Assume the conditional distribution is a member of exponential family, i.e. f(y|) = A()B(y) exp(c()T(y) 2 for some functions A, B and T. Show that a prior of the form g() (A())k exp(dc()) for some functions A and constants d and k is the conjugate prior.

(ii) Assume (the geometric distribution) f(y|) = (1 ) y I(y {0, 1, 2, . . .}). Use part (i) and find the conjugate prior and then calculate the posterior distribution for .

(iii) Suppose we observe y = 8, the number of failures until the first success in a sequence of independent Bernoulli trials where is the probability of success. Find the posterior distribution assuming a uniform prior on . Draw the graph of the posterior and find the shortest 95% credible region for . Calculate the mean and standard deviation of |y = 8.