Mathematical statistics - Bayesian approach

Let X1,.... X, be a random sample from an exponential distribution on (0,co) with scale parameter 1. Suppose that we observe T = Xi+ ...+ Xe, where 0 is an unknown positive integer. Consider estimation of 0 under the loss function L(0, a) = (0 - a)2/0 and a geometric distribution with mean 1/p as the prior distribution for d, where p e (0, 1) is known. (a) Show that the posterior expected loss is E[L(0, a) |T = t] = 1+5 - 2a + (1 -e-)a?/5, where { = (1 - p)t. (b) Find the Bayes estimator of e and show that its posterior expected loss is 1 - Em=lems. (c) Find the marginal distribution of (1 -p), unconditional on d. (d) Obtain an explicit expression for the Bayes risk of the Bayes estimator in part (b )