Question: 7. Suppose we are interested in a count process Nt with unknown random rate A > 0. Given the rate, the number of arrivals Nt

7. Suppose we are interested in a count process Nt with unknown random rate A > 0. Given the rate, the number of arrivals Nt is Poisson, Pr(Nt = n | A = x) = (At)"e-At n! , n = 0, 1,2,... . Suppose we put a Gamma(a, B) prior on A, TT ( 1 ) = Ba I(x) where a > 0 and B > 0 are known hyperparameters. (a) Find the posterior distribution, 7() | Nt = n), of the rate and then find the Bayes estimate of A, which is the mean of the posterior distribution. (b) Find Pr(Nt = n) for n = 0, 1,2,.... Does Nt have a Poisson distribution

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