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## An election is being held. There are two candidates, A and B

An election is being held. There are two candidates, A and B, and thereare n voters. The probability of voting for Candidate A varies by city. There arem cities, labeled 1, 2, . . . , m. The jth city has njvoters, so n1+ n2+ Â· Â· Â· + nm= n.Let Xjbe the number of people in the jth city who vote for Candidate A, withXj|pj~ Bin(nj, pj). To re?ect our uncertainty about the probability of voting ineach city, we treat p1, . . . , pmas r.v.s, with prior distribution asserting that they arei.i.d. Unif(0, 1). Assume that X1, . . . , Xmare independent, both unconditionally andconditional on p1, . . . , pm. Let X = X1+ Â· Â· Â· + Xm.(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 = n12+ n22+â¦+ nm2

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