Let Y be the sum of the observations of a random sample from a Poisson distribution with

QUESTION:

Let Y be the sum of the observations of a random sample from a Poisson distribution with mean . Let the prior pdf of be gamma with parameters and . (a) Find the posterior pdf of , given that Y = y. (b) If the loss function is [w(y) ] 2, find the Bayesian point estimate w(y). (c) Show that w(y) found in (b) is a weighted average of the maximum likelihood estimate y/n and the prior mean , with respective weights of n/(n + 1/) and (1/)/(n + 1/)