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_j voters, so n₁ + n₂ + + n_m = n.Let X_j be 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_m as r.v.s, with prior distribution asserting that they are i.i.d. Unif(0, 1). Assume that X₁, . . . , X_m are independent, both unconditionally and conditional on p₁, . . . , p_m.

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

(b) Find E(X) and Var(X) in terms of n and s, where s = n²₁ + n²₂ + + n²_m.