1. You feel that ~p , the probability of heads on a toss of a particular coin is either 0.4, 0.5, or 0.6. Your prior probabilities are P(0.4) = 0.1, P(0.5) = 0.7, and P(0.6) = 0.2. You toss the coin three times and obtain heads once and tails twice. What are the posterior probabilities? If you then toss the coin three more times and once again obtain heads once and tails twice, what are the posterior probabilities? Also, compute the posterior probabilities by pooling the two samples and revising the original probabilities just once; compare with your previous answer.
Suppose that you feel that the mean of your prior distribution is 1/2 and that the variance of the distribution is 1/20. If your prior distribution is a member of the beta family, find r and n and determine the posterior distribution following the sample of size six. Graph the density functions and find the mean and the variance of the posterior distribution.