Moment generating functions and central limit theorem. A moment generating function of a realvalued random variable Z is defined as M Z(s) = lE[eSZ]. It is closely related to the generating function Z(s) = lE[sZ] we have been using so far. (a) Show that Mg(s) = odes). (b) Assuming Z is a random integer in Z, show that MZ(0) = 1,Mg(0) = IE[Z] and 1145(0) = lE[Z2]. (In general, for a real valued Z we have M $1) = ]E[Z"]. This is why M is called the moment generating function.) (e) Either using (a) or by direct computation, show that if X and Y are independent realvalued random variables, then M X+y = M XMy, and that for a deterministic constant A E R we have MAXCS) = MXOLS). One way to prove the central limit theorem is by using moment generating functional. Here we sketch the proof in the case when the random variable is a centred integer and has a moment generating function. Let X a random number in Z with mean IE[X] = O and variance Var(X) = 1. Denote by H Sn 2 j=1 Xj the sum of n independent copies of X. Do at least one of the following steps. ((1) Show that the moment generating function of N\" = is.\" can be written as Mme) = (MX(%))H