1. (a) Let c > 0 and L(0, a) = co - a), 0

(a) Let c > 0 and cleal, IOal, Assume that e has a continuous distribution. Show that the Bayes estimator of e is the 1/(1 + c) quantile of the posterior distribution of e. [15 marks] (b) Suppose that the population has a Poisson distribution with mean e which is un- known. Suppose that the prior distribution of e is a gamma distribution with parame- ters a and . Show that the posterior distribution of 9 is again a gamma distribution with parame- ters n is the sample size and Rn is the sample mean. Total: 25 marks [10 marks]