Question
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
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 nj voters, so n1+n2+...+nm=n. Let Xj be the number of people in the jth city who vote for candidate A, with Xj|pj ~Bin(nj, pj). To reflect our uncertainty about the probability of voting in each city, we treat p1,...,pm as rvs, with prior distribution asserting that they are iid Unif(0, 1). Assume that X1,...,Xm are independent, both unconditionally and conditional on p1,...,pm. Let X be the total number of votes for candidate A.
a)Find the marginal distribution of X1 and the posterior distribution of p1|(1=k1).
b)Find E(X) and Var(X) in terms of n and s, where s= n1^2+n2^2+...+nm^2.
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