) ¼

SS (v)e jva SS (v) þ SN (v)

(22:2:5)

If N(t) ¼ 0 in Eq. (22.1.7), then X(t) ¼ S(t), RX(t) ¼ RS(t), and Eq. (22.2.5) becomes Hopt (v) ¼ e jva

(22:2:6)

a pure prediction if a . 0 and a pure delay if a , 0. The minimum mean-square error is obtained from Eq. (22.1.17) as ð1 hopt (t u)RXS (t u þ a)du Jmin ¼ RS (0) 1

ð1

hopt (t)RXS (t þ a)dt

¼ RS (0)

(22:2:7)

1

and using Parseval’s theorem, we obtain the following in the frequency domain: ð 1 1 ½SS (v) SXS (v)Hopt (v)e jva dv (22:2:8) Jmin ¼ 2p 1 Substituting Eq. (22.2.3) in Eq. (22.2.8), we obtain ð 1 1 SXS (v) Jmin ¼ SS (v) SXS (v) dv 2p 1 SX (v) ð 1 1 SS (v)SX (v) jSXS (v)j2 dv ¼ SX (v) 2p 1 ð 1 1 SS (v)SX (v) jSXS (v)j2 ¼ dv; since SX (v) ¼ SX (v) SX (v) 2p 1 and if S(t) and N(t) are independent, Eq. (22.2.9) reduces to ð 1 1 SS (v)SN (v) Jmin ¼ dv 2p 1 SS (v) þ SN (v)

(22:2:9)

(22:2:10)

If the noise N(t) ¼ 0, then we obtain Jmin ¼ 0 from Eq. (22.2.10). It should be noted that the optimum transfer function Hopt(v) is noncausal and hence unrealizable. In the following examples we will assume that the signal S(t) and the noise N(t) are independent and both are zero mean stationary random processes.

22.2 Wiener Filtering

Example 22.2.1 to a system are

629

The autocorrelation functions of the input signal S(t) and the noise N(t) RS (t) ¼ ejtj=2

and

RN (t) ¼ 4d(t)

We will estimate the signal S(t þ a) by ﬁnding the unrealizable ﬁlter Hopt(v). The power spectral densities (psd) of signal and noise are SS (v) ¼

4 1 þ 4v2

and

SN (v) ¼ 4

Since the signal and noise are independent, we can use Eq. (22.2.5) to obtain the optimum transfer function Hopt(v) as 4 e jva 1 þ 4v2 e jva ¼ Hopt (v) ¼ 4 2(1 þ 2v2 ) þ4 2 1 þ 4v The optimum transfer function jHopt(v)j is shown in Fig. 22.2.1. The corresponding optimum impulse response hopt(t) is obtained by taking the inverse Fourier transform (FT) of the optimum transfer function: pﬃﬃ 1 hopt (t) ¼ pﬃﬃﬃ ejtaj= 2 4 2 hopt(t) is shown in Fig. 22.2.2 for values of the lag a ¼ 6,3,0,23. It is seen from the ﬁgure that hopt(t) is unrealizable because it is noncausal; however, it is approximately causal for lags a 6. It can be truly causal only for inﬁnite lags. The minimum mean-square error is obtained from Eq. (22.2.10) as follows:

Jmin

1 ¼ 2p

4 ð 4 1 1 2 1 1 þ 4v2 dv ¼ dv ¼ pﬃﬃﬃ 4 2p 1 1 þ 2v2 2 1 þ4 1 þ 4v2

ð1

FIGURE 22.2.1

630

Weiner and Kalman Filters

FIGURE 22.2.2

Example 22.2.2 This example will show that even for an inﬁnite lag, hopt(t) cannot be causal. The autocorrelation functions of the input signal S(t) and the noise N(t) to a system are RS (t) ¼ ejtj=2

and

RN (t) ¼ e2jtj

We have to estimate the signal S(t þ a). We will determine the unrealizable optimum ﬁlter Hopt(v) and the corresponding minimum mean-square error. The psd’s of the signal and noise are SS (v) ¼

4 1 þ 4v2

and

SN (v) ¼

4 4 þ v2

Since the signal and noise are independent, we can use Eq. (22.2.5) to obtain the transfer function Hopt(v) as follows: 4 e jva 2 4 þ v2 jva 1 þ 4v e Hopt (v) ¼ ¼ 4 4 5(1 þ v2 ) þ 1 þ 4v2 4 þ v2 The optimum transfer function is shown in Fig. 22.2.3.

FIGURE 22.2.3

22.2 Wiener Filtering

631

FIGURE 22.2.4

The transfer function Hopt(v) is not physically realizable since it violates the Paley – Wiener criterion given by Eq. (21.3.4): ð1 ln jH(v)j dv , 1 (21:3:4) 2 1 1 þ v Taking the inverse FT, we obtain the optimum impulse response hopt(t) as 1 3 hopt (t) ¼ d(t) þ ejtaj 5 10 hopt(t) is shown in Fig. 22.2.4 for values of a ¼ 6,3,0,23. The impulse response is also noncausal for any value of the lag a because of the presence of the impulse function. The minimum mean-square error is obtained from Eq. (22.2.9) and is given by

Jmin

1 ¼ 2p

4 4 ð1 4 2 1 þ 4v2 4 þ v2 dv ¼ 1 dv ¼ 2 4 4 2p 1 5(1 þ v ) 5 1 þ 1 þ 4v2 4 þ v2

ð1

Causal Weiner Filtering (Realizable Filter)

In the last section we derived an integral equation [Eq. (22.2.2)] for an unrealizable ﬁlter. We now impose the condition that the system is causal, that is, h(t) ¼ 0, t , 0. The estimation interval is the real line T ¼ f21 , t , 1g, and the observation interval is now U ¼ fu tg. The ranges for S(t) and N(t) are T. Under this condition, Eq. (22.1.13) becomes ðt hopt (t u)RX (u v)du ¼ RXS (t v), v [ U (22:2:11) 1

Making the change of variables t 2 u ¼ j and t 2 v ¼ t in Eq. (22.2.11), we obtain ð1 hopt (j)RX (t j)dj ¼ RXS (t), t 0 (22:2:12) 0

632

Weiner and Kalman Filters

Equation (22.2.12) is called the Wiener –Hopf integral equation. This equation is a necessary and sufﬁcient condition for the causal optimum ﬁlter hopt(t) ¼ 0, t , 0 to exist. The corresponding mean-square error is obtained from Eq. (22.1.14) as ð1 hopt (t)RXS (t)dt (22:2:13) Jmin ¼ RS (0) 0

Even though Eq. (22.2.12) is similar to Eq. (22.2.2) with a ¼ 0, the solution is far more complex because h(t) was required to vanish along the negative time axis. If we take the FT of both sides of Eq. (22.1.12), we obtain ð 1 ð 1 ð1 RXS (t)ejvt dt ¼ hopt (j)RX (t j)dj ejvt dt 0

0

0

or ð1

RXS (t)ejvt dt ¼

0

ð 1 ð 1 0

hopt (j)ejvj RX (t j)dj

0

ejv(tj) dt Substituting t 2 j ¼ b in Eq. (22.2.14), we obtain ð1 ð1 ð1 RXS (t)ejvt dt ¼ hopt (j)ejvj dj RX (b)ejvb db 0

0

(22:2:14)

(22:2:15)

j

The presence of j in the lower limit of the RHS of Eq. (22.2.15) poses a serious problem since it is a variable involved in the integration. We have to seek other methods for solving the Weiner – Hopf integral equation. We note that Eq. (22.2.11) or (22.2.12) is not dependent on the exact nature of the signal X(t) and S(t) but only on their autocorrelation functions, which are not unique. Thus, other signals that have the same autocorrelation functions will give the same result. Equation (22.2.12) will be solved by spectral factorization techniques for which we require some intermediate results. Whitening Filter

We are given an input stationary random process X(t) and want to ﬁnd a causal impulse response g(t)u(t) such that the output process Z(t) given by ð1 g(t)X(t t)dt (22:2:16) Z(t) ¼ 0

is a white-noise process with autocorrelation RZ(t) ¼ d(t). We will make the assumption that the psd SX(v) of X(t) is a real rational function of v2 for real v so that all the poles and zeros of SX(v) in the complex v plane occur as complex conjugate pairs. From Eq. (21.2.9) we have SZ (v) ¼ 1 ¼ SX (v)jG(v)j2

(22:2:17)

Hence the required transfer function is given by 1 1 ¼ SX (v) G(v) G (v)

(22:2:18)

22.2 Wiener Filtering

633

where G(v) has poles in the top half and G (v) has poles in the bottom half of the v plane. The corresponding impulse responses g(t) and g 21(t), given by ð ð 1 1 1 1 1 g(t) ¼ G(v)e jvt dv; g1 (t) ¼ (22:2:19) e jvt dv 2p 1 2p 1 G(v) satisfy the causality and stability criteria: g(t) and g1 (t) ¼ 0, t , 0;

g(t) and g1 (t) ! 0 as t ! 1:

Under these conditions X(t) can be generated from the following equation: ð1 g1 (t)Z(t t)dt X(t) ¼

(22:2:20)

0

Spectral Factorization

As mentioned in the previous section, the psd SX(v) is a rational function of v2, and some properties are listed in Chapter 19, Eqs. (19.5.8)– (19.5.11). Additional properties relevant to Wiener ﬁltering will be enumerated: 1. SX(v) is bounded because X(t) is assumed to be a zero mean stationary process: SX (v) , 1

(22:2:21)

2. As a consequence of property 1, poles of SX(v) cannot be real. 3. Since RX(t) is real, SX(v) ¼ SX (v) and hence it has complex conjugate poles and zeros (symmetry about the real axis). 4. From Eq. (19.5.8), SX(v) ¼ SX(2v) and hence it has symmetry about the imaginary axis. 5. As a consequence of properties 3 and 4, the poles and zeros have symmetry about both axes as shown in Fig. 22.2.5. Two methods of factoring SX(v) will be discussed. The ﬁrst is called multiplicative factoring, given by SX (v) ¼ k

(v z1 )(v z2 ) (v zM ) (v z1 )(v z2 ) (v zM ) k (v p1 )(v p2 ) (v pN ) (v p1 )(v p2 ) (v pN )

FIGURE 22.2.5

(22:2:22)

634

Weiner and Kalman Filters

where pn, n ¼ 1, . . . , N are poles with positive imaginary parts and zn, n ¼ 1, . . . , M are zeros with positive imaginary parts. Thus, SX(v) can be represented by SX (v) ¼ ½SX (v)þ ½SX (v)

(22:2:23)

þ

where ½SX (v) has poles and zeros on the upper and ½SX (v) has poles and zeros in the lower half of the v plane as shown in Fig. 22.2.5. The second one is additive factoring, where we have a partial fraction expansion and separate the positive and negative roots in the top and bottom halves of the v plane. In this case SX(v) can be represented by SX (v) ¼ ½SX (v)þ þ ½SX (v)

(22:2:24)

In multiplicative factoring poles and zeros are involved, whereas in additive factoring only the poles are involved. We can now identify from Eq. (22.2.18) the transfer functions G(v) and G (v) as follows: G(v) ¼

Example 22.2.3

1 ; ½SX (v)þ

G (v) ¼

1 ½SX (v)

(22:2:25)

The psd SX(v) is given by SX (v) ¼ 25

(v2 þ 4) (v2 þ 9)(v2 þ 16)

The multiplicative factoring is obtained from the form of SX(v) and is given by ½SX (v)þ ¼ 5

(v þ 2j) ; (v þ 3j)(v þ 4j)

½SX (v) ¼ 5

(v 2j) (v 3j)(v 4j)

The additive factoring is obtained by obtaining a partial fraction expansion and expressing SX(v) as 25 12 5 SX (v) ¼ 7 v2 þ 16 v2 þ 9 25 3 5 3 5 þ ¼ 7 2(4 þ jv) 6(3 þ jv) 2(4 jv) 6(3 jv) and identifying 25 3 5 ; 7 2(4 þ jv) 6(3 þ jv) 25 3 5 ½SX (v) ¼ 7 2(4 jv) 6(3 jv) ½SX (v)þ ¼

The differences between these two factorings are clear.

Solution for Wiener– Hopf Filter Equation [Eq. (22.2.12)]

The method for ﬁnding the causal solution is to pass the input process X(t) through the causal whitening ﬁlter g(t) of Eq. (22.2.16), and then through another optimal causal ﬁlter gopt(t) so that these two ﬁlters are in tandem, will yield the desired hopt(t), which will yield

22.2 Wiener Filtering

635

the optimum Yo(t) as shown in Fig. 22.2.6: Yo (t) ¼ gopt (t) g(t) X(t)

(22:2:26)

From Eq. (22.2.26) the ﬁlter gopt(t) can be obtained as hopt (t) ¼ g(t) gopt (t) or gopt (t) ¼ hopt (t) g1 (t)

(22:2:27)

Taking FT on both sides of Eq. (22.2.27), we obtain Hopt (v) ¼ G(v)Gopt (v) or

Gopt (v) ¼

Hopt (v) G(v)

(22:2:28)

The ﬁlter gopt(t) is the optimal ﬁlter obtained by passing an input white noise to obtain the desired optimal response Yo(t). The Wiener –Hopf equation corresponding to gopt(t) is obtained from Eq. (22.2.2) with a ¼ 0: ð1 gopt (j)RZ (t j)dj ¼ RZS (t), t 0 (22:2:29) 0

In Eq. (22.2.29), we have RZ (t j) ¼ d(t j);

RZS (t) ¼ RXS (t) g(t)

(22:2:30)

Substituting Eq. (22.2.30) into Eq. (22.2.29) yields t0

gopt (t) ¼ RXS (t) g(t),

(22:2:31)

In other words, gopt(t) is the causal part of the noncausal term RXS(t) * g(2t). We can deﬁne the FT of RXS (t) g(t) in Eq. (22.2.31) as K(v). From the partial fraction expansion, K(v) can be additively factored by using Eq. (22.2.24) as K(v) ¼ Gopt (v) þ Gopt;L (v), where Gopt (v) has poles in the upper and Gopt;L (v) has poles in the lower half of the v plane, or K(v) ¼ SXS (v)G (v) ¼ Gopt (v) þ Gopt,L (v)

(22:2:32)

where SXS (v) Gopt (v) ¼ ½SXS (v)G (v)þ ¼ ½SX (v) þ SXS (v) Gopt,L (v) ¼ ½SXS (v)G (v)þ ¼ ½SX (v)

FIGURE 22.2.6

(22:2:33a) (22:2:33b)

636

Weiner and Kalman Filters

and IFT½K(v) ¼ k(t) ¼ gopt (t)u(t) þ gopt,L (t)u(t)

(22:2:34)

Substituting Eqs. (2.22.25) and (22.2.33a) in Eq. (22.2.28), we obtain the ﬁnal form for the optimal causal transfer function Hopt(v) as 1 SXS (v) Hopt (v) ¼ (22:2:35) ½SX (v)þ ½SX (v) þ The minimum mean-square error associated with the causal optimal impulse response hopt(t) is given by Eq. (22.2.13): ð1 hopt (t)RXS (t)dt (22:2:13) Jmin ¼ RS (0) 0

The integral in the minimum mean-square error [Eq. (21.2.13)] can be written in the frequency domain using Parseval’s theorem: ð ð1 1 1 hopt (t)RXS (t)dt ¼ Hopt (v)SXS (v)dv (Parseval’s theorem) 2p 1 0 ð 1 1 ¼ Gopt (v)½G(v)SXS (v)dv ½Eq. (22:2:33) 2p 1 ð 1 1 SXS (v) Gopt (v) dv ¼ 2p 1 ½SX (v)þ ð 1 1 ¼ Gopt (v)Gopt (v)dv 2p 1 ð1 ¼ g2opt (t)dt (Parseval’s theorem) (22:2:36) 0

Substituting Eq. (22.2.36) in Eq. (22.2.13), the minimum mean-square error Jmin can be written in terms of the optimal ﬁlter gopt(t) for a white-noise input as ð1 g2opt (t)dt (22:2:37) Jmin ¼ RS (0) 0

If signal S(t) and noise N(t) are independent, then Eqs. (22.2.35) and (22.2.13) can be further simpliﬁed using Eq. (22.2.4). Equation (22.2.35) can be written as 1 SS (v) Hopt (v) ¼ ½SS (v) þ SN (v)þ ½SS (v) þ SN (v) þ (22:2:38) Gopt (v)

G(v)

Equation (22.2.13) can be written as ð1 Jmin ¼ RS (0)

hopt (t)RS (t)dt

(22:2:39)

0

Further simpliﬁcations are possible [21] if additionally N(t) is white noise with psd N0. Under this condition and from Eq. (22.2.4) we have SXS (v) ¼ SS (v) ¼ SX (v) N0

(22:2:40)

22.2 Wiener Filtering

637

Substituting Eq. (22.2.40) in ﬁlter equation (22.2.35), we can write 1 SX (v) N0 Hopt (v) ¼ ½SX (v)þ ½SX (v) ½SX (v) þ 1 ½SX (v)þ ½SX (v) ½N0 þ ½N0 ¼ ½SX (v) ½SX (v) þ ½SX (v)þ 1 ½N0 þ ½N0 þ ¼ ½SX (v) ½SX (v) þ ½SX (v)þ ½N0 þ N0 ¼1 (22:2:41) ½SS (v) þ N0 þ SS (v) þ N0 þ pﬃﬃﬃﬃﬃﬃ The multiplicative factoring yields ½N0 þ ¼ N0 and the factor within the braces in Eq. (22.2.41) can be simpliﬁed as follows: N0 N0 N0 D(v) ¼ ¼ SS (v) þ N0 N(v) N(v) þ N0 D(v) þ N0 D(v)

(22:2:42)

In Eq. (22.4.42) the degree of N(v) is less than or equal to the degree of D(v), and hence N0 D(v) ¼ 1 þ ½ (A(v)) þ ¼ 1 (22:2:43) N(v) þ N0 D(v) þ Since all the poles of (A(v)) are in the lower half of the v plane, the factor ½(A(v)) þ ¼ 0 in Eq. (22.2.43). Substituting Eq. (22.2.43) in Eq. (22.2.41), we obtain a simpliﬁed expression for the optimal ﬁlter: pﬃﬃﬃﬃﬃﬃ N0 (22:2:44) Hopt (v) ¼ 1 ½SS (v) þ N0 þ In a similar manner, Eq. (22.2.39) can also be simpliﬁed. Since RX(t) ¼ RS(t) þ N0d(t) the Wiener – Hopf equation (22.2.12) can be written as ð1 hopt (j)RS (t j)dj þ N0 hopt (t) ¼ RS (t), t 0 (22:2:45) 0

and Eq. (22.2.45) evaluated at t ¼ 0 becomes ð1 hopt (j)RS (j)dj ¼ RS (0) N0 hopt (0)

(22:2:46)

0

Substituting Eq. (22.2.46) in Eq. (22.2.39) we obtain Jmin ¼ N0 hopt (0)

(22:2:47)

Solution for Wiener– Hopf Prediction– Smoothing Equation

The form of the Wiener –Hopf (W – H) equation for prediction and smoothing is obtained by replacing RXS(t) with RXS(t þ a) in Eq. (22.2.12): ð1 hopt (j)RX (t j)dj ¼ RXS (t þ a), t 0 (22:2:48) 0

638

Weiner and Kalman Filters

As mentioned earlier, if a ¼ 0, we have a ﬁltering problem whose solution is given by Eq. (22.2.35). If a . 0, we have a prediction problem, and if a , 0, we have a smoothing problem. Corresponding to the factor a, the optimal transfer function will be given by jva 1 e SXS (v) 1 ¼ ½K(v)e jva þ (22:2:49) Hopt (v) ¼ ½SX (v)þ ½SX (v) þ ½SX (v)þ and the minimum mean-square error by ð1 hopt (t)RXS (t þ a)dt

Jmin ¼ RS (0)

(22:2:50)

0

Similar to Eq. (22.2.34), we can deﬁne the following IFT jva e SXS (v) ¼ IFT½e jva K(v) IFT ½SX (v) ¼ k(t þ a) ¼ gopt (t þ a)u(t þ a) þ gopt, L (t a)u(t a) (22:2:51) and the minimum mean-square error Jmin will be given by an equation similar to Eq. (22.2.37) as ð1 ð1 k2 (t þ a)dt ¼ RS (0) k2 (t)dt (22:2:52) Jmin ¼ RS (0) 0

0

From Eq. (22.2.52) we can observe that the error is a maximum when the prediction time a ¼ 1, and a minimum for inﬁnite lag time a ¼ 21. When a ¼ 0, we have Eq. (22.2.37). We will now determine a realizable ﬁlter for Examples 22.2.1 and 22.2.2 with the same assumption that S(t) and the noise N(t) are independent and both are zero mean stationary random processes. Example 22.2.4 We will consider Example 22.2.1, where the autocorrelation functions of the input signal S(t) and the noise N(t) to a system are RS (t) ¼ eðjtj=2Þ

and

RN (t) ¼ 4d(t)

The power spectral densities (psd’s) of signal and noise are SS (v) ¼

4 1 þ 4v2

and

SN (v) ¼ 4

1. Filtering Problem: a ¼ 0. We will ﬁrst estimate the signal S(t) by a realizable ﬁlter Hopt(v) using Eq. (22.2.38): pﬃﬃﬃ pﬃﬃﬃ 8 þ 16v2 2 þ 2jv 2 2jv 2 ¼2 SS (v) þ SN (v) ¼ 1 þ 4v2 1 þ 2jv 1 2jv pﬃﬃﬃ 21 1 1 þ 2jv 1 G(v) ¼ ¼ pﬃﬃﬃ þ ¼ pﬃﬃﬃ 2 2 2 2jv ½SS (v) þ SN (v) 2 2 þ 2jv pﬃﬃﬃ 1 1 2jv 1 21 G (v) ¼ ¼ pﬃﬃﬃ ¼ pﬃﬃﬃ ½SS (v) þ SN (v) 2 2 2jv) 2 2 2 2jv

22.2 Wiener Filtering

639

We ﬁnd K(v) as follows: K(v) ¼

SS (v) ¼ SS (v)G (v) ½SS (v) þ SN (v)

1 2jv 4 2 ¼ pﬃﬃﬃ ¼ pﬃﬃﬃ 2 2 2 2jv 1 þ 4v 2 2jv (1 þ 2jv) Partial fraction expansion of K(v) yields

pﬃﬃﬃ

pﬃﬃﬃ 2 21 2 21 K(v) ¼ þ pﬃﬃﬃ (1 þ 2jv) 2 2jv

(A)

Taking the IFT of Eq. (A) above, we obtain k(t) ¼ gopt (t)u(t) þ gopt, L (t)u(t) ¼

i pﬃﬃ

pﬃﬃﬃ h 2 1 e(t=2) u(t) þ e(t= 2) u(t)

(B)

We can identify Gopt(v) and Gopt,L(v) as follows:

pﬃﬃﬃ 2 21 SS (v) Gopt (v) ¼ ¼ ½K(v)þ ¼ ½SS (v) þ SN (v) þ (1 þ 2jv)

pﬃﬃﬃ 2 21 SS (v) p ﬃﬃ ﬃ ¼ ½K(v) ¼

Gopt,L (v) ¼ ½SS (v) þ SN (v) 2 2jv

Hence Hopt(v) is given by

pﬃﬃﬃ pﬃﬃﬃ 2 21 1 þ 2jv 21 ¼ pﬃﬃﬃ Hopt (v) ¼ G(v)Gopt (v) ¼ pﬃﬃﬃ

1 þ 2jv 2 2 þ 2jv 2 þ 2jv This result pcan Eq. ﬃﬃﬃ also be obtained frompﬃﬃﬃﬃﬃ ﬃ (22.2.44) as follows. Substituting ½SS (v) þ N0 þ ¼ 2½ 2 þ 2jv =(1 þ 2jv) and N0 ¼ 2 in Eq. (22.2.44), we obtain pﬃﬃﬃ pﬃﬃﬃ 2(1 þ 2jv) 2 þ 2jv 1 2jv 21 pﬃﬃﬃ ¼ pﬃﬃﬃ Hopt (v) ¼ 1 pﬃﬃﬃ ¼ 2( 2 þ 2jv 2 þ 2jv 2 þ 2jv which is the same as before. The inverse Fourier transforms of G(v), G (v), and Hopt(v) are given by pﬃﬃﬃ pﬃﬃﬃ 2 1 (t=pﬃﬃ2) 2 1 ðt=pﬃﬃ2Þ 1 1 e e u(t); g(t) ¼ d(t) u(t) g(t) ¼ d(t) 4 4 2 2 pﬃﬃﬃ 2 1 ðt=pﬃﬃ2Þ e u(t) hopt (t) ¼ 2 The time function k(t) in Eq. (B), and g(t) and g(2t) corresponding to the poles and zeros of [SX (v)þ and ½SX (v) , and hopt(t), the optimum realizable impulse response, are shown in Fig. 22.2.7. To determine the minimum mean-square error Jmin, we need the function k 2(t), which is given by pﬃﬃﬃ

pﬃﬃﬃ 2 2 1 ½et u(t) þ e 2t u(t) (C) k2 (t) ¼

640

Weiner and Kalman Filters

FIGURE 22.2.7

The minimum mean-square error is calculated by substituting the causal part of Eq. (C) in Eq. (22.2.52): ð1 ð1 pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 2 2 k (t)dt ¼ 1 ( 2 1) et dt ¼ 1 ( 2 1)2 ¼ 2( 2 1) Jmin ¼ RS (0) 0

0

We can also calculate Jmin using Eqs. (22.2.39) or (22.2.47): ð1 ð1 pﬃﬃ 1 pﬃﬃﬃ e(t= 2) e(t=2) dt hopt (t)RS (t)dt ¼ 1 Jmin ¼ RS (0) 8 0 0 2þ h pﬃﬃﬃ i1 pﬃﬃ pﬃﬃﬃ ¼ 1 (2 2 3)e(t= 2) e(t=2) ¼ 2( 2 1) 0

pﬃﬃﬃ 4 2 pﬃﬃﬃ ¼ pﬃﬃﬃ ¼ 2( 2 1) 2þ 8 1þ 2 pﬃﬃﬃ All three equations yield the same result: Jmin ¼ 2( 2 1) ¼ 0:8284. The causal part of the noncausal solution of Example 22.2.1 given by hopt pﬃﬃ pﬃﬃ noncausal(t) ¼ 82 e(t= 2) u(t) and the causal solution of this example, hoptcausal (t) ¼ pﬃﬃ pﬃﬃﬃ ½( 2 1=2)e(1= 2) u(t), are shown in Fig. 22.2.8. The minimum mean-square errors in ¼ N0 hopt (0) ¼

FIGURE 22.2.8

22.2 Wiener Filtering

641

both examples are given by Jmin noncausal ¼ 0.7071 and Jmin causal ¼ 0.8284. The differences between the causal and noncausal solutions are not that signiﬁcant for this problem, but the complexity of getting a causal solution is much greater. From K(v) in Eq. (A) for a delay a, we can write pﬃﬃﬃ pﬃﬃﬃ 2( 2 1) 2( 2 1) jva þ pﬃﬃﬃ e K(v)e jva ¼ (1 þ 2jv) ( 2 2jv) i pﬃﬃ

pﬃﬃﬃ h () 2 1 e½(tþa)=2 u(t þ a) þ e½(tþa)= 2 u(t a) and the time function k(t þ a) is shown for both a . 0 and a , 0 in Fig. 22.2.9. 2. Prediction Problem: a . 0. For the prediction problem of a . 0, the causal part of k(t þ a) from Fig. 22.2.9 is given by

pﬃﬃﬃ 2 1 e½(tþa)=2 u(t), a . 0 k(t þ a)casual ¼ and the FT of this equation yields,

pﬃﬃﬃ

pﬃﬃﬃ 2 1 e(a=2) 2 2 1 e(a=2) jva ¼ ½K(v)e þ ¼ 1 1 þ 2jv 2 þ jv and substituting the preceding equation in Eq. (22.2.49), we obtain the optimum transfer function for the prediction problem pﬃﬃﬃ 1 þ 2jv 2( 2 1Þe(a=2) jva Hopt (v) ¼ G(v)½K(v)e þ ¼ pﬃﬃﬃ 1 þ 2jv 2( 2 þ 2jv) pﬃﬃﬃ (a=2) ( 2 1)e ¼ pﬃﬃﬃ 2 þ 2jv and the optimum impulse response given by pﬃﬃﬃ 2 1 ½(1=pﬃﬃ2)½tþ(a=2) hopt (t) ¼ u(t), e 2

FIGURE 22.2.9

a.0

642

Weiner and Kalman Filters

Since a . 0, the minimum mean-square error is obtained by substituting the causal part of pﬃﬃﬃ Eq. (C), k2 (t) ¼ ( 2 1)2 et u(t), in Eq. (22.2.52) to obtain ð1 ð 1 pﬃﬃﬃ pﬃﬃﬃ Jmin ¼ RS (0) k2 (t)dt ¼ 1 ( 2 1)2 et dt ¼ 1 ( 2 1)2 ea , a . 0 a

a

3. Smoothing Problem: a , 0. For the smoothing problem of a , 0, the causal part of k(t þ a) from Fig. 22.2.9 is h i pﬃﬃ pﬃﬃﬃ k(t þ a)causal ¼ ( 2 1) e½(tþa)=2 u(t þ a) þ e½(tþa)= 2 ½u(t a) u(t) , a , 0 and the FT of this equation yields ½K(v)e

jva

pﬃﬃ # e jva e jva ea= 2 þ pﬃﬃﬃ 1 þ 2jv 2 2jv " pﬃﬃﬃ pﬃﬃ # pﬃﬃﬃ ( 2 þ 1)e jva (1 þ 2jv)ea= 2 pﬃﬃﬃ ¼ 2( 2 1) (1 þ 2jv)( 2 2jv)

pﬃﬃﬃ þ ¼ 2( 2 1)

"

and substituting this equation in Eq. (22.2.49), we obtain the optimum transfer function for the smoothing problem: " pﬃﬃﬃ pﬃﬃ # pﬃﬃﬃ 1 þ 2jv ( 2 þ 1)e jva (1 þ 2jv)ea= 2 pﬃﬃﬃ 2( 2 1) Hopt (v) ¼ pﬃﬃﬃ 2( 2 þ 2jvÞ (1 þ 2jv)( 2 2jvÞ " pﬃﬃﬃ pﬃﬃ # pﬃﬃﬃ ( 2 þ 1)e jva (1 þ 2jv)ea= 2 ¼ ( 2 1) (2 þ 4v2 ) To obtain the minimum mean-square error, we note that since a , 0 we have to use the entire Eq. (C) in Eq. (22.2.52) and write ð 0 pﬃﬃ ð1 ð1 pﬃﬃﬃ k2 (t)dt ¼ 1 ( 2 1)2 e 2t dt þ et dt Jmin ¼ Rs (0) a

pﬃﬃ pﬃﬃﬃ 2 1 2a ¼ 1 ( 2 1) pﬃﬃﬃ (1 e ) þ 1 , 2

a

0

a,0

The graph of the minimum mean-square error Jmin(a) for values of a ranging from 21 to 1 is shown in Fig. 22.2.10. The MSE Jmin reaches a minimum value for inﬁnite lag, and this value is given by pﬃﬃ pﬃﬃﬃ 2 1 2a Jmin (1) ¼ 1 ( 2 1) pﬃﬃﬃ (1 e ) þ 1 2 a!1 pﬃﬃﬃ pﬃﬃﬃ 2þ1 1 ¼ 1 ( 2 1)2 pﬃﬃﬃ ¼ pﬃﬃﬃ 2 2 and the maximum error for inﬁnite prediction time is pﬃﬃﬃ Jmin (1) ¼ 1 ½( 2 1)2 ea

a!1

¼1

Example 22.2.5 This example will ﬁnd a causal ﬁlter for Example 22.2.2, where the autocorrelation functions of the input signal S(t) and the independent noise N(t) to a

22.2 Wiener Filtering

643

FIGURE 22.2.10

system are given by RS (t) ¼ eðjtj=2Þ

and

RN (t) ¼ e2jtj

The power spectral densities of signal and noise are SS (v) ¼

4 1 þ 4v2

and

SN (v) ¼

4 4 þ v2

1. Filtering Problem: a . 0. From Example 22.2.4 we can write 1 þ v2 (1 þ 4v2 )(4 þ v2 ) pﬃﬃﬃ pﬃﬃﬃ 2 5(1 þ jv) 2 5(1 jv) ¼ (1 þ 2jv)(2 þ jv) (1 2jv)(2 jv)

SX (v) ¼ SS (v) þ SN (v) ¼ 20

G(v) ¼

(1 þ 2jv)(2 þ jv) (1 2jv)(2 jv) pﬃﬃﬃ pﬃﬃﬃ ; G (v) ¼ 2 5(1 þ jv) 2 5(1 jv)

We ﬁnd K(v) as follows: K(v) ¼ SS (v)G (v) ¼

4 (1 2jv)(2 jv) 2(2 jv) pﬃﬃﬃ ¼ pﬃﬃﬃ 2 1 þ 4v 5(1 þ 2jv)(1 jv) 2 5(1 jv)

Partial fraction expansion of K(v) yields pﬃﬃﬃ 2 5 1 1 K(v) ¼ þ 3 (1 þ 2jv) 5(1 jv)

(A)

Taking the IFT of Eq. (A), we obtain pﬃﬃﬃ 5 (t=2) 2 e u(t) þ et u(t) k(t) ¼ gopt (t)u(t) þ gopt,L (t)u(t) ¼ 3 5 k(t) is shown in Fig. 22.2.11.

(B)

644

Weiner and Kalman Filters

FIGURE 22.2.11

We can identify Gopt(v) and Gopt,L(v) as pﬃﬃﬃ 2 5 3(1 þ 2jv) þ pﬃﬃﬃ SS (v) 2 5 Gopt;L (v) ¼ ¼ ½K(v) ¼ ½SS (v) þ SN (v) 15(1 jv)

Gopt (v) ¼

SS (v) ½SS (v) þ SN (v)

¼ ½K(v)þ ¼

Hence Hopt(v) is given by pﬃﬃﬃ (1 þ 2jv)(2 þ jv) 2 5 (2 þ jv) pﬃﬃﬃ Hopt (v) ¼ G(v)Gopt (v) ¼ ¼ 3(1 þ 2jv) 3(1 þ jv) 2 5(1 þ jv) The inverse Fourier transforms of G(v), G (v), and Hopt(v) are given by pﬃﬃﬃ 5 t g(t) ¼ e u(t) 3d(t) þ 2d0 (t); 10 1 1 hopt (t) ¼ et u(t) þ d(t) 3 3

pﬃﬃﬃ 5 t g(t) ¼ e u(t) 3d(t) 2d0 (t) 10

The optimum impulse response may not be realizable because of the presence of the impulse function. To determine the minimum mean-square error Jmin, we need the function k 2(t), which is given by k2 (t) ¼

5 t 4 e u(t) þ e2t u(t) 9 25

(C)

The minimum mean-square error is calculated by substituting the causal part of Eq. (C) in Eq. (22.2.52): ð1 Jmin ¼ RS (0) 0

k2 (t)dt ¼ 1

5 9

ð1 0

et dt ¼

4 9

22.2 Wiener Filtering

645

We can also calculate Jmin using Eqs. (22.2.39) or (22.2.47): ð1

1 t 1 e þ d(t) eðt=2Þ dt 3 0 3 1 2 5 4 þ ¼1 ¼1 ¼ 3 9 9 9

ð1 Jmin ¼ RS (0)

hopt (t)RS (t)dt ¼ 1 0

The two equations yield the same result Jmin ¼ 49 ¼ 0:4444. The causal part of the noncausal solution of Example 22.2.2 for a ¼ 0 is given by 3 2t e u(t), and the causal solution of this example is hoptcausal hopt noncausal (t) ¼ 15d(t) þ 10 1 1 2t (t) ¼ 3d(t) þ 3e u(t), shown in Fig. 22.2.12. The minimum mean-square errors in both examples are given by Jminnoncausal ¼ 0.4 and Jmincausal ¼ 0.4444. The differences between the causal and noncausal solutions in this example also are not that signiﬁcant for this problem, but the complexity of getting a causal solution is much greater. From K(v) in Eq. (A) for a delay a, we can write K(v)e

jva

pﬃﬃﬃ 2 5 1 1 þ ¼ e jva 3 (1 þ 2jv) 5(1 jv) pﬃﬃﬃ 5 ½(tþa)=2 2 ½tþa e , u(t þ a) þ e u(t a) 3 5

2. Prediction Problem: a . 0. For the prediction problem of a . 0, the causal part of k(t þ a) is given by k(t þ a)causal

pﬃﬃﬃ 5 ½(tþa)=2 e ¼ u(t), 3

a.0

and the FT of the preceding equation yields ½K(v)e

jva)

pﬃﬃﬃ (a=2) pﬃﬃﬃ 5e 2 5e(a=2) ¼ þ ¼ 1 3(1 þ 2jv) 3 2 þ jv

FIGURE 22.2.12

646

Weiner and Kalman Filters

and substituting the preceding equation in Eq. (22.2.49), we obtain the optimum transfer function for the prediction problem

Hopt (v) ¼ G(v)½K(v)e ¼

jva

pﬃﬃﬃ (1 þ 2jv)(2 þ jv) 2 5e(a=2) pﬃﬃﬃ þ ¼ 3(1 þ 2jv) 2 5(1 þ jv)

(2 þ jv) (a=2) e 3(1 þ jv)

and the impulse response, given by 1 hopt (t) ¼ e½tþ(a=2) u(t) þ d(t), 3

a.0

Since a . 0, the minimum mean-square error is obtained by substituting the causal part of Eq. (C), k2 (t) ¼ 59 et u(t), in Eq. (22.2.52) to obtain ð1 Jmin ¼ RS (0)

2

ð1

k (t)dt ¼ 1 a

a

5 t 5 e dt ¼ 1 ea , 9 9

a.0

3. Smoothing Problem: a , 0. For the smoothing problem of a , 0, the causal part of k(t þ a) is

k(t þ a)causal

pﬃﬃﬃ 5 ½(tþa)=2 2 e ¼ u(t þ a) þ etþa ½u(t a) u(t) , 3 5

a,0

and the FT of the equation above yields

½K(v)e

jva

pﬃﬃﬃ 2 5 e jva 1 e jva ea þ þ ¼ 3 1 þ 2jv 5 (1 jv) pﬃﬃﬃ 2 5 3e jva (2 jv) (1 þ 2jv)ea ¼ 5(1 þ 2jv)(1 jv) 3

and substituting the preceding equation in Eq. (22.2.49), we obtain the optimum transfer function for the smoothing problem: pﬃﬃﬃ (1 þ 2jv)(2 þ jv) 2 5 3e jva (2 jv) (1 þ 2jv)ea pﬃﬃﬃ Hopt (v) ¼ 5(1 þ 2jv)(1 jv) 3 2 5(1 þ jv) jva 1 3e (4 þ v2 ) (2 þ jv)(1 þ 2jv)ea ¼ 5(1 þ v2 ) 3

22.3 Discrete Kalman Filter

647

FIGURE 22.2.13

For obtaining the minimum mean-square error, we note that since a , 0, we have to use the entire Eq. (C) in Eq. (22.2.52) and write ð ð1 ða 5 4 0 2t 2 t k (t)dt ¼ 1 e dt þ e dt Jmin ¼ RS (0) 9 25 a a 0 5 2 4 2 ¼1 (1 e2a Þ þ 1 ¼ (1 e2a ), a , 0 9 25 9 45 The graph of the minimum mean-square error Jmin (a) for values of a ranging from 21 to 1 is shown in Fig. 22.2.13. Jmin reaches a minimum value for inﬁnite lag, and this value is given by 4 2 4 2 2 2a Jmin (1) ¼ ½(1 e ) ¼ ¼ 9 45 9 49 5 a!1 and the maximum error for inﬁnite prediction time is 5 ¼1 Jmin (1) ¼ 1 ea 9 a!1

B 22.3

DISCRETE KALMAN FILTER

The Wiener ﬁlter discussed in the previous sections was derived under the imposition of the physical realizability condition on the ﬁlter. As a consequence of this condition, Eq. (22.2.12) is not a convolution, and transform techniques could not be directly used, so we had to resort to spectral factorization. Further, the input processes are assumed to be stationary, and the system is linear time-invariant. To circumvent these and other difﬁculties, Kalman and Bucy [27] proposed a recursive linear digital ﬁlter that accommodates nonstationary processes. We will motivate the concept of recursive estimation.

Sections 22.3 and 22.4 are adapted with permission from Chapter 9 of the book, Nonlinear FItering and Smoothing, by Venkatarama Krishnan, originally published by John Wiley and Sons in 1984, and republished by Dover Publications in 2005.

648

Weiner and Kalman Filters

Recursive Estimation

We will assume that fXi, i ¼ 1,2, . . . , kg is a discrete signal random process and that fYi, i ¼ 1,2, . . . , kg is an observation or measurement random process related to the signal process by the equation i ¼ 1, 2, . . . , k

Y i ¼ X i þ Vi ,

(22:3:1)

where fVi, i ¼ 1,2, . . . , kg is a zero mean white-noise sequence with E½Vi Vj ¼ s2V dij , i, j ¼ 1,2, . . . , k. We will further assume that Xi ¼ X for all i. The schematic is shown in Fig. 22.3.1. We want to estimate the process fXi ¼ Xg given the observation process fYi, i ¼ 1, ^ the con2, . . . , kg. From Eq. (18.3.10) the minimum variance unbiased estimator is X, ditional expectation of X given fYi, i ¼ 1,2, . . . , kg, which can be written as 1 X^ k ¼ k

k X

Yi

(22:3:2)

i¼1

An additional observation Ykþ1 is available at time k þ 1. The new estimate is given by X^ kþ1 ¼

kþ1 1 X Yi k þ 1 i¼1

(22:3:3)

However, the estimate given by Eq. (22.3.3) processes all observations fYi, i ¼ 1, . . . , k þ 1g again, and this is unacceptable for machine computation. We want to use the prior estimate X^ from Eq. (22.3.2) to save on computation time. Hence we can rewrite Eq. (22.3.3) as follows: kþ1 k 1X Ykþ1 k ^ 1 ¼ Xk þ Ykþ1 Yi þ kþ1 kþ1 k þ 1 k i¼1 kþ1 1 1 1 Ykþ1 ¼ X^ k þ (Ykþ1 X^ k ) ¼ 1 X^ k þ kþ1 kþ1 kþ1

X^ kþ1 ¼

(22:3:4)

In Eq. (22.3.4) the term (Ykþ1 X^ k ) is the innovations process, so called because it contains new information. We have utilized the past estimate to update the present estimate, thus achieving recursive action. More generally, the ﬁrst equation in Eq. (22.3.4) can be expressed in the form X^ kþ1 ¼ Kk0 X^ k þ Kk Ykþ1

(22:3:5)

where the weighting coefﬁcients Kk0 and Kk have to be determined using the optimality criteria of unbiasedness and minimum variance. We will reformulate this problem so that the solution in the general case will be clear. The time just before the observation

FIGURE 22.3.1

22.3 Discrete Kalman Filter

649

FIGURE 22.3.2

Ykþ1 is denoted by (k þ 1)2, and the time just after the observation is denoted by (k þ 1)þ. Hence X^ kþ1 () is the estimate just before the observation, and X^ kþ1 (þ) is the estimate just after the observation. The timing diagram of Fig. 22.3.2 illustrates these quantities. The evolution of the state between the times kþ and (k þ 1)2 is governed by the dynamics of the system, and in this case, since Xk is a constant X for all times, we have X^ k (þ) ¼ X^ kþ1 ()

Xk (þ) ¼ X and Xkþ1 () ¼ X and hence

(22:3:6)

Thus there are two distinct zones: (1) between observations, where the estimate is governed by the dynamics of the system; and (2) across observations, where the estimate is updated by the new observation. The recursive formulation of Eq. (22.3.4) can be written in two stages as Between observations:

X^ kþ1 () ¼ X^ k (þ)

Across observations: X^ kþ1 (þ) ¼ X^ kþ1 () þ

1 ½Ykþ1 X^ kþ1 () kþ1 Innovations

(22:3:7)

The two equations in Eq. (22.3.7) can be combined into a single equation as X^ kþ1 (þ) ¼ X^ k (þ) þ

1 ½Ykþ1 X^ k (þ) kþ1

(22:3:8)

which is Eq. (22.3.4) in a more precise form. Formulation of Discrete Kalman Filter

We can now formulate the recursive discrete nonstationary ﬁltering problem. Boldface will represent both matrices and vectors. A discrete linear nonstationary signal process is represented by the vector difference equation Signal process:

Xk ¼ Fk, k1 Xk1 þ Gk1 Wk1 ,

k [ {1, 2, . . . , N}, X0

(22:3:9)

The observation process is another discrete linear process represented by Observation process:

Yk ¼ Hk Xk þ Vk ,

k [ {1, 2, . . . , N}

(22:3:10)

^ k for the Given the observations fYk, k [ f1, 2, . . . , Ng, we have to ﬁnd the estimate X signal Xk under the optimality criteria of unbiasedness and minimum variance, and this ^ k ¼ E½X j Yk , k [ {1, 2, . . . , N}. estimate is given by the conditional expectation X

650

Weiner and Kalman Filters

The following assumptions are made regarding Eqs. (22.3.9) and (22.3.10). Assumptions 1. The signal process fXk, k ¼ [1, 2, . . ., N ]g with initial condition X0 is an n-vector, and the observation process fYk, k ¼ [1, 2, . . ., N ]g with initial condition Y0 ¼ 0 is an m-vector at time t ¼ tk. 2. Fk,k21 is an n n nonsingular discrete signal transition matrix from time t ¼ tk21 to time t ¼ tk. 3. Gk is an n r discrete nonrandom rectangular matrix at time t ¼ tk. 4. Wk is an r-vector of zero mean white Gaussian signal noise sequence with covariance matrix E½Wk WTk ¼ Qk . 5. Vk is an m-vector of zero mean white Gaussian observation noise sequence with covariance matrix E½Vk VTk ¼ Rk where Wk and Vk are independent, E½Wk VTk ¼ 0. All quantities are evaluated at time t ¼ tk. 6. Hk is an m n discrete nonrandom rectangular weighting matrix at time t ¼ tk. 7. The Gaussian initial condition vector X0 is assumed independent of both the noise vector sequences fWkg and fVkg. ^ k is the expectation of Uk 8. If Uk is any random vector sequence, then the estimate U conditioned on the observations fYk, k ¼ [1, 2, . . . , N ]g and will be represented by ^ k ¼ E½Uk j Yk , k ¼ 1, 2, . . . , N. Ek ½Uk ¼ U ^ k Xk ) and the noise vector 9. The estimation error vector sequence Xk ¼ (X sequences fWkg and fVkg are independent. 10. The innovations sequence fnk, k ¼ [1,2, . . . , N ]g is also an m-vector, given by ^ k, nk ¼ Yk Hk X

k ¼ 1, 2, . . . , N

(22:3:11)

^ k Xk ½X ^ k Xk T is 11. The n n error covariance matrix deﬁned by, Pk ¼ E½X positive deﬁnite. Under the these assumptions, the discrete Kalman ﬁlter takes the form as shown: Across Observations. At the observation time tk, the conditional expectations ^ k (þ) and Ek ½Xk ^ k () satisfy ¼ X Ek ½Xk tk (þ) ¼ X tk () ^ k (), ^ k () þ Kk ½Yk Hk X ^ k (þ) ¼ X X ^ 0 ¼ X0 k [ {1, 2, . . . , N}, X

(22:3:12a)

and the covariance matrix Pk(þ) satisﬁes Pk (þ) ¼ (I Kk Hk )Pk (),

k [ {1, 2, . . . , N}

(22:3:12b)

where the n m matrix Kk, called the Kalman gain, is given by Kk ¼ Pk ()HTk ½Hk Pk ()HTk þ Rk 1 ,

k [ {1, 2, . . . , N}

(22:3:13)

Between Observations. Between observation times tk21 and tk the estimates satisfy ^ k1 (þ), ^ k () ¼ Fk, k1 X X

k [ {1, 2, . . . , N}

(22:3:14a)

22.3 Discrete Kalman Filter

651

and the covariance matrix satisﬁes Pk () ¼ Fk, k1 Pk1 (þ)FTk, k1 þ Gk1 Qk1 GTk1 , k [ {1, 2, . . . , N}

(22:3:14b)

^ k () be the We can gain insight into the problem by deriving these equations. Let X estimate determined at time tk(2) and Yk, the observation at time tk. We will use the prior ^ k (þ) at time tk(þ). We ^ k () and the observation Yk to get a new estimate to X estimate X can write an equation analogous to the scalar Eq. (22.3.5) as ^ ^ k (þ) ¼ K0 X X k k () þ Kk Yk

(22:3:15)

0

where the weighting n n matrix Kk and weighting n m matrix Kk have to be determined using the two criteria of unbiasedness and minimum variance. If Xk is the true unobservable value of the state at time tk, we can deﬁne estimation ˜ k () at time tk () as ˜ k (þ) at time tk(þ) and X errors X ^ k (þ) Xk ˜ k (þ) ¼ X X ˜ k () ¼ X ^ k () Xk X

(22:3:16)

We can convert Eq. (22.3.15) into an error equation by subtracting Xk from both sides: ^ k () Xk þ K0 Xk Xk þ Kk Yk ^ k (þ) Xk ¼ K0 ½X X k k

(22:3:17)

Substituting for Yk from Eq. (22.3.10) and error matrices from Eq. (22.3.16) in Eq. (22.3.17), we obtain 0 ˜ ˜ k (þ) ¼ K0 X X k k () þ ½Kk þ Kk Hk I Xk þ Kk Vk

(22:3:18)

Taking expected value of both sides of Eq. (22.3.18), we obtain ˜ k (þ) ¼ K0 E½X ˜ k () þ ½K0 þ Kk Hk IE½Xk þ Kk E½Vk E½X k k

(22:3:19)

^ k is to be unbiased, then we must have If the estimator X ˜ k (þ) ¼ 0 ¼ E½X ˜ k () E½X

(22:3:20)

Hence, with E[Vk] ¼ 0 and E[Xk] = 0 in Eq. (22.3.19), we deduce K0k þ Kk Hk I ¼ 0

or K0k ¼ I Kk Hk

(22:3:21)

We have thus expressed the matrix K0k in terms of the matrix Kk. Across Observations. Substituting Eq. (22.3.21) in Eq. (22.3.15), we establish the signal update equation: ^ k (), ^ k () þ Kk ½Yk Hk X ^ k (þ) ¼ X X

^0 ¼X ^ 0 (22:3:22) k [ {1, 2, . . . , N}, X

Thus yet unknown matrix Kk is determined using the minimum variance criterion. The estimation error equation [Eq. (22.3.18)] can be rewritten by substituting K0k as ˜ k (þ) ¼ ½I Kk Hk X ˜ k () þ Kk Vk X ˜ k () þ Kk ½Vk Hk X ˜ k () ¼X

(22:3:23)

652

Weiner and Kalman Filters

Equation (22.3.12a) represents the estimate update, whereas Eq. (22.3.23) represents the estimate error update. We can now deﬁne the error covariance matrices Pk(þ) and Pk(2) as T

˜ k (þ)X ˜ (þ); Pk (þ) ¼ E½X k

T

˜ k ()X ˜ () Pk () ¼ E½X k

(22:3:24)

Substituting Eq. (22.3.23) in Eq. (22.3.24), we obtain ˜ k () ˜ k () þ Kk ½Vk Hk X Pk ðþÞ ¼ E X ˜ k ())T ˜ k () þ Kk ½Vk Hk X (X

¼ (I Kk Hk )Pk ()(I Kk Hk )T þ Kk Rk KTk

(22:3:25)

The variance function that should be minimized is T

˜ k (þ) ¼ Tr Pk (þ) ˜ (þ)X E½X k

(22:3:26)

where Tr Pk (þ) is the trace of the covariance matrix Pk (þ). We can minimize Eq. (22.3.26) with respect to the matrix Kk by solving

@ @ Tr Pk (þ) ¼ Tr (I Kk Hk )Pk ()(I Kk Hk )T þ Kk Rk KTk @Kk @Kk ¼0 (22:3:27) Equation (22.3.27) is solved by using the result from Eq. (16.3.23) stating that if a matrix B is symmetric and A is any other matrix, then @ Tr½ABAT ¼ 2AB @A

(16:3:23)

Applying Eq. (16.3.23) to Eq. (22.3.27) results in @ Tr Pk (þ) ¼ 2(I Kk Hk )Pk ()HTk þ Kk Rk ¼ 0 @Kk

(22:3:29)

Solving for Kk in Eq. (22.3.29), we obtain the equation for the Kalman gain matrix Kk ¼ Pk ()HTk ½Hk Pk ()HTk þ Rk 1 ,

k [ {1, 2, . . . , N}

(22:3:30)

which is Eq. (22.3.13). Substituting Kk from Eq. (22.3.30), in Eq. (22.3.25), we obtain Eq. (22.3.12b): Pk (þ) ¼ (I Kk Hk )Pk (),

k [ {1, 2, . . . , N}

(22:3:31)

We have shown the result for “across observations”. Between Observations. The conditional expectation of the signal process shown below Xk ¼ Fk, k1 Xk1 þ Gk1 Wk1 ;

k [ {1, 2, . . . , N}, X0

(22:3:32)

conditioned on the observations fYk21g, is given by ^ k1 (þ) þ Gk1 W ^ k1 , ^ k () ¼ Fk, k1 X X

k [ {1, 2, . . . , N}

(22:3:33)

^ k () ¼ E½Xk j {Yk1 }. Since Wk21 is independent of the observations where X ^ k1 ¼ E½Wk1 j {Yk1 } ¼ E½Wk1 ¼ 0, and Eq. (22.3.33) reduces to fYk21g, W Eq. (22.3.14a): ^ k1 (þ), ^ k () ¼ Fk, k1 X X

k [ {1, 2, . . . , N}

(22:3:34)

22.3 Discrete Kalman Filter

653

FIGURE 22.3.3

To demonstrate Eq. (22.3.14b) we subtract Eq. (22.3.32) from Eq. (22.3.34) to obtain the error equation ˜ k1 (þ) Gk1 Wk1 ˜ k () ¼ Fk, k1 X X

(22:3:35)

and form the error covariance matrix Pk () ¼ E½Xk ()X()Tk , given by ˜ k1 (þ) Gk1 Wk1 ½Fk, k1 X ˜ k1 (þ) Gk1 Wk1 T Pk () ¼ E ½Fk, k1 X T T ˜ k1 (þ)X ˜ T (þ)FT ¼ E Fk, k1 X k, k1 þ Gk1 Wk1 Wk1 Gk1 k1 ˜ k1 (þ)WT GT Gk1 Wk1 X ˜ T (þ)FT (22:3:36) Fk, k1 X k, k1 k1 k1 k1 From assumption 9 (above) that the estimation errors are independent of the noise sequences, Eq. (22.3.36) can be simpliﬁed to Eq. (22.3.14b): Pk () ¼ Fk, k1 Pk1 (þ)FTk, k1 þ Gk1 Qk1 GTk1 ,

k [ {1, 2, . . . , N}

(22:3:37)

It is to be particularly noted that the variance update equation [Eq. (22.3.31)] does not depend on either the estimate or the observations, whereas the signal update [Eq. (22.3.22)] is dependent on the covariance matrix via the Kalman gain. The timing diagram similar to Fig. 22.3.2 for the realization of the Kalman ﬁlter is shown in Fig. 22.3.3. The block diagram for the implementation of the discrete Kalman ﬁlter is shown in Fig. 22.3.4.

FIGURE 22.3.4

654

Weiner and Kalman Filters

Comments 1. The equations for across and between measurements can be combined into a single equation for both the estimated mean and the unconditional variance, and written as follows: ^ k1 () þ Kk1 ½Yk1 Hk1 X ^ k1 () , ^ k () ¼ Fk, k1 X X k [ {1, 2, . . . , N} Pk () ¼ Fk, k1 ½(I

(22:3:38)

Kk1 Hk1 )Pk1 ()FTk, k1

þ Gk1 Qk1 GTk1 ,

k [ {1, 2, . . . , N}

(22:3:39)

where, from Eq. (22.3.30), we obtain Kk1 ¼ Pk1 ()HTk1 ½Hk1 Pk1 ()HTk1 þ Rk1 1 , k [ {1, 2, . . . , N}

(22:3:40)

2. The error covariance update equation [Eq. (22.3.31)] Pk (þ) ¼ (I Kk Hk )Pk (),

k [ {1, 2, . . . , N}

(22:3:31)

can be expressed in an alternate form as 1 T 1 P1 k (þ) ¼ Pk () þ Hk Rk Hk

(22:3:41)

3. The Kalman gain matrix Kk given by Eq. (22.3.30) Kk ¼ Pk ()HTk ½Hk Pk ()HTk þ Rk 1 ,

k [ {1,2, . . . , N}

(22:3:30)

can also be expressed in an alternate form as Kk ¼ Pk (þ)HTk R1 k

(22:3:42)

4. If the signal and observation noises are not assumed Gaussian, then the variance equation will be a conditional variance that will be dependent on the third conditional moment. The Gaussian assumption renders the third moments zero, resulting in an unconditional variance. Hence the signal estimate equation [Eq. (22.3.38)] and the variance equation [Eq. (22.3.39)] are not coupled as stated earlier. As a consequence, the Kalman gain and covariance matrix can be

FIGURE 22.3.5

22.3 Discrete Kalman Filter

655

processed independently of the observations or the state estimate. This is of immense value in navigation problems. The propagation of the covariance matrix Pk(þ) and the evaluation of the Kalman gain Kk are shown in Fig. 22.3.5. The Kalman ﬁlter algorithm is summarized in the following list: Summary of Discrete Kalman Filter Algorithm In this list N stands for the discrete data set N ¼ f1, 2, . . . , Ng. Signal or State Process: Xk ¼ Fk, k1 Xk1 þ Gk1 Wk1 ;

k [ N; X0

(22:3:9)

Observation or Measurement Process: Yk ¼ Hk Xk þ Vk ,

k [ N; Y0 ¼ 0

(22:3:10)

Signal Error Process: ^ k Xk Xk ¼ X Prior Statistics: E½Wk ¼ E½Vk ¼ 0; E½Wk WTk ¼ Qk , E½Vk VTk ¼ Rk E½Wk VTk ¼ E½Wk XT0 ¼ E½Vk XT0 ¼ 0

(22:3:43)

cov½X0 ¼ E½X0 XT0 ¼ P0 ; cov½Xk ¼ E½Xk XTk ¼ Pk Filter Algorithm Across Observations Variance Update Equation: Pk (þ) ¼ (I Kk Hk )Pk (1) or T 1 ¼ P1 k () þ Hk Rk Hk ,

P1 k (þ) k[N

(22:3:12b)

Kalman Gain Update Equation: Kk ¼ Pk ()HTk ½Hk Pk ()HTk þ Rk 1 ¼ Pk (þ)HTk R1 k ,

k[N

(22:3:13)

Signal Estimate Update Equation: ^ k (þ) ¼ X ^ k (), ^ k () þ Kk ½Yk Hk X X

^ 0 ¼ X0 k [ N, X

(22:3:12a)

Between Observations Variance Extrapolation: Pk () ¼ Fk, k1 Pk1 (þ)FTk, k1 þ Gk1 Qk1 GTk1 ,

k[N

(22:3:14b)

Signal Estimate Extrapolation: ^ k1 (þ), ^ k () ¼ Fk, k1 X X Example 22.3.1

k[N

The observation process is given by a scalar equation: Yk1 ¼ Xk1 þ Vk1 , k [ {1, 2, . . . , N}

(22:3:14a)

656

Weiner and Kalman Filters

where the variance of the noise sequence fVkg is Rk ¼ var½Vk ¼ s2V and the scalar covariance matrices Pk(þ) ¼ pk(þ) and Pk(2) ¼ pk(2). We will investigate the propagation of the error variance when the next observation Yk is available. Using the more convenient covariance equation [Eq. (22.3.40)], we obtain 1 1 1 ¼ þ 2 pk (þ) pk () sV

or

pk (þ) ¼

pk () 1 þ pk ()=s2V

We observe that if the observation uncertainty s2V is low, then the additional observation Yk reduces the estimation uncertainty pk(þ) drastically, or pk(þ) pk (2). On the other hand, if s2V is high, then the estimation uncertainty remains practically the same: pk (þ) pk(2). An important fact to note is that an additional observation does not degrade the estimation accuracy! The Kalman gain kk is computed from Eq. (22.3.41) as follows: kk ¼ pk (þ) 1

1 pk () 1 pk () 1 ¼ ¼ ¼ s2V 1 þ pk ()=s2V s2V s2V þ pk () 1 þ s2V =pk ()

For low measurement variance, the Kalman gain is approximately 1; for high measurement variance, the gain becomes 0. As stated earlier, the variance and the gain can be processed independently of the observations.

Example 22.3.2

A zero mean signal and observation processes are given by

Xk ¼ Xk1 ,

k [ {1, 2, . . . , N};

Y k ¼ Xk þ Vk ,

k [ {1, 2, . . . , N}

where fVkg is a zero mean Gaussian white noise sequence with variance s2V . The initial error variance is given by E½X^ 0 X0 2 ¼ p0 . Across Observations. The Kalman gain kk is obtained from Eq. (22.3.13) as kk ¼ pk () 1 ½1 pk () 1 þ s2V 1 ¼

pk () pk () þ s2V

and from Eq. (22.3.41) it can also be given by kk ¼ pk (þ) 1

1 pk (þ) ¼ 2 sV s2V

The variance is obtained from Eq. (22.3.12b) as pk () pk ()s2V pk () ¼ pk () ¼ pk (þ) ¼ 1 2 pk () þ s2V 1 þ pk ()=s2V pk () þ sV

(A)

which is the same as obtained in the previous example. The signal estimate is obtained from Eq. (22.3.12a) as pk (þ) pk () ½Yk X^ k () ¼ X^ k () þ ½Yk X^ k () X^ k (þ) ¼ X^ k () þ 2 sV pk () þ s2V Between Observations. From Eqs. (22.3.14), we obtain X^ kþ1 () ¼ X^ k (þ);

pkþ1 () ¼ pk (þ)

(B)

22.3 Discrete Kalman Filter

657

Closed-Form Solution. Substituting pk () ¼ pk1 (þ) in Eq. (A), we obtain pk (þ) ¼

pkþ1 (þ) 1 þ pk1 (þ)=s2V

The preceding difference equation can be solved as follows: p0 p1 (þ) ¼ 1 þ p0 =s2V p0 p1 (þ) p0 1 þ p0 =s2V p2 (þ) ¼ ¼ ¼ p0 1 þ 2p0 =s2V 1 þ p1 (þ)=s2V 1 þ s2V (1 þ p0 =s2V ) and by induction the solution is pk (þ) ¼

p0 1 þ kp0 =s2V

and from Eq. (B)

pkþ1 () ¼

p0 1 þ kp0 =s2V

Hence the Kalman gain will be given by kk ¼

pk (þ) p0 ¼ 2 2 sV þ kp0 sV

The signal estimate equation can be rewritten as X^ k (þ) ¼ X^ k () þ

p0 ½Yk X^ k () s2V þ kp0

and from Eq. (B) X^ kþ1 () ¼ X^ k () þ

p0 ½Yk X^ k () s2V þ kp0

Remarks. As the number k of observations increases, pkþ1(2) decreases and the estimation accuracy improves. Kalman Filter—Scalar Case

For a clearer understanding of the Kalman ﬁlter, we shall take a scalar case and derive the ﬁlter equations. Let N represent the observation interval given by N ¼ f1,2, . . . ,Ng. The signal and the observation processes for a scalar case are given by Xk ¼ fXk1 þ Wk , ¼ h k X k þ Vk ,

k[N k[N

(22:3:44)

where fWkg and fVkg are zero mean Gaussian white-noise sequences with variances s2W and s2V respectively. The initial error variance is given by E½X^ 0 X0 2 ¼ p0 . Across Observations. The Kalman gain kk is obtained from Eq. (22.3.13) as, kk ¼ pk ()hk ½h2k pk () þ s2V 1 ¼

pk ()hk pk (þ)hk ¼ h2k pk () þ s2V s2V

(22:3:45)

658

Weiner and Kalman Filters

The variance is obtained from Eq. (22.3.12b) as follows: pk ()hk hk pk ()s2V pk (þ) ¼ 1 2 pk () ¼ 2 2 hk pk () þ sV hk pk () þ s2V pk () ¼ 1 þ h2k pk ()=s2V

(22:3:46)

The signal estimate is obtained from Eq. (22.3.12a) as follows: X^ k (þ) ¼ X^ k () þ kk ½Yk hk X^ k () ¼ X^ k () þ

pk ()hk 2 hk pk () þ s2V

½Yk hk X^ k ()

(22:3:47)

Between Observations. From Eqs. (22.3.14a) and (22.3.14b), we obtain X^ k () ¼ fX^ kþ1 (þ);

pk () ¼ f2 pk1 (þ) þ s2W

(22:3:48)

Substituting for pk(2) from Eq. (22.3.48) in Eq. (22.3.46), we obtain pk (þ) ¼

1þ

f2 pk1 (þ) þ s2W 2 hk ½f2 pk1 (þ) þ s2w =s2V

(22:3:49)

Since the ﬁrst observation comes at t ¼ 1, we have p0(2) ¼ p0(þ) ¼ p0 and p1(þ) is p1 (þ) ¼

f2 p0 þ s2W 1 þ h2k ½f2 p0 þ s2W =s2V

(22:3:50)

Hence pk(þ) can be solved recursively. The Kalman gain kk is given by Eq. (22.3.45). The signal estimate extrapolation is obtained from Eq. (22.3.14a) as X^ k () ¼ fX^ k1 (þ)

(22:3:51)

Example 22.3.3 The signal and observation processes are given by Eq. (22.3.44) with f ¼ 45 and hk ¼ 65. The zero mean white-noise signals have variances s2W ¼ 1 and s2V ¼ 1. The initial variance p0 ¼ 45. The initial value X0 ¼ 2.5403, and the observations fYkg for N ¼ 20 are given in Table 22.3.1. TABLE 22.3.1 k

Yk

k

Yk

0 1 2 3 4 5 6 7 8 9 10

5.5629 0.5982 2.9623 3.4911 2.3531 20.3591 20.3427 2.4036 1.3624 2.8077 3.2718

11 12 13 14 15 16 17 18 19 20

0.7010 4.2405 0.9243 1.3428 3.5725 1.3205 0.9066 2.9516 2.7597 1.8991

22.3 Discrete Kalman Filter

659

We will formulate the Kalman ﬁlter for two cycles and then use the recursive formulas. Since the ﬁrst observation arrives at t ¼ 1, we have p0(2) ¼ p0(þ) ¼ p0 ¼ 45, and substituting this value of p0 in Eq. (22.3.50), we obtain Cycle 1: 2 4 4 þ1 4725 5 5 ¼ 0:4759 ¼ p1 (þ) ¼ 2 9929 4 4 2 þ1 6 5 5 1þ 1 5 The Kalman gain k1 is obtained from Eq. (22.3.45) as k1 ¼

p1 (þ)hk 4725 6 5670 ¼ ¼ ¼ 0:5711 1 9929 5 9929

Since X^ 0 (þ) ¼ X0 ¼ 2:5403, we can determine X^ 1 () from Eq. (22.3.48) as X^ 1 () ¼ 0:8 2:5403 ¼ 2:0322. Hence X^ 1 (þ) is obtained from Eq. (22.3.47) as follows: X^ 1 (þ) ¼ X^ 1 () þ k1 ½Y1 h1 X^ 1 () ¼ 2:0322 þ 0:5711 (0:5982 1:2 2:0322) ¼ 0:9811 Cycle 2. From Eq. (22.3.49), we have 2 4 4725 þ1 323, 825 5 9929 ¼ ¼ 0:4532 p2 (þ) ¼ 2 714, 533 4 4725 2 þ1 6 5 9929 1þ 1 5 The Kalman gain k2 is obtained from Eq. (22.3.45) as k2 ¼

p2 (þ)hk 323, 825 6 388, 590 ¼ ¼ ¼ 0:5438 1 714, 533 5 714, 533

Since X^ 1 (þ) ¼ 0:9811, we can determine X^ 2 () from Eq. (22.3.48) as X^ 2 () ¼ 0:8 0:9811 ¼ 0:7849 Hence X^ 2 (þ) is obtained from Eq. (22.3.47) as follows: X^ 2 (þ) ¼ X^ 2 () þ k2 ½Y2 h2 X^ 2 () ¼ 0:7849 þ 0:5438 (2:9623 1:2 0:7849) ¼ 1:8836 We can now use the recursive formulas [Eqs. (22.3.49), (22.3.45), and (22.3.47)] to obtain pk(þ), kk and X^ k (þ) respectively for observation interval N ¼ 20. Figure 22.3.6 shows the simulated signal process Xk, the observation data Yk, and the estimated data X^ k (þ) for k ¼ 0, . . . , 20. Figure 22.3.7 shows the Kalman gain kk for k ¼ 0, . . . ,10. The ﬁgure shows that from the ﬁfth observation the gain is a constant at 0.5415 and further observations do not improve the estimation accuracy.

660

Weiner and Kalman Filters

FIGURE 22.3.6

FIGURE 22.3.7

B 22.4

CONTINUOUS KALMAN FILTER

We can now derive the analogous Kalman ﬁlter for the continuous-time problem. The continuous linear signal process can be represented in the time interval T by an n-vector stochastic differential equation dX(t) ¼ F(t)X(t) þ G(t)W(t), dt

t [ T, X0

(22:4:1)

The corresponding linear observation process is given by Y(t) ¼ H(t)X(t) þ V(t),

t[T

(22:4:2)

^ for the signal X(t) Given the observations fY(t), t [ Tg, we have to ﬁnd the estimate X(t) under the optimality criteria of unbiasedness and minimum variance, and this estimate is ^ ¼ E½X(t) j Y(t), t [ T. will be given by the conditional expectation X(t) Assumptions 1. The n-vector process {X(t), t [ T} and the m-vector process {Y(t), t [ T} are random processes deﬁned on a suitable probability space.

22.4

Continuous Kalman Filter

661

2. F(t) is a nonsingular n n matrix, G(t) is an n r rectangular matrix, and H(t) is an m n rectangular matrix such that ð jFij (t)jdt , 1, i ¼ 1, 2, . . . , n, j ¼ 1, 2, . . . , n t[T

ð

jGij (t)j2 dt , 1, i ¼ 1, 2, . . . , n,

j ¼ 1, 2, . . . , r

jHij (t)jdt , 1,

j ¼ 1, 2, . . . , n

(22:4:3)

t[T

ð

i ¼ 1, 2, . . . , m,

t[T

3. W(t) is an r-vector of zero mean Gaussian white-noise processes with covariance parameter matrix Q(t), given by Q(t) ¼ E[W(t)W T(t)]dt. 4. V(t) is an m-vector of zero mean Gaussian white-noise processes with covariance parameter matrix R(t), given by R(t) ¼ E[V(t)V T(t)]dt. 5. The noise vectors W(t) and V(t) are independent, or E[W(t)V T(t)] ¼ 0. 6. The Gaussian initial condition vector X0 is independent of both the noise vectors W(t) and V(t), or E[W(t)X T0 ] ¼ 0 and E[V(t)X T0 ] ¼ 0. ^ is the expectation of 7. If U(t) is any random process vector, then the estimate U(t) U(t) conditioned on the observation fY(s), s tg and will be represented by ^ ¼ E½U(t) j Y(s), Et ½U(t) ¼ U(t)

s t:

8. The innovations process fn(t), t [ Tg is an m-vector white-noise process given by ^ n(t) ¼ Y(t) H(t)X(t),

t[T

(22:4:4)

n o ^ X(t)½X(t) ^ X(t)T is positive 9. The error covariance matrix P(t) ¼ E ½X(t) deﬁnite. Under these assumptions the continuous Kalman ﬁlter can be given by the following equations. Since this is a continuous-time process, we cannot distinguish between ^ satisﬁes across and between observations. The conditional expectation Et ½X(t) ¼ X(t) the stochastic differential equation d ^ ^ þ K(t)½Y(t) H(t)X(t), ^ X(t) ¼ F(t)X(t) dt

t[T

(22:4:5)

and the covariance matrix P(t) satisﬁes the stochastic matrix Riccati equation d P(t) ¼ F(t)P(t) þ P(t)FT (t) þ G(t)Q(t)GT (t) K(t)H(t)P(t), dt

t[T

(22:4:6)

where the Kalman gain matrix K(t) satisﬁes K(t) ¼ P(t)HT (t)R1 (t),

t[T

(22:4:7)

We can show these results by discretizing the differential equations and comparing them to the discrete Kalman ﬁlter. Assuming a uniform discretization interval of Dt ¼ tk 2 tk21, k ¼ 1, . . .,N, we can write Eq. (22.4.1) as Xk ¼ (I þ Fk1 Dt)Xk1 þ Gk1 Wk1 Dt,

k [ {1, 2, . . . , N}, X0

(22:4:8)

662

Weiner and Kalman Filters

where the subscript k stands for the value of the processes at time t ¼ tk. Comparing Eq. (22.4.8) to Eq. (22.3.9), we can identify I þ Fk1 Dt ¼ Fk, k1 ;

Gk1 Dt ¼ Gk1

(22:4:9Þ

From assumptions 3 and 4, the noise covariance matrices are given by E½W(t)WT (t) ¼

Q(t) ; dt

E½V(t)VT (t) ¼

R(t) dt

(22:4:10)

As the discretization interval is Dt, the covariance matrices of W(t) and V(t) at time tk can be given in terms of the discrete covariance matrices Qk and Rk as Q(tk ) ¼ Qk ; Dt

R(tk ) ¼ Rk Dt

(22:4:11)

Since the limit will be taken as Dt ! 0, the distinction between tk(þ) and tk(2) will be eliminated and written as tk(þ) ¼ tk(2) ¼ tk. After substituting Eqs. (22.4.9) and (22.4.11), the Kalman gain equation [Eq. (22.3.40)] can be rewritten for k ¼ 1, . . .,N: R(tk1 ) 1 Kk1 ¼ Pk1 HTk1 Hk1 Pk1 HTk1 þ Dt

1 ¼ Pk1 HTk1 Hk1 Pk1 HTk1 Dt þ Rk1 Dt

(22:4:12)

Dividing Eq. (22.4.12) by Dt and taking the limit as Dt ! 0, we obtain lim

Dt!0

Kk1 ¼ K(t) ¼ P(t)HT (t)R1 (t), Dt

t[T

(22:4:13)

which is Eq. (22.4.7). Similarly, substituting Eqs. (22.4.9) and (22.4.11) in the covariance equation [Eq. (22.3.39)], we can write Pk ¼ ½I þ Fk1 Dt½ðI Kk1 Hk1 )Pk1 ½I þ Fk1 DtT þ Gk1 Dt

Q(tk ) T Gk1 Dt Dt

¼ ½I þ Fk1 DtPk1 ½I þ Fk1 DtT ½I þ Fk1 Dt½Kk1 Hk1 Pk1 ½I þ Fk1 DtT þ Gk1 Q(tk )GTk1 Dt,

k [ {1, 2, . . . , N}

(22:4:14)

Substituting for Kk21 ¼ K(t)Dt from Eq. (22.4.13) in Eq. (22.4.14), we obtain Pk Pk1 ¼ Pk1 FTk1 Dt þ Fk1 Pk1 Dt þ Gk1 Q(tk )GTk1 Dt K(t)Hk1 Pk1 Dt þ o(Dt)

(22:4:15)

where limDt!0 o(Dt) ! 0. Dividing Eq. (22.4.15) by Dt and taking the limit as Dt ! 0, we obtain lim

Dt!0

Pk Pk1 d ¼ P(t) ¼ P(t)FT (t) þ F(t)P(t) þ G(t)Q(t)GT (t) Dt dt K(t)H(t)P(t),

which is Eq. (22.4.6).

t[T

(22:4:16)

22.4

Continuous Kalman Filter

663

FIGURE 22.4.1

Finally, substituting the same Eqs. (22.4.9) and (22.4.11) in the state estimate equations [Eq. (22.3.38)], we can write ^ k1 } ^ k ¼ ½I þ Fk1 Dt{X ^ k1 þ Kk1 ½Yk1 Hk1 X X ^ k1 þ Fk1 X ^ k1 Dt þ Kk1 ½Yk1 Hk1 X ^ k1 ¼X ^ k1 , þ Fk1 DtKk1 ½Yk1 Hk1 X

k [ {1, 2, . . . ,N}

(22:4:17)

Substituting for Kk21 ¼ K(t)Dt from Eq. (22.4.13) in Eq. (22.4.17), we obtain ^k X ^ k1 ¼ Fk1 X ^ k1 Dt þ K(t)Dt½Yk1 Hk1 X ^ k1 þ o(Dt) X

(22:4:18)

Dividing Eq. (22.4.18) by Dt and taking the limit as Dt ! 0, we obtain ^ k1 ^k X d ^ X ^ þ K(t)½Y(t) H(t)X(t), ^ ¼ X(t) ¼ F(t)X(t) Dt!0 Dt dt lim

t[T

(22:4:19)

which is Eq. (22.4.5). The block diagram implementing the continuous Kalman ﬁlter is shown in Fig. 22.4.1. Example 22.4.1 This example is the continuous analog of the discrete example (Example 22.3.2). The zero mean signal and observation processes are given by d X(t) ¼ 0, dt

t [ T;

Y(t) ¼ X(t) þ V(t),

t[T

where V(t) is a zero mean white-noise process with variance s2V. The variance of the initial state is p0. In this example F(t) ¼ 0, G(t) ¼ 0 and H(t) ¼ 1. The Kalman gain k(t) is obtained from Eq. (22.4.7): k(t) ¼

p(t) s2V

The variance is obtained from Eq. (22.4.6): d p2 (t) p(t) ¼ 2 dt sV

or

dp(t) dt ¼ 2 2 p (t) sV

664

Weiner and Kalman Filters

Integrating both sides of the equation from p0 to p(t), we have ð p(t) p0

dp(t) 1 ¼ 2 p2 (t) sV

ðt dt

or p(t) ¼

0

p0 1 þ p0 t=s2V

The signal estimate from Eq. (22.4.5) is given by d ^ p0 ^ ½Y(t) X(t) X(t) ¼ 2 dt sV þ p0 t Example 22.4.2 We will formulate a continuous version of Example 22.3.3. The signal and observation processes are given by d X(t) ¼ X(t) þ W(t), X0 ; dt

Y(t) ¼ 2X(t) þ V(t)

The zero mean noise signals W(t) and V(t) are independent Gaussian-distributed processes with the variance parameters E[W 2(t)]dt ¼ 1 and E[V 2(t)]dt ¼1. The error variance from Eq. (22.4.6) is d p(t) ¼ 2p(t) 2p2 (t) þ 1, p0 dt

(A)

This equation is a Riccati equation, and one method of solving it is to convert it to a linear differential equation. Otherwise, it can be solved numerically. The Kalman gain and the state estimate hinge on ﬁnding this soution. Following the procedure in Ref. 57, we will assume a solution to the preceding equation in the form p(t) ¼ a þ b tanh(ct þ d) where a,b,c,d are constants to be determined. Differentiating p(t) and substituting in Eq. (A), we obtain bc½1 tanh2 (ct þ d) ¼ 1 2½a þ b tanh(ct þ d) 2½a þ b tanh(ct þ d)2 ¼ 1 2a 2a2 2b(1 þ 2a) tanh(ct þ d) 2b2 tanh2 (ct þ d) Equating like coefﬁcients of tanh(ct þ d ) on both sides of the equation above, we obtain pﬃﬃﬃ pﬃﬃﬃ 3 1 ; c¼+ 3 a¼ ; b¼+ 2 2 Since p(t) as a variance function has to be positive, we use the positive values for b and c in p(t) and write pﬃﬃﬃ pﬃﬃﬃﬃ 3 1 tanh( 3t þ d) p(t) ¼ þ 2 2 where the unknown constant d has to be obtained from the initial condition p0. Thus 2p0 þ 1 2p0 þ 1 pﬃﬃﬃ tanh(d) ¼ pﬃﬃﬃ ; d ¼ tanh1 3 3

22.4

Continuous Kalman Filter

665

FIGURE 22.4.2

Hence pﬃﬃﬃ pﬃﬃﬃﬃ 3 1 1 2p0 þ 1 pﬃﬃﬃ p(t) ¼ þ tanh 3t þ tanh , 2 2 3

0t,1

(B)

p(t) is shown in Fig. 22.4.2 for values of the initial condition p0 ¼ 0,0.2,0.4,0.6,0.8,1. From Eq. (B) and from Fig. 22.4.2 we see that as t ! 1, p(t) tends to constant given by pﬃﬃﬃ pﬃﬃﬃ 3 31 1 tanh½1 ¼ lim p(t) ¼ þ t!1 2 2 2 which is independent of the initial condition. This is a desirable situation where the ﬁnal value of the uncertainty is a constant independent of the initial conditions because they may not be known a priori to any degree of certainty. From Eq. (22.4.7), the Kalman gain k(t) is given by k(t) ¼ 2p(t) The signal estimate is obtained from Eq. (22.4.5): d ^ ^ þ 2p(t)½Y(t) 2X(t) ^ X(t) ¼ X(t) dt ^ will be independent of the initial Here also as t ! 1, the ﬁnal value of the estimate X(t) condition.

CHAPTER

23

Probabilistic Methods in Transmission Tomography

Probabilistic methods are used extensively in medical imaging. In this chapter Bayesian reconstruction algorithms for transmission tomographic images are presented. The reconstruction is based on maximum a posteriori (MAP) estimation using the expectation – maximization (EM) algorithm. Gibbs’ density is used as the prior distribution with a sigmoidal potential function. The reconstruction problem itself is solved through a one-step-late expectation –maximization algorithm (EM-OSL) proposed by Green [9].† Suitable approximations are made to render the problem more tractable for parallel computation. Computer-simulated phantoms are used for image reconstruction. These images are compared to those reconstructed from currently available reconstruction algorithms in terms of image quality and convergence rate. This algorithm works quite well for low-photon-count and low-contrast cases in which the widely used convolution backprojection algorithm performs rather poorly.

B 23.1

INTRODUCTION

In transmission tomography (TT) the acquired projection data contain high statistical variations due to limited dose of radiation introduced in patients. Deterministic reconstruction techniques such as convolution backprojection and other Fourier-based methods introduce various types of artifacts. Transmission tomography models for reconstruction takes into account the probabilistic nature of the problem by incorporating statistical variations and other physical phenomena in a stochastic model. Application of the maximum-likelihood estimation to image reconstruction problem was recognized by Rockmore and Macovski [1,2]. An alternative to the maximum-likelihood approach was provided by Dempster et al. [3] with a general iterative method known as the expectation –maximization (EM) algorithm. The relevance of the EM algorithm to maximum-likelihood image reconstruction This chapter is based on the work [14] carried out by the author and his student at the Center of Advanced Computation and Telecommunications, Univ. Massachusetts Lowell and reported in an internal memorandum, Modiﬁed EM Algorithm for Transmission Tomography, T94-I, 1994. † References cited in this chapter are listed in Section 23.8 (not in the References list at the end of the book).

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

666

23.2

Stochastic Model

667

was noticed by Shepp and Vardi [4], who applied it to a stochastic model of positron emission tomography (PET), and independently by Lange and Carson [5], who applied it to general models of emission and transmission tomography. The initial optimism of the EM method of reconstruction in emission tomography has been tempered by the grainy, speckled appearance of the reconstructed images. This graininess is a manifestation of low correlations between parameter estimates for neighboring pixels. A satisfying solution is to incorporate some smoothness criterion in the reconstruction process itself. In emission tomography, this perspective has been advocated by Geman and McClure [6,7], Hebert and Leahy [8], and Green [9], who build on previous works of Geman and Geman [10] and Besag [11]. In particular, Geman and McClure [6,7] introduce Gibbs’ priors with nearest-neighbor interactions. Green [9] devises an approximate EM algorithm incorporating nearest-neighbor interactions with the one-step-late expectation– maximization (EM-OSL) algorithm. Lange [12] has demonstrated the conditions under which the EM-OSL algorithm converges to the unique maximum posterior point in image space for both emission and transmission cases. In this chapter Gibbs’ prior with sigmoidal potential function is developed that applies the EM-OSL scheme of Green [9] to the transmission algorithm proposed by Lange [12]. In simplifying the resultant transcendental equation, three terms have been used in the series expansion rather than the two in Lange [12]. The solution of the resulting quadratic equation yields more accurate results. The sigmoidal potential function smooths out signiﬁcant differences in neighboring pixel values while retaining sharp-edged features of images.

B 23.2

STOCHASTIC MODEL

Notation

Before describing the model in detail, some of the notations are deﬁned in Fig. 23.2.1:

i

Projection subscript

K j

Number of projection paths Ordered pixel subscript for each projection path i starting from the pixel farthest from the detector i

fIig fJjg

Set of (m 2 1) pixels contributing to the projection path i Set of projections to which pixel j contributes

Yi

Random variable for detected photons from projection i, or Yi ¼ Xim

Xij X^ ij

Random variable for photons from projection i entering pixel j Random variable for photons from projection i leaving pixel j, or entering pixel j þ 1, that is, X^ ij ¼ Xijþ1

Mj mj

Random variable for attenuation coefﬁcient of pixel j Realization of Mj

lij N

Pathlength of projection i through pixel j Neighboring pixels

The subscript i in mi has been omitted for clarity of notation.

668

Probabilistic Methods in Transmission Tomography

FIGURE 23.2.1 Probability Model

Corresponding to each projection i, the source photons are assumed to travel along a thin path en route to the detector. This path cuts through a set fIig of (m 21) pixels ordered by the index j; that is, j ¼ 1 is the pixel nearest to the source, and pixel j ¼ m21 is the last pixel along the path. The attenuation of pixel j is a random variable Mj taking the value mj and the length of the path inside pixel j is lij. Photons traveling along path i are counted as Xij at the entrance to pixel j and as X^ ij ¼ Xijþ1 at the exit, with Xij X^ ij photons becoming attenuated in pixel j. The number of photons Xim leaving the last pixel (m 2 1) is the same as the number of photons Yi being detected by the detector i, or Xim ¼ Yi. The attenuation coefﬁcients fmjg have to be estimated from the observed photon count Xim ¼ Yi. The random variable Xi1 of photons leaving the source at projection i is assumed to be Poisson-distributed as P(Xi1 ¼ xi1 ) ¼ edi

dixi1 xi1 !

(23:2:1)

where di ¼ E[Xi1] is the mean value of photons leaving the source in projection i. The probability that a photon entering pixel j along path i will survive the attenuation at pixel j, conditioned on the attenuation coefﬁcient Mj ¼ mj, is assumed to be Pj ¼ emj lij . The probability that X^ ij photons will emanate from pixel j conditioned on both Xij ¼ xij photons entering pixel j, and on the attenuation coefﬁcient Mj ¼ mj, can be characterized by a binomial distribution: xij ^ P{Xij ¼ x^ ij j Xij ¼ xij j Mj ¼ mj } ¼ (23:2:2) (emj lij )x^ ij (1 emj lij )(xij ^xij ) x^ ij From Eq. (23.2.1) the conditional mean value, gij ¼ E[Xij j m1, m2, . . . , mj21] of the photons, Xij entering pixel j along the projection i can be given by j1

gij ¼ E½Xij j m j1 ¼ di (em1 li1 )(em2 li2 ) (em j1 lij1 ) ¼ di eSk¼1 mk lik

(23:2:3)

where mj21 represents the attenuation coefﬁcient vector mj21 ¼ [m1, m2, . . . , mj21]T. Similarly, the conditional mean value, gim ¼ E[Xim j m1, m2, . . . , mm21] of the detected photons Xim is given by m1

gim ¼ E½Xim j mm1 ¼ di (em1 li1 )(em2 li2 ) (emm1 lim1 ) ¼ di eSk¼1 mk lik

(23:2:4)

23.2

Stochastic Model

669

where mm21 represents the attenuation coefﬁcient vector mm21 ¼ [m1, m2, . . . , mm21]T. The ratio gim/gij can be expressed as follows: Pm1 Pm1 gim E½Xim j mm1 di e k¼1 mk lik mk lik k¼j ¼ ¼e ¼ P j1 gij m l ik k k¼1 di e E½Xij j mj1

(23:2:5)

Since the input random variable Xij to pixel j conditioned on mj21 ¼ [m1, m2, . . . , mj21]T is Poisson-distributed with mean gij given by Eq. (23.2.3), the conditional distribution of Xij is x

P½Xij ¼ xij j m j1 ¼ egij

gijij xij !

(23:2:6)

The probability of the detected photons Xim conditioned on Xij photons entering pixel j, and the attenuation vector mm21 ¼ [m1, m2, . . . , mm21]T, can again be given by a binomial distribution similar to Eq. (23.2.2): P{Xim ¼ xim j Xij ¼ xij j mm1 } Pm1 xim ðxij xim Þ Pm1 xij mk lik mk lik k¼j k¼j e 1e ¼ xim

(23:2:7)

The ratio gim =gij from Eq. (23.2.5) is substituted in Eq. (23.2.7) and rewritten as P{Xim ¼ xim j Xij ¼ xij j mm1 } ¼

xij xim

gim gij

!xim

g 1 im gij

!(xij xim ) (23:2:8)

Using Bayes’ theorem [Eq. (2.4.4)] the probability of Xij conditioned on the detected photons Xim ¼ xim ¼ yi, and the attenuation coefﬁcient vector mm21 can be written as P{Xij ¼ xij j Xim ¼ xim j mm1 } ¼

¼

P{Xim ¼ xim j Xij ¼ xij j mm1 }P{Xij ¼ xij j mm1 } P{Xim ¼ xim j mm1 } !xim !(xij xim ) x g ij xij ! gim gim gij ij 1 e xim !(xij xim )! gij gij xij ! egim

gximim xim !

(23:2:9)

Simplifying Eq. (23.2.9), we obtain a conditional Poisson distribution P{Xij ¼ xij j Xim ¼ xim j mm1 } ¼

1 (g gim )(xij xim ) e(gij gim ) (xij xim )! ij

(23:2:10)

The mean of the photons Xij conditioned on the observed detected photons Xim ¼ xim ¼ yi, and attenuation coefﬁcient vector mm21 can be obtained from by taking the expected

670

Probabilistic Methods in Transmission Tomography

value of Eq. (23.2.10): E½Xij j Xim ¼ Yi ¼ xim j mm1 ¼

1 X xij

¼

xij (gij gim )(xij xim ) e(gij gim ) (x x )! ij im ¼0

1 X xij xim þ xim xij ¼0

(xij xim Þ!

(gij gim )(xij xim ) e(gij gim )

¼ gij gim þ xim ¼ gij gim þ yi 4 Nij ¼

(23:2:11)

We will now ﬁnd the joint conditional probability of the photons from projection i entering pixels f j, j ¼ m, m 2 1, . . . , 1g conditioned on the attenuation coefﬁcient vector mm21. Deﬁning an m-vector Xi ¼ fXim, Xim21, . . . , Xi1gT, the joint conditional probability, PfXi j mm21g of Xi conditioned on the attenuation coefﬁcient vector mm21, can be evaluated by using the Markov property of Xi: P{Xi j mm1 } ¼ P{Xim , Xim1 , . . . , Xi1 j mm1 } ¼ P{Xim j Xim1 j mm1 }P{Xim1 j Xim2 j mm2 } P{Xi2 j Xi1 j m1 }P{Xi1 }

(23:2:12)

But by analogy to Eq. (23.2.2), the conditional probability of Xikþ1 conditioned on Xik and mk is given by a binomial distribution: P{Xikþ1 ¼ xikþ1 j Xik ¼ xik j Mk ¼ mk } xik ¼ (emk lik )xikþ1 (1 emk lik )(xik xikþ1 ) xikþ1 Substituting Eqs. (23.2.13) and (23.2.1) into Eq. (23.2.12), we obtain " # m 1 Y dxi1 xik mk lik x^ ik mk lik (xik ^xik ) edi 1 ) (1 e ) (e P{Xi j mm1 } ¼ x^ ik xi1 ! k¼1

(23:2:13)

(23:2:14)

where we have used x^ ik ¼ xikþ1 . Equation (23.2.14) represents the conditional probability for the projection path i. Similar conditional probabilities can be formulated for all the K projection paths. We deﬁne matrices X ¼ {X1 , X2 , . . . , XK }T , Y ¼ {Y1 , Y2 , . . . , YK }T , and M ¼ {the set of attenuation coefficient vectors mmi 1 for all projections i ¼ 1, . . . , K}. Summing Eq. (23.2.14) over all projections i, the following conditional probability can be written as " # K m 1 Y Y d xi1 xik mk lik x^ ik mk lik (xik ^xik ) edi i (23:2:15) ) (1 e ) (e P{X j M} ¼ x^ ik xi1 ! i¼1 k¼1 and the loglikelihood function ln P{X j M} can be written as ln P{X j M} ¼

K X m1 X

x^ ik ln (emk lik ) þ (xik x^ ik ) ln (1 emk lik ) þ C

(23:2:16)

i¼1 k¼1

where the constant C represents the logarithm of terms independent of mk. The estimate of the attenuation matrix M is obtained by maximizing the posterior probability of M conditioned on the observed but incomplete data Y, that is P{M j Y}

23.3

Stochastic Estimation Algorithm

671

or the corresponding loglikelihood function ln P{M j Y}. However, maximizing ln P{M j Y} is an intractable problem. The problem is mitigated by embedding the observed but incomplete space Y into the complete but unobservable space X. Since X is connected to Y by a many-to-one mapping, the richer space X provides more ﬂexibility in data manipulation, resulting in an easier maximization of ln P{M j X}. Using Bayes’ theorem, the likelihood function P{M j X} will be given by P{M j X} ¼

P{X j M}P{M} P{X}

(23:2:17)

and the corresponding loglikelihood function is ln P{M j X} ¼ ln P{X j M} þ ln P{M} ln P{X}

(23:2:18)

Equations (23.2.17) or (23.2.18) will form the basis for the estimation procedure.

B 23.3

STOCHASTIC ESTIMATION ALGORITHM

The maximum-likelihood estimator for M is obtained by assuming that the attenuation coefﬁcients {mk } are equiprobable, in which case PfMg is constant. Since P{X} is independent of M, maximizing P{M j X} amounts to maximizing the loglikelihood function ln P{X j M} given in Eq. (23.2.16) with respect to M. Since it contains the unknown values of random variables Xik and X^ ik, direct maximization of Eq. (23.2.16) will not give meaningful results. Hence its expected value will be maximized with respect to {mk }. From Eq. (23.2.11), the conditional expectations of Xik and X^ ik are functions of {mk } via {gik }, and hence an iterative method of maximizing Eq. (23.2.18) has to be developed. The projection matrix Y is some function g(X) where g(.) is a many-to-one mapping from the space of X to the space of Y. Hence the iterative solution to the maximumlikelihood estimation problem will proceed in alternating steps of expectation and maximization. First the expectation of ln P{X j M} is taken conditioned on the observation matrix Y and the current estimate M(n). This conditional expected value E½ln P{X j M} j Y, M(n) is maximized with respect to M to yield an updated estimate M(nþ1). The updated conditional expected value E½ln P{X j M} j Y, M(nþ1) is again maximized and the process repeated. According to Dempster [3], the loglikelihood function ln P{M j Y} or equivalently ln P{Y j M} will increase in value at each iteration step with the result that ln P{Y j M(nþ1) } ln P{Y j M(n) }

(23:3:1)

and the equality holds only when the maximum-likelihood solution Mmax is reached and the maximum value is ln P{Y j Mmax }. Expectation Step

If M(n) is the estimate of the attenuation matrix M at the nth iteration, the expectation of ln P{X j M} at the (n þ 1)st iteration, conditioned on the observation Y and M(n), can be written as K X m1 h i X N^ ik ln (emk lik ) þ (Nik N^ ik ) ln (1 emk lik ) þ C1 E ln P{X j M} j Y, M(n) ¼ i¼1 k¼1

(23:3:2)

672

Probabilistic Methods in Transmission Tomography

^ conditioned on Y and M(n) are obtained where the conditional expectations Nik and Nik from Eq. (23.2.11) as Nik ¼ E Xik j Yi j M(n) ¼ gik gim þ yi h i N^ ik ¼ E X^ ik j Yi j M(n) ¼ gikþ1 gim þ yi

(23:3:3)

where Xˆik ¼ Xikþ1 and C1 is yet another constant independent of {mk }. Maximization Step

Since Nik and N^ ik are known, E½ln P{X j M} j Y, M(n) can now be maximized by differentiating Eq. (23.3.2) with respect to mk and setting it equal to 0: X @ lik E ln P{X j M} j Y, M(n) ¼ N^ ik lik þ (Nik N^ ik ) m l ¼0 e k ik 1 @mk i[Jk

(23:3:4)

Equation (23.3.4) is a transcendental equation that can be solved by approximating the exponential term by using the ﬁrst three terms in the series expansion: 1 emk lik

1

¼

1 1 1 þ mk lik þ o½(mk lik )2 mk lik 2 12

(23:3:5)

Expectation– Maximization Algorithm

Substitution of Eq. (23.3.5) in Eq. (23.3.4) results in "

" # # " # X 1 X 1X 2 2 ^ ^ ^ (Nik Nik )lik mk (Nik þ Nik )lik mk þ (Nik Nik ) ¼ 0 12 i[Jk 2 i[Jk i[Jk B

A

or

C

Am2k Bmk þ C ¼ 0

for all k

(23:3:6)

Solving for mk in the quadratic equation [Eq. (23.3.6)] yields the (n þ 1)st estimate mk(nþ1) mk(nþ1)

¼

B+

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B2 4AC 2A

j

m(n) k

for all k

(23:3:7)

where the coefﬁcients A, B, and C are dependent on m(n) k . The smaller root has to be chosen as the solution in this algorithm so that for lik mk , 6 this solution will always fall inside the solution bounds established by Lange [5] for all values of k and i. max As n ! 1 in Eq. (23.3.7), m(n) k ! mk , and hence it is the desired expectation – maximization (EM) algorithm. EM-One-Step-Late (OSL) Algorithm

Equation (23.3.7) represents the maximum-likelihood solution with equiprobable attenuation coefﬁcients mk(nþ1) . This solution can be considerably reﬁned by using prior information PfMg about the attenuation coefﬁcients M, giving rise to the maximum

23.3

Stochastic Estimation Algorithm

673

a posteriori solution. In this case, in Eq. (23.2.18) only P{X} does not depend on M. Applying the same iterative techniques to Eq. (23.2.18) as used in deriving Eq. (23.3.4), there results an equation similar to Eq. (23.3.4) as shown below: @ @ @ E ln P{M j X} j Y, M(n) ¼ E ln P{X j M} j Y, M(n) þ ln P{M} @mk @mk @mk X lik @ ^ ^ ¼ Nik lik þ (Nik Nik ) m l ln P{M} ¼ 0 þ ik k e 1 @mk i[Jk (23:3:8) Using the same series approximation for the exponential in Eq. (23.3.8) as in Eq. (23.3.5) leads to the equation "

" # # 1 X 1X @ 2 2 ^ ^ (Nik Nik )lik mk (Nik Nik )lik ln P{M} mk 12 i[Jk 2 i[Jk @mk " þ

A

X

#

B

(Nik N^ ik ) ¼ 0

(23:3:9)

i[Jk C

where A, B, and C are the same quantities deﬁned in Eq. (23.3.6), Eq. (23.3.9) can be rewritten as follows: @ Am2k B ln P{M} mk þ C ¼ 0 @mk

for all k

(23:3:10)

Solving for mk in the preceding quadratic yields the result vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !2 u u @ @ t B ðnþ1Þ ln P{M} + B ðnþ1Þ ln P{M} 4AC @mk @mk ¼ (n) 2A mk !

mk(nþ1)

(23:3:11)

Equation (23.3.11) cannot be solved since it involves the derivative of the log-prior distribution at mk(nþ1) . Green [9] resolves the problem by using the derivative at m(n) k yielding the EM one-step-late algorithm (EM-OSL). Additionally, the derivative of the prior P(M) is multiplied with a parameter b to add ﬂexibility on the inﬂuence of the prior on the algorithm. With these adjustments, Eq. (23.3.11) can be rewritten as follows: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ !2 u u @ @ B b ðnÞ ln P{M} + t B b ðnÞ ln P{M} 4AC @mk @mk ¼ (n) 2A mk !

mk(nþ1)

(23:3:12)

In Eq. (23.3.12) the smaller root of Eq. (23.3.12) is chosen in this iterative algorithm for the same convergence reasons as in Eq. (23.3.7). The optimum value of b will depend on both the characteristics of the image and the prior used. If b ¼ 0, then Eq. (23.3.12) is the EM algorithm of Eq. (23.3.7).

674

Probabilistic Methods in Transmission Tomography

B 23.4

PRIOR DISTRIBUTION PfMg

The prior distribution PfMg in Eq. (23.3.12) has to be carefully chosen. From consideration of Markovian ﬁelds, a proper choice is the exponential distribution given by Ð Pfmg ¼ (1/z)e 2V(m), m 0, where z ¼ m0 eV(m) dm is the normalization constraint. This is known as the Gibbs distribution discussed in Section 7.15. The potential function V(m) is designed to penalize large differences in estimated parameters for neighboring pixels. The constraints on the potential functions in Section 7.15 are discussed by Lange [12]. In this application, a ﬁrst-order neighborhood interaction of the form X w jk v(mj mk ), (23:4:1) V(m) ¼ ð j, kÞ[N N neighboring pixels

is considered where the weighting functions wjk for a ﬁrst-order Markovian ﬁeld are deﬁned by 8 < pﬃﬃ1:ﬃ orthogonal pixel w jk ¼ 1= 2: diagonal pixel : 0: otherwise and are shown in Fig. 23.4.1. With Eq. (23.4.1), the prior distribution Pfmg is of the form P 1 (j, k)[N wjk v(mj mk ) , m0 P{m} ¼ e z

(23:4:2)

Two potential functions v(m) are considered here and their performances compared. The ﬁrst is the proposed logarithm of hyperbolic cosine proposed by Green [9] with a parameter j given by v(m) ¼ ln cosh(jm) ¼ ln

ejm þ ejm 2

(23:4:3)

The second is the proposed sigmoidal function with a positive parameter j given by v(jm) ¼

2 1 1 þ ejm2

FIGURE 23.4.1

(23:4:4)

23.4

Prior Distribution PfMg

675

The parameter j ¼ 0 in both of these potential functions amounts to a straight EM algorithm without any prior. Figures 23.4.2 show the sigmoid function and its derivatives for values of the adjustable parameter j ¼ 2000, 3000, 4000, 5000, 6000. By varying this parameter, it is possible to smooth out speckles in the reconstructed images while retaining the sharp edges. This fact will be demonstrated analytically from the series expansion of Eq. (23.3.12). Deﬁning, for convenience B1 ¼ B b

@ ln P{M} @m(n) k

(23:4:5)

Eq. (23.3.12) can be expanded in series in AC/B21 and written as follows: " pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ # B1 + B21 4AC B1 4AC 1=2 (nþ1) 1 1 2 ¼ ¼ mk 2A 2A B1 2 2 3 3 C AC 2A C 5A C 14A4 C4 42A5 C 5 1þ 2 þ þ þ þ þ ¼ B1 B1 B41 B81 B10 B61 1

(23:4:6)

Real roots of Eq. (23.4.6) mandate that (AC/B21) , 14, and the ﬁrst-order approximation for can be given by m(nþ1) k P (Nik N^ ik ) C i[Jk ¼ X (23:4:7) mk(nþ1) ¼ 1 @ B1 (Nik þ N^ ik )lik b ln P{M} 2 i[Jk @mk (n) mk

Taking the derivative of the logarithm of Eq. (23.4.2) with respect to m(n) k and substituting it in Eq. (23.4.7), we obtain P (Nik N^ ik ) i[Jk (23:4:8) mk(nþ1) ¼ X h i X 1 @ (n) (n) (Nik þ N^ ik )lik b w jk (n) v(mj mk ) 2 i[Jk @mk (n) j[N mk

The denominator of Eq. (23.4.8) involves the sum of two terms, one of which is the derivative of the potential function. This sum directly determines the penalty imposed on successive iterations in this algorithm. The relatively higher intensity of pixel k than those of its nearest neighbors is reﬂected by a positive value of the derivative and hence a larger value of the denominator. As a consequence, the penalty for the next iteration of mk(nþ1) is higher, resulting in a lower estimate for the intensity of pixel k. On the other hand, the intensity of pixel k lower than those of its nearest neighbors results in a higher estimate of the intensity of pixel k. Figure 23.4.2b shows that the derivative of the sigmoidal function exhibits a band of high (0 –0.05) and low (20.05– 0) values dependent on the parameter j. Because of this characteristic, the penalty imposed is effective only when intensity differences fall into this band. As a consequence, the unusually high differences in intensity such as sharp-edged features of the image are preserved while the speckled appearance of the image due to Poisson noise is smoothed. As shown in Fig. 23.4.2, the degree of edge separation needed can be adjusted with the parameter j for any particular image reconstruction.

676

Probabilistic Methods in Transmission Tomography

FIGURE 23.4.2

The derivative of the lncosh function, on the other hand, monotonically increases. As a consequence, there is no discrimination between edges and artifacts, resulting in uniformly smoothed edges. Hence the sigmoidal potential function should give a better reconstruction of the images.

B 23.5

COMPUTER SIMULATION

A computer simulation was performed to investigate the quality of reconstructed images using the sigmoid potential function and to compare it with lncosh potential function. Comparisons are also made with the conventional convolution backprojection (CBP) and the straight expectation – maximization algorithm. The simulation block diagram is shown in Fig. 23.5.1a. The simulation to obtain the four images for comparison is shown in Fig. 23.5.1b. A phantom was created with parameters as shown in Fig. 23.5.2. The numerical values show the uniform attenuation coefﬁcients in that region. This phantom is fairly complicated in detail, including a high-contrast outer shielding ring. In the usual reconstructions, especially from projections with a low number of photon counts, this shielding ring obscures the delicate features present inside it. The intention in this study was to assess how well the sigmoidal potential function handled the presence of the shielding ring. The photon source was assumed to emit a relatively low number of di ¼ 104 expected number of photons per projection path i. The phantom was divided into cells of 64 64 pixels. A major problem was in the implementation of the coefﬁcient matrix [lij] with a size of 642 642 . [lij] is a collection of lengths of the path along the projection path i inside pixel j. This is a highly sparse matrix, as each path i passes through (Fig. 23.2.1) only a small fraction of the total number of pixels in the image, and each pixel has only a small fraction of the total number of paths passing through it. Furthermore, computation for each projection path i requires access to all corresponding [lij]. Even though only a fraction of elements of this matrix is nonzero, calculation of all of them on the ﬂy for each iteration would still require an excessive amount of computation. To reduce this computational burden, the pathlengths are calculated only once and stored in a form that is easily accessible. These stored data are deﬁned as the pathlength matrix. A semisparse matrix scheme is devised to handle the problem of computational burden. Two sets of path coefﬁcients are stored. The ﬁrst set is accessible in the

23.5

Computer Simulation

677

FIGURE 23.5.1

“forward” direction; that is, given the path i, all coefﬁcients associated with it are stored. This is deﬁned as the forward pathlength matrix. The second set is accessible in the “backward” direction; that is, given pixel j, all coefﬁcients associated with it are stored. This is deﬁned as the backward pathlength matrix. The forward pathlength matrix is used to generate noise-free projection data. This matrix, in conjunction with a Poisson random-number generator routine, is used to

FIGURE 23.5.2

678

Probabilistic Methods in Transmission Tomography

generate Poisson-randomized data. The Poisson random number is generated for a given intensity parameter and acquisition time. This random number is equal to a total number of events that have exponentially P distributed interarrival times with mean equal to the given m21 intensity parameter gim ¼ die 2 k¼1 mklik occurring during the acquisition time. Exponential random variates are generated from a set of random variates uniformly distributed over [0,1] by probability transformation discussed in Chapter 15. The data from the Poisson randomizer are the projection matrix Y. From this projection matrix Y four different types of reconstruction images are generated. The ﬁrst of the reconstructed images used the conventional deterministic convolution backprojection algorithm. In the other three images using the EM algorithm, a combination of forward and backward pathlength matrices are used in reconstructing the image. Both Nik and N^ ik deﬁned in Eq. (23.3.3) are computed by using the forward pathlength matrix. The computed Nik and N^ ik are then in turn utilized to determine the updated estimated attenuation in Eqs. (23.3.7) and (23.3.11) using the backward pathlength matrix. coefﬁcient m(nþ1) k These three images are reconstructed using (1) the conventional EM algorithm proposed by Lange [12], (2) the modiﬁed EM with lncosh potential function proposed by Green [9], and (3) the modiﬁed EM with the proposed sigmoidal potential function. Performances of these algorithms and their reconstructions are then compared. The criteria used for comparison are

.

Convergence and convergence rates Image quality (primarily visual tempered with analytical criteria)

.

Complexity (actual execution time)

.

Noise-free projection data were generated over 64 equally spaced projection angles between 0 and 1808 using parallel beamscanner geometry. Each projection consisted of 64 equally spaced rays over the ﬁeld of view of 30 cm in length. Poisson-randomized projection data were generated using the previous noise-free data as the mean value. The Poisson random-number generator algorithm as described previously is used for this purpose. The CBP method was implemented on an Apollo Domain series 3500 system. All EM-based algorithms were implemented on the Alliant FX-2800 series system, which has both concurrent and vector processing capability. Each iteration took about 18 s. This may be slow but is not deemed a serious drawback because the speed of iteration can be considerably enhanced with multiprocessor machines. The grayscale images presented herein are windowed between 0.150 and 0.275 cm21 for better visual contrast.

B 23.6

RESULTS AND CONCLUSIONS

Figure 23.6.1 shows (a) the original phantom image, (b) reconstruction with the convolution backprojection (CBP), (c) reconstruction with the conventional EM, (d) reconstruction with EM-OSL using lncosh potential function, and (e) reconstruction with EM-OSL using sigmoidal potential function. Cross-sectional histograms of reconstructed images sectioned vertically through the center with the cross-sectional histogram of the phantom superimposed are shown in Fig. 23.6.2. In Fig. 23.6.3, the progression of RMS errors between the reconstructed images and the phantom as the iteration proceeds are shown for CBP, EM, lncosh, and sigmoid realizations. The conventional convolution backprojection (CBP) reconstruction shown in Fig. 23.6.1b is achieved in a single step. It is the quickest in execution time because of

23.6

Results and Conclusions

679

FIGURE 23.6.1

its deterministic nature. However, it is the poorest reconstruction among all the four. This poor performance is caused by factors such as intrinsic statistical variation of data, low photon counts, low angular and spatial sampling rates, and shielding ring effects. The cross-sectional histogram of Fig. 23.6.2a shows well-rounded edges, and the RMS error shown in Fig. 23.6.3 is quite high. By using a probabilistic model, better results are obtained for the EM algorithms as shown in Figs. 23.6.1c – 23.6.1e. However, the straight EM algorithm in Fig. 23.6.1c has a grainy and speckled appearance. This graininess is simply a manifestation of low correlation between parameter estimates for neighboring pixels. The cross-sectional histogram in Fig. 23.6.2b shows severely jagged edges, and the RMS error shown in Fig. 23.6.3 remains fairly constant after about 60 iterations. There is also the problem that this algorithm tends to diverge [13] after a certain number of iterations, and stopping rules are to be devised for the optimum number of iterations. To smooth the speckles in the EM procedure, the algorithm was modiﬁed using Gibbs’ prior with a lncosh potential function to penalize large differences between

680

Probabilistic Methods in Transmission Tomography

FIGURE 23.6.2

adjacent pixels. The lncosh reconstruction of Fig. 23.6.1d shows a smoother result, but the cross-sectional histogram of Fig. 23.6.2c indicates smoothed edges. The RMS error curve of Fig. 23.6.3 is also better than CBP and EM reconstructions but tends to saturate at a slightly higher error level. The difﬁculty stems from the fact that the lncosh potential function tends to oversmooth the image, resulting in the loss of sharp-edged features.

FIGURE 23.6.3

23.8

References for Chapter 23

681

To overcome the oversmoothing difﬁculty, a sigmoidal potential function that preserves sharp edges while retaining the smoothing effect is proposed. Convergence curves for increasing iterations were plotted using the loglikelihood function and RMS error for different values of b and j; b was varied between 0 and 0.0004, and j was varied between 0 and 7000. From these curves the optimum value of b was found to be 0.0002 and the optimum value of j was 5000. With these values, the reconstructed image of Fig. 23.6.1e is superior in quality to the rest. The image is visually very close to the phantom. The cross-sectional histogram of Fig. 23.6.2d also indicates a better edge performance than the lncosh potential function. The RMS error curve monotonically decreases as the iteration proceeds and at the 120th iteration is almost negligible. Thus, the modiﬁed EM algorithm with the sigmoidal potential function clearly performs better with regard to accuracy and smoothness of reconstruction.

B 23.7

DISCUSSION OF RESULTS

In comparing reconstructions in medical imaging, visual accuracy is the most important. Analytical criteria such as root-mean-square (RMS) error is secondary. The RMS error curves of Fig. 23.6.3 indicate major differences between the CBP reconstruction and EM reconstructions. The visual inspection of the reconstructed images also show discernible differences. On the other hand, the error curves for EM and lncosh reconstructions are not that far apart; but visually the differences are quite discernible. The sigmoidal reconstruction is clearly more appealing both visually and analytically. The error curves also show that during the ﬁrst 30 iterations all the EM algorithms converge almost linearly at the same rate. But after that, their convergence becomes sublinear and their graphs begin to fan out separately. Among all the four convergence rates presented, the sigmoidal algorithm again performs the best. The speed of each iteration is dependent on the number of processors in the computer and can be considerably enhanced with more processors. The structure of this algorithm is more conducive to a multiprocessor machine. The convergence rates and the quality of reconstruction indicate the decided superiority of this modiﬁed EM algorithm using sigmoidal potential functions.

B 23.8

REFERENCES FOR CHAPTER 23

1. Rockmore, A. J., and A. Macovski, A maximum likelihood approach to emission image reconstruction from projections, IEEE Trans. Nucl. Sci. NS-23, 1428– 1432 (1976). 2. Rockmore, A. J., and A. Macovski, A maximum likelihood approach to transmission image reconstruction from projections, IEEE Trans. Nucl. Sci. NS-24, 1929– 1935 (1977). 3. Dempster, A. P., N. M. Laird, and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Statist. Soc. B 39, 1 – 38 (1977). 4. Shepp, L. A., and Y. Vardi, Maximum likelihood reconstruction for emission tomography, IEEE Trans. Med. Imag. MI-1, 113– 121 (1982). 5. Lange, K., and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assist. Tomogr. 8(2), 306 –316 (1984). 6. Geman, S., and D. E. McClure, Bayesian image analysis: An application to single photon emission tomography, Proc. Am. Statist. Assoc. Statist. Comput. 12 – 18 (1985). 7. Geman, S., and D. E. McClure, Statistical methods for tomographic image reconstruction, Bull. Int. Statist. Inst. LII-4, 5 – 21 (1987).

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8. Hebert, T., and R. Leahy, A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs’ priors, IEEE Trans. Med. Imag. 8(2), 194 – 202 (1989). 9. Green, P. J., Bayesian reconstructions from emission tomography data using a modiﬁed EM algorithm, IEEE Trans. Med. Imag. 9(1), 84 – 93 (1990). 10. Geman, S., and D. Geman, Stochastic relaxation, Gibbs’ distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 721 – 741 (1984). 11. Besag, J., Spatial interaction and the statistical analysis of lattice system, J. Roy. Statist. Soc. B 36, 192– 236 (1974). 12. Lange, K., Convergence of EM image reconstruction algorithms with Gibbs’ smoothing, IEEE Trans. Med. Imag. 9(4), 439– 446 (1990). 13. Rajeevan, N., A Stochastic Estimation Approach to Emission Tomography, Ph.D. thesis, Electrical Engineering Dept., Indian Institute of Science, Bangalore, India, 1991. 14. Kamonpet, P., and V. Krishnan, Modiﬁed EM Algorithm for Transmission Tomography, Internal Memo T94-1, Center for Advanced computation, Univ. Massachusetts Lowell, 1994.

APPENDIX

A

A Fourier Transform Tables

The governing equations are as follows: ð1 Fourier transform:

x(t)ejvt dt

X(v) ¼ 1

Inverse Fourier transform:

x(t) ¼

1 2p

ð1

X(v)e jvt dv

1

Tables give, in the following order, the time function x(t), the plot of the time function, the plot of the Fourier transform, and the frequency function X(v).

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684

Appendix A

APPENDIX A Fourier transform tables

(Continued)

Appendix A

APPENDIX A

685

Continued

(Continued)

686

Appendix A

APPENDIX A Continued

APPENDIX

B

Cumulative Gaussian Tables

The governing equations are as follows: Gaussian density: Gaussian distribution function:

1 2 f (x) ¼ pﬃﬃﬃﬃﬃﬃ e(1=2)x 2p ðx 2 1 pﬃﬃﬃﬃﬃﬃ e(1=2)j dj ¼ p F(x) ¼ 2p 1

Tables give the values of the probability p ¼ F(x) for values of x between 25.00 and þ5.00. Figure B.1 shows the standard Gaussian density and distribution with x along the horizontal axis (abscissa) and f (x), F(x) ¼ p along the vertical axis (ordinate). Representative point is shown for x ¼ 0.5744 and F(0:5744) ¼ 0:7:

FIGURE B.1

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687

688

Appendix B

APPENDIX B

Cumulative Gaussian tables

(Continued)

Appendix B

APPENDIX B

689

Continued

(Continued)

690

Appendix B

APPENDIX B

Continued

(Continued)

Appendix B

APPENDIX B

Continued

691

APPENDIX

C

Inverse Cumulative Gaussian Tables

The governing equations are as follows: ðx 2 1 pﬃﬃﬃﬃﬃﬃ e(1=2)j dj ¼ p and F(x) ¼ 2p 1

x ¼ F1 (p)

Tables give the values of x in x ¼ F1 ( p) for values of p between (0.01, 0.99) grouped in such a manner that adjacent column values of p add to 1. Figure C.1 shows f (x), p along the horizontal axis and x, F1 ( p) along the vertical axis. Since f (x) is an even function, F1 ( p) has the symmetry F1 ( p) ¼ F1 (1 p). Representative points are shown for p ¼ 0.1 and 0.9 with F1 (0:1) ¼ 1:2816 and F1 (0:9) ¼ 1:2816.

FIGURE C.1

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

692

Appendix C

APPENDIX C

Inverse cumulative Gaussian tables

693

APPENDIX

D

Inverse Chi-Square Tables

The governing equations are as follows: Chi-square density:

Chi-square distribution function:

x(n=2)1 e(x=2) n u(x) 2n=2 G 2 ð x (n=2)1 (j=2) j e n dj ¼ p F(x) ¼ 0 2n=2 G 2

f (x) ¼

Tables give the values of x in x ¼ F 21( p) for values of p between (0.005, 0.995) grouped in such a manner that adjacent column values of p add to 1. Figure D.1 shows f (x), p along the horizontal axis and x, F 21( p) along the vertical axis. Representative points are shown for number of degrees of freedom n ¼ 4, and p ¼ 0.1 and 0.9 with F 21(0.1) ¼ 1.06362 and F 21(0.9) ¼ 7.7794. The asymmetry of the distributions is to be noted and F 1( p) = F 1(1 p).

FIGURE D.1

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

694

Appendix D

695

APPENDIX D Inverse chi-square tables

(Continued)

696

Appendix D

APPENDIX D Continued

(Continued)

Appendix D

APPENDIX D Continued

(Continued)

697

698

Appendix D

APPENDIX D Continued

(Continued)

Appendix D

699

APPENDIX D Continued

(Continued)

700

Appendix D

APPENDIX D Continued

APPENDIX

E

Inverse Student-t Tables

The governing equations are as follows: nþ1 ½(nþ1)=2 G t2 2 Student-t density: f (t) ¼ pﬃﬃﬃﬃﬃﬃ n 1 þ n np G 2 nþ1 ½(nþ1)=2 ðt G t2 2 Student-t distribution function: F(t) ¼ dt pﬃﬃﬃﬃﬃﬃ n 1 þ n 1 npG 2 Since f (x) is an even function, F 21( p) has the symmetry F 1 ( p) ¼ F 1 (p). The absolute values of t for p between (0.005,0.45) are the same as the values for p between (0.55, 0.999) with t ¼ F 1 (0:5) ¼ 0 for all n. Hence the tables give the values of t ¼ F 1 ( p) for p only between (0.55, 0.999). Figure E.1 shows f (t), p along the horizontal axis and t, F 1 ( p) along the vertical axis. Representative points are shown for number of degrees of freedom n ¼ 5, and p ¼ 0.1 and 0.9 with F 1 (0:1) ¼ 1:4759 and F 1 (0:9) ¼ 1:4759.

FIGURE E.1

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

701

702 APPENDIX E Inverse student-t tables

(Continued)

Appendix E

APPENDIX E Continued

703

APPENDIX

F

Cumulative Poisson Distribution

The governing equations are as Follows: Poisson mass function: p(k; l) ¼ el

Poisson distribution function:

F(k; l) ¼

lk , k!

k X m¼0

el

k0 lm , k0 m!

Tables give the values of the probability F(k;l) for values of l between (0.01, 10) and k between (0,20). Figure F.1 shows the Poisson mass function and distribution with k along the horizontal axis and p(k;l), F(k;l) along the vertical axis for l ¼ 10. The representative point is shown for k ¼ 10 and p(k;10) ¼ 0.1251, F(k;10) ¼ 0.5830.

FIGURE F.1

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

704

Appendix F

705

APPENDIX F Cumulative Poisson distribution

(Continued)

706

Appendix F

APPENDIX F Continued

(Continued)

Appendix F

APPENDIX F Continued

707

APPENDIX

G

Cumulative Binomial Distribution

The governing equations are as follows: Binomial mass function:

b(k; n, p) ¼

n! k!(n k)! pk (1 p)nk ,

Binomial distribution function:

F(k; n, p) ¼

0kn

k X

n! m!(n m)! m¼0 pm (1 p)nm ,

0kn

Tables give the values of the probability F(k; n, p) for values of p between (0.05, 0.9) and n between (2, 20). Figure G.1 shows the binomial mass function and distribution with k along the horizontal axis and b(k; n, p), F(k; n, p) along the vertical axis for p ¼ 0.3 and n ¼ 10. The representative point is shown for k ¼ 3 and b(k; 10, 0.3) = 0.2668, F (k; 10, 0.3) ¼ 0.6496. The range of k is only upto n.

FIGURE G.1 Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

708

Appendix G

APPENDIX G

709

Cumulative binomial distribution

(Continued)

710

Appendix G

APPENDIX G

Continued

(Continued)

Appendix G

APPENDIX G

711

Continued

(Continued)

712

Appendix G

APPENDIX G

Continued

(Continued)

Appendix G

APPENDIX G

Continued

713

REFERENCES

1. Abromowitz, M., and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1972. 2. Akansu, A. N., and R. A. Haddad, Multiresolution Signal Processing, Academic Press, 1992. 3. Anderson, R. L., Distribution of the serial correlation coefﬁcient, Annals Math. Statist., 13, 1– 13, 1942. 4. Aunon, J. I., and V. Chandrasekhar, Introduction to Probability and Random Processes, Mc-Graw-Hill, 1997. 5. Benford, F., The law of anomalous numbers, Proc. Am. Phil. Soc., 78, 551– 572, 1938. 6. Bharucha-Reid, A. T., Elements of the Theory of Markov Processes and Their Applications, Dover Publications, 1988. 7. Black, F., and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 1973. 8. Bo¨rjesson, P. O., and C. E. W. Sundberg, Simple approximations of the error function Q(x) for communications applications, IEEE Trans. Commun., 27, 639– 643, 1979. 9. Box, G. E. P., G. M. Jenkins, and G. C. Reinsel, Time Series Analysis, Prentice-Hall, 1994. 10. Chatﬁeld, C., The Analysis of Time Series: An Introduction, 2nd ed., Chapman & Hall, 1980. 11. Childers, D. G., Probability and Random Processes, Irwin, 1997. 12. Cooper, G. R., and C. D. McGillem, Probabilistic Methods of Signal and System Analysis, 3rd ed., Oxford Univ. Press, 1999. 13. Cox, D. R., and H. D. Miller, The Theory of Stochastic Processes, Chapman & Hall/CRC, 1977. 14. Davenport, W. B., and W. L. Root, An Introduction to the Theory of Random Signals and Noise, IEEE Press, 1987.

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

714

15. Davis, J., Performance Models for Correlated Trafﬁc Data, Ph.D. thesis, Univ. Massachusetts—Lowell, 2000. 16. Doob, J. L., Stochastic Processes, Wiley, 1953. 17. Feller, W., An Introduction to Probability Theory and Its Applications, 2nd ed., Wiley, 1968, Vol. I. 18. Fogiel, M., Handbook of Mathematical, Scientiﬁc, and Engineering, Formulas, Tables, Functions, Graphs, Transforms, Staff of Research and Education Assoc., 1988. 19. Fujisaki, M., G. Kallinapur, and H. Kunita, Stochastic differential equations for nonlinear ﬁltering problem, Osaka J. Math., 9, 19 – 40, 1972. 20. Gantmacher, F., The Theory of Matrices, Chelsea Publishing, 1959. 21. Gardner, W. A., Introduction to Random Processes, 2nd ed., Mc-Graw-Hill, 1990. 22. Grimmett G., and D. Stirzaker, Probability and Random Processes, 3rd ed., Oxford Univ. Press, 2001. 23. Gross, D., and C. M. Harris, Queuing Theory, Wiley, 1998. 24. Higgns, J. J., and S. Keller-McNulty, Concepts in Probability and Stochastic Modeling, Duxbury Press, 1995. 25. Jenkins, G. M., and D. G. Watts, Spectral Analysis and Its Applications, Holden-Day, 1968. 26. Kallianpur, G., Stochastic Filtering Theory, Springer-Verlag, 1980. 27. Kalman, R. E., and R. S. Bucy, New results in linear ﬁltering and prediction theory, Trans. ASME, J. Basic Eng., ser. D, 83, 95 – 108, 1961. 28. Kamonpet, P., and V. Krishnan, Modiﬁed EM Algorithm for Transmission Tomography, Internal Memo T94-1, Center for Advanced Computation, Univ. Massachusetts Lowell, 1994.

References

29. Kamonpet, P., Probabilistic Image Reconstruction from Noisy and Incomplete Projections, Ph.D. thesis, Univ. Massachusetts—Lowell, 1995. 30. Kleinrock, L., Queuing Systems, Theory, Wiley, 1975, Vol. II. 31. Kleinrock, L., Queuing Systems, Computer Applications, John Wiley, 1976, Vol. II. 32. Krishnan, V., Estimation of Autocovariance Functions, Technical Report M-215U, Dynamics Research Corporation, 1974. 33. Krishnan, V., Conﬁdence Limits and Equivalence of Statistical Tests, Technical Report M-256U, Dynamics Research Corporation, 1976. 34. Krishnan, V., Distribution of Spectral Estimators, Technical Report E-9334U, Dynamics Research Corporation, 1984. 35. Krishnan, V., Nonlinear Filtering and Smoothing, Wiley, 1984. 36. Krishnan, V., Linear Systems Properties, CRC Press, 1998. 37. Krishnan, V., Nonlinear Filtering and Smoothing, Dover Publications, 2005. 38. Lange, K., Convergence of EM image reconstruction algorithms with Gibbs’ smoothing, IEEE Trans. Med. Imag., 9, 439– 446, 1990. 39. Leon-Garcia, A., Probability and Random Processes for Electrical Engineering, AddisonWesley, 1993. 40. Lipster, R. S., and A. N. Shiryaev, Statistics of Random Processes, Springer-Verlag, 1977, Vols. I and II. 41. Meyer, P. A., A decomposition theorem for supermartingales, Ill. J. Math. 6, 193– 205, 1962. 42. Mood, A. M., and F. A. Graybill, Introduction to the Theory of Statistics, McGraw-Hill, 1963. 43. Naidu, P. S. Modern Spectrum Analysis of Time Series, CRC Press, 1996. 44. Ostle, B., Statistics in Research, Iowa State Univ. Press, 1963. 45. Papoulis, A., Fourier Integral and Its Applications, Mc-Graw Hill, 1962. 46. Papoulis, A., Probability, Random Variables, and Stochastic Processes, 3rd ed., McGrawHill, 1991. 47. Peebles Jr., Peyton Z., Probability, Random Variables and Random Signal Principles, 4th ed., Mc-Graw-Hill, 2001. 48. Priestley, M. B., Spectral Analysis and Time Series, Academic Press, 1981.

715

49. Proakis, J. G., Digital Communications, McGraw-Hill, 1995. 50. Rice, S. O., Mathematical analysis of random noise, in Selected Papers on Noise and Stochastic Processes, N. Wax, ed., Dover Publications, 1945, 1954. 51. Rome, H. J., and V. Krishnan, Causal probability model for evaluating future transoceanic airlane separations, IEEE Trans. Aerospace Electron. Syst., 26, 804 – 817, 1990. 52. Ross, S. M., A First Course in Probability, 6th ed., Prentice-Hall, 2002. 53. Ross, S. M., Stochastic Processes, 2nd ed., John-Wiley, 1996. 54. Ross, S. M., Introduction to Probability Models, 7th ed., Academic Press, 2000. 55. Shanmugan, S., and A. M. Breiphol, Random Signals, Detection, Estimation and Data Analysis, Wiley, 1988. 56. Shilov, G. E., An Introduction to the Theory of Linear Spaces, Prentice-Hall, 1961. 57. Srinath, M. D., P. K. Rajasekaran, and R. Viswanathan, Introduction to Statistical Signal Processing with Applications, Prentice-Hall, 1996. 58. Stark, H., and J. W. Woods, Probability and Random Processes with Applications to Signal Processing, 3rd ed., Prentice-Hall, 2002. 59. Therrien, C. W., Discrete Random Signals and Statistical Signal Processing, Prentice-Hall, 1992. 60. Thomas, J. B., An Introduction to Communication Theory and Systems, Springer-Verlag, 1988. 61. Trivedi, K. S., Probability and Statistics with Reliability, Queuing and Computer Science Applications, Prentice-Hall, 1982. 62. Viniotis, Y., Probability and Random Processes for Electrical Engineers, McGraw-Hill, 1998. 63. Xu, L., Probabilistic Reconstruction of Tomographic Images, Ph.D. thesis, Univ. Massachusetts—Lowell, 1997. 64. Yates, R. D., and D. J. Goodman, Probability and Stochastic Processes, Wiley, 1999.

Web References W1. http://www.itl.nist.gov/div898/handbook/ eda/section3/eda36.htm. W2. http://www.geocities.com/mikemclaughlin/ math_stat/Dists/Compendium.html.

Index

acceptance –rejecztion, 280 algorithm, 281 beta density, 282 comparison function, 280 adaptive coefﬁcients, 307 aperiodic random processes, 563 discrete Karhunen – Loeve (K– L) expansion, 571 Karhunen – Loeve (K– L) expansion, 565 eigenfunctions, 565 eigenvalues, 565 nonstationary process, 566 AR(1) process, 522 arcsine law, 192, 194 attenuation coefﬁcient, 668 autoregressive process, 439 bandpass signals, 606 analytic signal representation, 607 natural envelope, 606 lower side band, 607 upper side band, 606 complex envelope, 610 quadrature representation, 607 Bayesian estimation, 384 discrete probability, 398 MAP Criterion, 395 ML criterion, 396 Bayes’ theorem, 20, 669 Bayes’ theorem for densities, 132 Bernoulli distribution, 37 trials, 37 independent identically distributed, 37 Bernoulli process, 505 modiﬁed, 510 Bessel function of zero order modiﬁed, 107, 217, 621 bilinear form, 302 binomial coefﬁcient, 29

Pascal’s identity, 29 triangle, 30 properties, 29 sum, 39 Vandermonde’s identity, 30 binomial process, 506, 509 birthdays problem, 27, 53 bivariate Gaussian distribution, 144, 154 block matrices, 308 properties, 309 Bose Einstein statistics, 32 bound, 242 Chebyshev, 242 alternate forms, 244 geometric derivation, 243 Chernoff, 248 Brownian motion (see Wiener process), 492, 523 Catalan numbers, 31 Cauchy density, 232 distribution, 115, 221, 226 Cauchy – Riemann conditions, 303 Cayley Hamilton theorem, 294 central limit theorem, 260 characteristic functions, 155, 212, 213 examples, 157 continuous random variables, 159 discrete random variables, 157 existence, 155 joint characteristic functions, 157 moment generating properties, 156 chi-squared test, 267 combinations, 28 sampling with replacement, 32 sampling without replacement, 29 comparison of Gaussian and Poisson approximations, 93 conditional densities, 122

Probability and Random Processes, by Venkatarama Krishnan Copyright # 2006 John Wiley & Sons, Inc.

716

Index

conditional distributions, 122 properties, 122 conditional distributions and densities, 122, 313 clariﬁcation, 315 P(A) ¼ 0, 126 P(A) = 0, 122 conﬁdence coefﬁcient, 351, 364 conﬁdence interval, 351, 364 coefﬁcients of simple linear regression, 371 unknown mean of population, 364 known variance, 364 unknown variance, 366 unknown variance with unknown mean, 369 conﬁdence region simple linear regression line, 372 congruential generator, 264 linear, 264 mixed, 264 multiplicative, 264 continuous distributions, 79, 95 beta distribution, 114 Cauchy distribution, 115 chi distribution, 104 Maxwell density, 106 Nakagami density, 108 Rayleigh density, 105 Rice’s density, 107 chi-square, 102 double Laplace distribution, 118 Erlang distribution, 97 exponential distribution, 80 hazard rate, 82 instantaneous, 82 Poisson arrival process, 81 gamma, 99 reproductive property, 100 Gaussian distribution, 84 Gibbs distribution, 118, 674 potential function, 118 Laplace distribution, 96 double Laplace distribution, 118 lognormal distribution, 112 Pareto distribution, 117 Snedecor F-distribution, 111 Student-t density, 110 summary, 119 triangular, 95 uniform distribution, 79 Weibull, 101 continuous Kalman ﬁlter, 660 assumptions, 660

717

matrix Riccati equation, 661 continuous random variables, 151 means and variances, 150 exponential distribution, 151 Gaussian distribution, 152 convergence, 256 almost sure, 257 Cauchy criterion, 257 in distribution, 258 in probability, 257 mean square, 258 pointwise, 256 properties, 258 quadratic mean, 258 convolution techniques, 279 Erlang distribution, 279 triangular distribution, 279 counting, 25 addition rule, 25 multiplication rule, 25 convex function, 254 covariance matrices, 320 diagonalization principles, 330 diagonalization to an identity matrix, 331 principles, 330 simultaneous diagonalization, 334 summary of procedures, 335 covariance of ACF estimators, 452 craps game, 16 cumulants, 167 Gaussian distribution, 169 Poisson distribution, 169 cumulant generating functions, 167 cyclostationary process, 560 density function, 65 differentiation, 301 complex variable, 303 Cauchy – Riemann conditions, 303 partial derivative, 304 regular point, 304 singular point, 304 total derivative, 303 complex vectors, 305 partial differentiation, 305 real matrices, 308 real vectors, 301 direct determination of density, 194 conditions, 196 steps, 196 direct determination of joint density, 227 discrete distributions Bernoulli, 37 Benford, 55

718

Index

discrete distributions (Continued) binomial, 38 approximation, 50 geometric, 42 hypergeometric, 46 logarithmic, 55 ﬁnite law, 55 ﬁrst signiﬁcant digit, 55 second signiﬁcant digit, 58 inﬁnite law, 60 multinomial, 41 negative binomial, 44 Pascal distribution, 44 Poisson, 48 approximation to binomial, 50 summary, 62 discrete Kalman ﬁlter, 647 across observations, 649, 651 algorithm, 655 assumptions, 650 between observations, 649, 652 comments, 654 covariance propagation, 655 Kalman gain, 650 scalar case, 657 timing diagram, 649, 653 discrete random variables, 150 means and variances, 150 binomial distribution, 150 Poisson distribution, 151 discrete time processes, 458 Anderson’s theorem, 470 estimation of autocorrelation functions, 467 estimation of covariance of NACF, 469 estimation of NACF, 468 large-lag approximations, 469 variance of autocovariance estimators, 464 white noise, 470 discrete time stationary processes, 437 moments, 437 distribution and density functions, 66 properties, 73 cumulative distribution function FX (x), 73 probability density function fX (x), 74 probability mass function, 74 distribution function, 66 from density, 75 distribution of Y ¼ g(X ), 174 steps for determination, 175 eigenvalues of matrices, 294 characteristic polynomial, 293

eigenvectors of matrices, 296 normalization, 297 emission tomography, 667 ergodic processes, 439 correlation-ergodic, 441 mean-ergodic, 439 power ergodic, 442 time average, 439 estimated regression coefﬁcients, 358 covariance, 359 from data, 358 estimated regression line, 360 variance, 360 estimation likelihood function, 389 likelihood ratio, 389 linear estimation of vector variables, 337 overdetermined system, 338 underdetermined system, 338 maximum a posteriori probability (MAP), 388 minimization of average probability of error, 389 maximum likelihood, ML, 391 properties, 391 parameters, 350 Bayesian estimation, 384 covariance, 355 higher order moments, 358 interval estimation, 351 mean, 351 unbiased, 351 minimum mean square, 353 point estimation, 351 variance, 354 standard error, 355 statistic, 351 estimation criteria, 340 consistent, 341 efﬁcient, 341 Rao – Cramer bound, 341 log-likelihood, 341 maximum likelihood, 341 mean square error, 340 minimum variance, 340 unbiased, 340 estimation of a random variable by a constant, 342 by a function of another random variable, 342 expectation maximization algorithm, 672 one step late, 672 RMS error, 680

Index

ﬁeld, 8, 11 Boolean ﬁeld, 9 Borel s-ﬁeld, 9 functions of a single random variable, 173 higher order moments, 203 moments, 202 random variable g(X ), 173 conditions, 173 function of two random variables, 206 image, 206 inverse image, 206 conditions, 206 steps for determining FZ(z), 207 gambler’s ruin, 518 gamma function, 99 incomplete, 100 Gaussian approximation to binomial distribution, 91 Demoivre – Laplace limit theorem, 91 Gaussian distribution, 84 density function, 84 distribution function, 84 mean, 84 standard, 85 properties, 87 tails function Q(x), 88 lower bound, 89 properties, 90 upper bound, 89 variance, 84 Gaussian process, 521 properties, 521 generating functions, 161 examples, 162 binomial distribution, 163 geometric distribution, 164 Poisson distribution, 163 moment generating properties, 162 Gibbs’ distribution, 118, 674 potential function, 674 lncosh, 674 sigmoid, 675 Gibbs’ prior, 667 higher order moments, 153 cross moments, 154 Hilbert transform, 607 table, 610 histograms, 266 hypothesis testing, 373 acceptance region, 374

alternate hypothesis, 373 binary, 375 composite hypothesis, 374 cost for errors, 401 mean m—known variance s2, 378 mean m—unknown variance s2, 379 ML rule, 394 number of samples n for given a and b, 382 null hypothesis, 373 operating characteristic, 376 power of the test, 376 procedure for testing, 378 rejection region, 374 signiﬁcance testing, 373, 374 simple hypothesis, 374 statistic, 373 image, 64 independence, 4, 18 functional, 4, 18 statistical, 4, 18 independent increment process, 507 stationary independent increment, 507 indicator function, 205, 243 properties, 205 inequalities, 242 Cauchy – Schwartz, 251 frequency domain, 253 Chebyshev, 242 geometric derivation, 243 generalized Chebyshev, 244 Jensen, 254 Markov, 246 innovations process, 648 interval estimation, 364 inverse Hilbert transform, 607 inverse image, 64 inverse problem g(x) from fX (x) and fY ( y), 200 inverse transformation, 269 discrete distributions, 270 continuous distributions, 272 exponential distribution, 272 Rayleigh distribution, 274 Weibull distribution, 273 tables, 278 Jacobian determinant, 227, 230, 233 Jacobian matrix, 227, 230, 233 joint cumulants, 169 joint distribution, 135 continuous distribution, 136 discrete distribution, 135

719

720

Index

joint distribution (Continued) joint probability mass function, 135 marginal distribution, 135 properties, 137 region bounded by a function, 140 regions bounded by constants, 138 regions bounded by functions, 141 joint distribution of two vectors, 312 joint moments of two random vectors, 321 cross correlation matrix, 321 cross covariance matrix, 322 joint Gaussian density, 213 Karhunen – Loeve transform, 565 continuous, 565 discrete, 571 matrix formulation, 572 Kolmogorov axioms, 11 Boolean additivity, 11 non negativity, 11 normalization, 11 sigma additivity, 11 kurtosis, 168 Lagrange multipliers, 351 likelihood function, 341 log likelihood function, 341 limit theorems, 259 central limit theorem, 260 strong law of large numbers, 260 weak law of large numbers, 259 linear ﬁlters, 592 carrier frequency, 592 cut-off frequency, 592 ideal band-pass, 592 ideal high-pass, 592 ideal low-pass, 592 matched ﬁlters, 596 linear systems, 574 causal system, 574 convolution integral, 574 impulse response, 574 state transition matrix, 575 transfer function, 575 linear regression, 343 evaluating coefﬁcients, 343 multiple, 343 simple, 343, 360 estimated regression line, 361 variance of regression coefﬁcients, 362 linear transformations, 328 marginal distributions, 142, 143 Markov chains, 536

classiﬁcation of states, 544 absorbing state, 544 aperiodic, 545 ergodic, 545 intercommunicating, 544 irreducible, 544 periodic, 545 recurrent, 545 reducible, 544 transient, 545 discrete, 536 ﬁrst passage probability, 544 ﬁrst return probability, 545 homogeneous, 536 mean ﬁrst passage time, 545 mean recurrence time, 545 null recurrent, 545 positive recurrent, 545 m-step transition probability, 536 steady state, 539 properties, 550 state occupancy probabilities, 538 stochastic matrix, 537 Markov process, 527 AR process, 534 Chapman – Kolmogorov equation, 528 homogeneous, 529 Markov property, 527 Poisson process, 532 properties, 530 transition probability density, 528 Wiener process, 533 martingale process, 551 continuous martingale, 552 discrete martingale, 552 likelihood ratio martingale, 555 Poisson martingale, 554 random walk, 553 Wiener martingale, 554 Mass megabucks, 46 matched ﬁlters, 596 input signal to noise ratio, 597 non-white Gaussian noise, 604 output signal to noise ratio, 597 properties, 599 rectangular pulse, 600 sinusoidal pulse, 600 matrices, 284 addition, 285 adjoint, 291 characteristic equation, 293 characteristic polynomial, 293 deﬁniteness, 295 negative deﬁnite, 295

Index

positive deﬁnite, 295 deﬁnition, 284 determinant, 286 properties, 287 diagonalization, 298 inverse, 290 properties, 292 multiplication, 285 orthogonal, 293 principal minors, 295 rank, 288 similarity transformation, 300 singular, 287 square, 284 symmetric, 299 properties, 299 trace, 288 transpose, 286 means and variances table, 170 means of estimated autocovariances, 449 minimum mean square estimator mean, 353 mixed distributions, 118 modal matrix, 297 modiﬁed Bessel function I0 (z), 107, 217, 621 modulation, 434 moment generating functions, 164 gamma distribution, 167 Gaussian distribution, 166 negative binomial, 165 Monty Hall problem, 22 multiple functions of random variables, 238 narrowband bandpass process, 612 analytic signal representation, 615 complex envelope, 615 natural envelope representation, 618 quadrature representation, 614 properties, 615 with additive sinusoid, 619 normal equations, 344 operating characteristic, 376 orthogonal increment processes, 508 orthogonality principle, 346 random processes, 626 random variables, 625 Paley– Wiener criterion, 593, 631 partial differentiation, 304 complex vectors, 305

real matrices, 308 real vectors, 301 periodic random processes, 557 stationary, 558 permutations, 26 sampling with replacement, 26 sampling without replacement, 26 Poisson process, 491, 492, 508 formal derivative, 503 generalized, 501 autocorrelation, 502 autocovariance, 503 mean, 502 interarrival time, 496 distribution, 496 waiting time, 496 distribution, 497 Poisson white noise, 504 Polya urn model, 534 power spectral density, 472 predictable process, 557 probability, 10 conditional, 14 reduced sample space, 14, 17 density, 13 distribution function, 13 measure, 11 space, 10 total, 20 probability mass function, 38, 74 pseudoinverse, 338 quadrature components properties, 615 quadratic error surface, 307 quadratic form, 302 random binary wave, 429 random number generator, 264 properties, 265 random processes, 406 bandpass processes, 606 classiﬁcation, 490 continuous state continuous time, 491 continuous state discrete time, 491 discrete state continuous time, 491 discrete state discrete-time process, 490 deﬁnition, 406 distribution and density, 408 estimation of parameters, 445 continuous time processes, 445 autocovariance, 447 covariance, 452 variance, 454

721

722

Index

random processes (Continued) mean, 446 variance, 446 discrete time processes, 458 autocovariance, 459 estimation, 461 Lagrange multiplier, 459 mean, 458 higher order distributions, 414 independent, 416 means and variances, 408 normalized autocovariance, 416 normalized cross covariance, 416 orthogonal, 417 second order moments, 415 autocorrelation, 415 autocovariance, 415 cross correlation, 416, 435 cross covariance, 416 normalized autocovariance, 416 normalized cross-covariance, 416 second order process, 414 through linear systems, 578 cross spectral density, 580 output power spectral density, 580 threshold ﬁlter, 588 uncorrelated, 417 random telegraph wave, 432 random variable, 64 conditional moments, 146 conditional expectation, 148 conditional variance, 149 continuous, 79 correlation coefﬁcient, 150 covariance, 150 deﬁnition, 64 degenerate, 241 discrete, 37 independent, 147 distribution and density, 66 properties, 73 moments, 146 expectation, 146 properties, 147 joint, 148 variance, 149 properties, 149 orthogonal, 147 uncorrelated, 147 random vectors, 311 Bayes theorem, 316 conditional density, 328 inner product, 320 independent, 323 joint density, 313, 326

moments, 319 correlation matrix, 319 properties, 324 covariance matrix, 320 properties, 324 expectation vector, 319 orthogonal, 323 outer product, 320 uncorrelated, 323 random walk process, 490, 512 elementary, 509 symmetric, 510 two absorbing barriers, 514 unrestricted, 512 Rayleigh density, 215, 621 Rayleigh distribution, 215, 232, 619 recursive estimation, 648 regression coefﬁcients, 343 Rice’s density, 217, 621 Rice’s distribution, 217, 621 sample space, 10 continuous, 10 discrete, 10 sample variance, 354 set, 1 cardinality, 2 unions and intersections, 5 Cartesian product, 7 complement, 6 De Morgan’s laws, 8 difference, 6 element, 1 empty set, 2 equal, 3 event, 9 ﬁnite, 1 inclusion – exclusion principle, 5 intersection, 4 mutually exclusive, 4 null, 11 partition, 7 power set, 3 relative complement, 6 subset, 2 symmetric difference, 7 table of set properties, 8 union, 3 universal, 2 Venn diagram, 2 shaping ﬁlter, 580 signal to noise ratio, 253, 597 signiﬁcance level, 351, 364, 374 similarity transformations, 300 skewness, 168

Index

solution of linear equations, 291 special methods, 274 Box– Mueller transformation, 275 Gaussian distribution, 275 Rayleigh distribution with phase, 277 spectral density– continuous time, 472 bandlimited process, 475, 476 bandpass process, 477 cross spectral density, 473 power spectral density, 473 alternate form, 480 estimation, 482 periodogram, 480 properties, 479 white noise, 478 Wiener – Khinchine theorem, 472 spectral density– discrete time, 484 power spectral density, 484 alternate form, 486 discrete white noise, 488 spectral factorization, 633 additive factoring, 634 multiplicative factoring, 633 standard error, 355 stationary random processes, 420 autocorrelation, 421 autocovariance, 421 cross correlation, 422 cross covariance, 422 ﬁrst order, 420 independent, 421 mean, 421 normalized autocovariance, 421 normalized cross covariance, 422 nth order stationary, 420 orthogonal, 421 periodic, 423 properties of correlation functions, 422 second order, 420 strict sense, 420 uncorrelated, 421 variance, 421 wide sense, 420 stochastic estimation algorithm, 671 expectation step, 671 maximization step, 672 strong law of large numbers, 260 submartingale, 552 decomposition, 557 supermartingale, 552 tables cumulative binomial, 708 cumulative Gaussian, 687

cumulative Poisson, 704 Fourier transform, 683 inverse chi-square, 694 inverse Gaussian, 692 inverse Student-t, 701 tails probability, 242 total probability for densities, 131 transmission tomography computer simulation, 676 pathlength matrix, 676 backward, 677 forward, 677 phantom description, 676 probability model, 668 reconstructed images, 679 cross sectional histograms, 680 RMS error, 680 two functions of two random variables, 222 conditions, 223 auxiliary random variable, 233 derivation of Student-t, 237 type I error, 374 false alarm, 374 producer’s risk, 374 type II error, 374 consumer’s risk, 374 missed alarm, 374 uncorrelated increment process, 508 uniform distribution, 264 variance of the estimated regression line, 360 vector Gaussian random variables, 323 weak law of large numbers, 259 weighted Poisson process, 501 white Gaussian noise, 521 whitening ﬁlter, 632 white noise process, 422 Wiener ﬁltering, 627 causal, 631 ﬁltering, 627, 638 noncausal, 627 prediction, 627, 641 smoothing, 627, 642 Wiener– Hopf integral equation, 632 ﬁltering solution, 634 prediction-smoothing solution, 637 Wiener process, 523 conditions, 523 derivation, 524 formal derivative, 526 properties, 526

723