We will use the Minitab macro BinoBP.mac or the equivalent R function to find the posterior distribution of the binomial probability when the observation

We will use the Minitab macro BinoBP.mac or the equivalent R function to find the posterior distribution of the binomial probability π when the observation distribution of Y |π is binomial(n, π) and we have a beta

(a,

b) prior for π. The beta family of priors is the conjugate family for binomial observations. That means that if we start with one member of the family as the prior distribution, we will get another member of the family as the posterior distribution. It is especially easy, for when we start with a beta

(a,

b) prior, we get a beta (a

, b

)

posterior where a
= a + y and b
= b + n − y.

Suppose we have 15 independent trials and each trial results in one of two possible outcomes, success or failure. The probability of success remains constant for each trial. In that case, Y |π is binomial (n = 15, π). Suppose that we observed y = 6 successes. Let us start with a beta (1, 1) prior. The details for invoking BinoBP.mac and the equivalent R function are given in Appendix 3 and Appendix 4, respectively. Store π, the prior g(π), the likelihood f(y|π), and the posterior g(π|y) in columns c1-c4 respectively.

(a) What are the posterior mean and standard deviation?

(b) Find a 95% credible interval for π.