Two basketball teams, A and B, play an n game match. Let Xj be the indicator of team A

Chapter 9, Problem 51

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Two basketball teams, A and B, play an n game match. Let Xj be the indicator of team A winning the jth game. Given p, the r.v.s X1,...,Xn are i.i.d. with Xj |p Bern(p). But p is unknown, so we will treat it as an r.v. Let the prior distribution be p Unif(0, 1), and let X be the number of wins for team A. (a) Find E(X) and Var(X). (b) Use Adams law to find the probability that team A will win game j + 1, given that they win exactly a of the first j games. (The previous exercise, which is closely related, involves an equivalent LOTP proof.) Hint: letting C be the event that team A wins exactly a of the first j games, P(Xj+1 = 1|C) = E(Xj+1|C) = E(E(Xj+1|C, p)|C) = E(p|C). (c) Find the PMF of X. (There are various ways to do this, including a very fast way to see it based on results from earlier chapters.)