Given a random variable X , the moment generating function of X is defined as the function MX (t ) = E[etX]. Moment generating functions, or MGFs for short, are immensely useful because of the Taylor expansion

Given a random variable X, the moment generating function of X is defined as the function Mx (t) = Efe ]. Moment generating functions, or MGFs for short, are immensely useful because of the Taylor expansion (tx) 2 (tx) 3 (tx )" =1+tX + + + . . . 2! 3! E n! By taking the kth derivative of the MGF of X with respect to f and evaluating at f = 0, we can gen- erate the &th moment of X, (i.e. the value of E X* ) without having to do any painful integration! (a) Compute the moment generating function My (t) of X, where X ~ Expo(1), fort