Problem 1 (Moment Generating Functions) Let X be exponentially distributed with parameter 1, i.e., fx (2 ) = JAe-x x20 otherwise. The moment generating function (as discussed in class) is defined as dx (t) = Eletx]. a. Find Dx (t) when X is the exponential random variable. Hint: [, e"du = e". b. If X and Y are independent exponentially distributed random variables, find the moment generating function of Z = X + Y. c. Find E[Z3]. d. Suppose V is a random variable whose p.d.f. is fv(v) = Kfx (v)fy (v), for some normalizing constant K; find K. e. Find the characteristic function of V in terms of the moment functions of X, Y