# Q. 5. Let X be any random variable, with moment generating function M(S) = E[es], and…

Q. 5. Let X be any random variable, with moment generating function M(S) = E[es], and assume M(s) < o for all s E R. The cumulant generating function of X is defined as A(s) = log Ele**] = log M(s), SER Show the following identities: (1) A'(0) = E[X]. (2) A”(0) = Var(X). (3) A"(0) = E[(X - E[X]))). Using the inversion theorem for MGFs, argue the following: (4) If A'(s) = 0 for all s ER, then P(X= 0) = 1. (5) If A"(s) = 0 for all s ER, then P(X = c) = 1 for some constant CER. (6) If A"(s) = 0 for all s ER and A"(0) + 0, then X is normally distributed; that is, X ~ N(H1, 02) for some ER and a > 0.