6. Given Z~ Normal(0,1) Using the definition of the moment generating function compute (derive) the moment generating function of Z2 [that is compute E(ez)]. Identify the distribution if possible.

Q5. A continuous r.v. X is said to have a Gamma (a, B) distribution if its moment generating function (m.g.f.) is given by M x (1) = (1- Bt). Let X1,X 2,.,X , be a random sample of size n from an exponential distribution with mean 1/0 and m.g.f. (1-0t). Let Y =EX; i =l 1 Ex [For p.d.f.s of Gamma and Exponential, and X i =1 (a) Find the m.g.f. My (t) of Y and state the distribution of Y specifying the parameters. (b) Find the m.g.f. M (t) of X and state the distribution of X specifying the parameters. (c) Suppose you have a random sample of size n = 100 from a Gamma (a, B) distribution with a = 3 and B = 5/3. Use the Central Limit Theorem to find P(4.5