## Statistics- Let p ∼ Beta(a, b), where a and b are positive real numbers

## Statistics- Let p ∼ Beta(a, b), where a and b are positive real numbers

(a) Let p ∼ Beta(a, b), where a and b are positive real numbers. Find E(p2(1 − p)2), fully simplified (Γ should not appear in your final answer).For the remaining parts, consider the following scenario. Two teams, A and B, have an upcoming match. They will play five games and the winner will be declared to be the team that wins the majority of games. Given p, the outcomes of games are independent, with probability p of team A winning and 1 − p of team B winning. But you donât know p, so you decide to model it as an r.v., with p ∼ Unif(0, 1) a priori (before you have observed any data).To learn more about p, you look through the historical records of previous games between these two teams, and find that the previous outcomes were, in chronological order, AAABBAABAB. (Assume that the true value of p has not been changing over time and will be the same for the match, though your beliefs about p may change over time.)(b) Does your posterior distribution for p, given the historical record of games between A and B, depend on the specific order of outcomes or only on the fact that A won exactly 6 of the 10 games on record? Explain.(c) Find the posterior distribution for p, given the historical data.The posterior distribution for p from (c) becomes your new prior distribution,and the match is about to begin!(d) Conditional on p, is the indicator of A winning the first game of the match positively correlated with, uncorrelated with, or negatively correlated of the indicator of A winning the second game of the match? What about if we only condition on the historical data?(e) Given the historical data, what is the expected value for the probability that the match is not yet decided when going into the fifth game (viewing this probability as an r.v. rather than a number, to reflect our uncertainty about it)?

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