An election is being held. There are two candidates, A and B, and there are n voters. The probability of voting for Candidate A varies by city. There are m cities, labeled 1, 2, . . . , m. The jth city has n_jvoters, so n₁+ n₂+ + n_m= n. Let X_jbe the number of people in the jth city who vote for Candidate A, with X_j| p_j Bin(n_j, p_j). To reflect our uncertainty about the probability of voting in each city, we treat p₁, . . . , p_mas r.v.s, with prior distribution asserting that they are i.i.d. Unif(0, 1). Assume that X₁, . . . , X_mare independent, both unconditionally and conditional on p₁, . . . , p_m.

(a) Find the marginal distribution of X₁and the posterior distribution of p₁|X₁= k₁.

(b) Find E(X) and Var(X) in terms of n and s, where s = n(from 1 to 2) + n (from 2 to 2) +...+ n(from m to 2)