In Section 16.1 and Exercise 16.6, we considered an example where the number of responders to a treatment for a virulent disease in a sample of size n had a binomial distribution with parameter p and used a beta prior for p with parameters α = 1 and β = 3.

a Find the Bayes estimator for p = the proportion of those with the virulent disease who respond to the therapy.

b Derive the mean and variance of the Bayes estimator found in part (a).

Exercise 16.6

Suppose that Y is a binomial random variable based on n trials and success probability p (this is the case for the virulent-disease example in Section 16.1). Use the conjugate beta prior with parameters α and β to derive the posterior distribution of p | y. Compare this posterior with that found in Example 16.1.

Example 16.1

Let Y1, Y2, . . . , Yn denote a random sample from a Bernoulli distribution where P(Yi = 1) = p and P(Yi = 0) = 1 − p and assume that the prior distribution for p is beta (α, β). Find the posterior distribution for p.