Let x and y be two random variables such that y has the

Question

Let X and Y be two random variables such that Y has the binomial distribution B_i(π, N) and, given Y = y, X has the binomial distribution B_i(p, y).

(a) Suppose that p ∈ (0, 1) and π ∈ (0, 1) are unknown and N is known. Show that (X, Y) is minimal sufficient for (p, π).

(b) Suppose that π and N are known and p ∈ (0, 1) is unknown. Show whether X is sufficient for p and whether Y is sufficient for p.