(a) Let c> 0 and L(0₁a) = {(0-al a|, 0 < a θ Σα Assume that has a continuous distribution. Show that the Bayes estimator of 0 is the 1/(1+c) quantile of the posterior distribution of 0. (b) Suppose that the population has a Poisson distribution with mean which is un- known. Suppose that the prior distribution of 0 is a gamma distribution with parame- ters a and B. Show that the posterior distribution of 0 is again a gamma distribution with parame- ters α = α+năn, ß' = ß+n, n is the sample size and X₁ is the sample mean.
