Exercise January 28
STA 4508S (Spring, 2014)
SM, Exercise 10.6.4. A model for over-dispersed binomial data can be obtained by assuming that R follows a Binomial(m, p) distribution, and p itself follows a beta distribution, with density f (p; α, β) =
Γ(α + β) α−1 p (1 − p)β−1 , Γ(α)Γ(β)
0 < p < 1, α, β > 0.
1. Show that this yields the beta-binomial density for R, given by f (r; α, β) =
Γ(m + 1)Γ(r + α)Γ(m − r + β)Γ(α + β) . Γ(r + 1)Γ(m − r + 1)Γ(α)Γ(β)Γ(m + α + β)
2. Show that E(R) =
mα , α+β
var(R) =
mαβ m−1 {1 + }. 2 (α + β) α+β+1
3. What is the condition for uniform over-dispersion?
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