9. Visitors arrive to your web server according to a Poisson process of rate A per hour. However, you do not know A. We will use the notation that N1 is the number of arrivals in the rst t hours. Rememba that (M )s e A: la! ' Also the following integrals may be helpful in one or more parts of the problem: Par. = r) = 5 5 j Ai'e-Mi = 2 7 3715-5 e 1.7507, ] Ase-MA = 6 7 2362'5 x 4.4098 c D [2|] points] (a) You observe the process for t = 1 hour, and count I: = '2 visitors. What is the Maximum Likelihood (ML) estimate of X! [5 points] (b) Suppose for this and all subsequent parts that you have a prior distribution on the rate of the poisson process. Specically A is a value that comes from the random variable A where A is uniformly distributed on the range [0,5]. As before, we suppose you observe 2 visitors during an hour of observation. What is the oonrlitional PMF of the number of arrivals in 1 hour, given an arrival rate A? In other words nd pNIIAUCIA). [2 points] (C) Now nd the joint distribution fNuAU': A) by taking your apression for PMIAU'H) and multiply- ing by Lil")- Remember to note what range of values for k and A your expression is valid for, and for what values of k and A the density function is equal to zero. (Err example, it is equal to zero when k