arlie%20Welch/Downloads/penultimate.pdf |CD Page view A Read aloud | T Add text | Draw Highlight (c) Find the conditional distribution of N, given that Paul's time to finish the marathon is t. (d) Find Var(N). (Your answer should depend only on n, since Paul's time is an r.v.) Hint: Let T be Paul's time, and use Eve's law to condition on T. Alternatively, use indicator r.v.S. 7. (BH 9.60) 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, voters, so n + 2 + . .. + 0m = n. Let X, be the number of people in the jth city who vote for Candidate A, with Xjp, ~ Bin(n,, p,). To reflect our uncertainty about the probability of voting in each city, we treat P1, . . . . Pm as r. v.S, with prior distribution asserting that they are i.i.d. Unif(0, 1). Assume that X1, ..., Xm are independent, both unconditionally and conditional on pi, . . ., Pm. Let X be the total number of votes for Candidate A. (a) Find the marginal distribution of X, and the posterior distribution of pilX1 = ki. (b) Find E(X) and Var(X) in terms of n and s, where s = n; + n; + .. . + n O Fi 39 ALIENWARE