Exercise 1. Let ? be a set, F be o-field on 2, and Ce F. Show that Fo = {CnA : Ac F} is a g-field on C.2. Find a relationship between the transform of a gamma random variable with parameters o and A and the transform of a gamma random variable with parameters o - 1 and A. What does this imply if X is an m-Erlang random variable? Note that m- Erlang is the special case of a gamma random variable when o is an integer m. This random variable can be obtained by taking the sum of m iid exponential random variables with parameter X.Prob. 4 Let X be an exponentially distributed random variable with probability densityr metion (PDF) given by _ 32"\". x a D f" (x) _ {0. otherwise Consider the random variable I" = I? . (a) Determine the hazard rate function for the random variable Y. (b) Give an algorithm for generating the random variable Y from a standard uniform random variable U. (c) Choose a value for the parameter 1 so that the mean of the random variable Y is 5, Le, E(Y) = 5