For iid Poisson variates y1,…, yn with parameter ????, suppose ???? ∼ gamma(????, k) (Recall Section 4.7.2).

a. Show that the posterior distribution of ???? is gamma (i.e., the prior is conjugate), with posterior mean that is a weighted average of the sample mean and the prior mean. Explain how the weights change as n increases.

When n is very large, show that the posterior distribution has approximate mean ȳ and approximate variance ȳ∕n.

b. Find the posterior predictive distribution.