If we consider the following Beta-Binomial setup:

0 ~ Beta(a, B), X1, . .., Xn | 0 ~ Bernoulli(0). Recall that the posterior distribution is 0 | X1, . .., Xn ~ Beta(a +T, B +n - T) where T = X1+ . .. + X, is the total number of successes. You can use any facts we've stated before about the Beta(o, 8) distribution, including: its density is Bad" (1-x)- and its mean is atB' (a) What is the posterior mean of 0, i.e. its expected value under the posterior distribution, E(0 | X1, . . ., Xn)? (b) Find the posterior mode of 0, i.e. the value of 0 maximizing the posterior density. (Hint: it might be easier to maximize the log of the posterior density.) (c) Give a 95% credible interval for 0. Use the notation Fa. for the CDF of a Beta(a, ) distribution. (d) Suppose that we will take another draw Xn+1 ~ Bernoulli(0). What is the probability that Xn+1 = 1, conditioning on all the data you've seen so far? In other words, calculate P(Xn+1 = 1 X1,..., Xn). Hint: use the tower law to handle 0