Use the Minitab macro BinoDP.mac to find the posterior distribution of the binomial probability when the

Use the Minitab macro BinoDP.mac 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 discrete prior for π.

Suppose we have 8 independent trials and each has one of two possible either success or failure. The probability of success remains constant for each trial.

In that case, Y |π is binomial (n = 8, π). Suppose we only allow that there are 6 possible values of π, 0, .2, .4, .6, .8, and 1.0. In that case we say that we have a discrete distribution for π. Initially we have no reason to favor one possible value over another. In that case our we would give all the possible values of π

probability equal to 1 6

Suppose we observe 3 "successes" in the 8 trials. Use BinoDP.mac or the equivalent R function to find the posterior distribution g(π|y). Details for invoking BinoDP.mac are in Appendix 3. The details for the equivalent R function are in Appendix 4.

(a) Identify the matrix of conditional probabilities from the output. Relate these conditional probabilities to the binomial probabilities in Table B.1.

(b) What column in the matrix contains the likelihoods?

(c) Identify the matrix of joint probabilities from the output. How are these joint probabilities found?

(d) Identify the marginal probabilities of Y from the output. How are these found?

(e) How are the posterior probabilities found?