1.17 Suppose that P(Y = 1) = 1 P(Y = 0) = , i = 1,..., n, where {Y;} are inde- pendent. Let Y

1.17 Suppose that P(Y₁ = 1) = 1 − P(Y₁ = 0) = π, i = 1,..., n, where {Y;} are inde-

pendent. Let Y = ΣΥ.

a. What is the distribution of Y? What are E(Y) and var(Y)?

b. When {Y;} instead have pairwise correlation p > 0, show that var(Y) > ηπ(1

π), overdispersion relative to the binomial. [Altham (1978) and Ochi and Prentice

(1984) discussed generalizations of the binomial that allow correlated trials.]

c. Suppose that heterogeneity exists: P(Y₁ = 1|π) = π for all i, but is a random variable with density function g(-) on [0, 1] having mean p and positive variance.

Show that var(Y) > η ρ(1 – ρ). (When π has a beta distribution, Y has the beta-

binomial distribution of Section 14.3.)