Chapter 16.2, Problem 6E

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, …, Y_n denote a random sample from a Bernoulli distribution where P(Y_i = 1) = p and P(Y_i = 0) = 1 − p and assume that the prior distribution for p is beta (α, β). Find the posterior distribution for p.