A random variable has the Pareto Distribution with parameters :30 > 0 and a > 0 if it has a continuous distribution with pdf, 0151:\" f(33|'130a0) : Slim-31 1 for m 2 330,01 > 0, and f(:c|:cg,a) = 0 for w 0,04 > 0. What is the Bayes estimator of 9 for the square error loss? 2. (10pts) Let X1, ...,Xn be a random sample from a Poisson distribution with unknown mean 6 and that the prior distribution of 6 is gamma with nite mean u > 0. Show that the Bayes estimator of 6 for square error loss converges in probability to 9. 3.(40 pts) Suppose that a population height of people in a town follows a normal distribution with mean [.L unknown and standard deviation 2.5 inches. Assume that the prior distribution of ,u is normal with mean 69 inches and standard deviation 1 inch. If the average height of a random sample of 12 individuals from the population is 69 inches, what is the posterior distribution of ,u. 4. (40pts) Let Xhz' = 1, ...,n be a random sample from an exponential distri- bution with parameter [.L. Assume that the prior distribution of ,u is Gamma (01,5), Oz, 5 > 0. Find the posterior distribution of of p