Suppose that P(Y i = 1) = 1 P(Y i = 0) = , i = 1, . . . , n, where {Y

Suppose that P(Y_i = 1) = 1 – P(Y_i = 0) = π, i = 1, . . . , n, where {Y_i} are independent. Let Y = ∑_i Y_i).

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

b. When {Y_i} instead have pairwise correlation ρ > 0, show that var(Y) > nπ(1 – π), overdispersion relative to the binomial. [Altharn (1978) discussed generalizations of the binomial that allow correlated trials.]

c. Suppose that heterogeneity exists: P(Y_i = 1|π) = π for all i, but π is a random variable with density function g(.) on [0, 11 having mean ρ and positive variance. Show that var(Y) > nρ(l – ρ). (When π has a beta distribution, Y has the beta-binomial distribution of Section 13.3.)

d. Suppose that P(Y_i = 1|π_i) = π_i, i = 1,.. .,n, where (π_i) are independent from g(). Explain why Y has a bin(n, ρ) distribution unconditionally but not conditionally on {π_i}.