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 My(t) = (1 - Bt). Let X1, X 2,...,x , be a random sample of
n
72
size n from an exponential distribution with mean 1/8 and m. g.f. (1 - 01)-°. Let Y = XX,
i=1
and = 2x;. [For p. d.f. s of Gamma and Exponential, see the book or lecture notes).
i=1
(a) Find the m. g.f. My () of Y and state the distribution of Y specifying the parameters.
(b) Find the m. g.f. My(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 < X <5.5).