1 MGFs (moment generating function

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