Chapter 16.2, Problem 9E

Suppose that we conduct independent Bernoulli trials and record Y, the number of the trial on which the first success occurs. As discussed in Section 3.5, the random variable Y has a geometric distribution with success probability p. A beta distribution is again a conjugate prior for p.

1. a If we choose a beta prior with parameters α and β, show that the posterior distribution of p | y is beta with parameters α* = α + 1 and β* = β + y − 1.
2. b Find the Bayes estimators for p and p(1 − p).