Chapter 16.2, Problem 8E

Refer to Exercise 16.6. If Y is a binomial random variable based on n trials and success probability p and p has the conjugate beta prior with parameters α = 1 and β = 1,

1. a determine the Bayes estimator for p, ${\hat{p}}_B$.
2. b what is another name for the beta distribution with α = 1 and β = 1?
3. c find the mean square for error (MSE) of the Bayes estimator found in part (a). [Hint: Recall Exercise 8.17].
4. d For what values of p is the MSE of the Bayes estimator smaller than that of the unbiased estimator $\hat{p}$ = Y/n?

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, …, 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.