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).

Suppose we have a set of small wooden blocks showing the 26 letters of the english alphabet, one letter per block. (think of scrabble tiles.) our set includes 10 copies of each letter. we place them into a bag and draw out one block at a time. (a) if we line up the letters on a rack as we draw them, how different ways coukl we fill a rack of 5 letters? (b) now suppose we just toss our chosen blocks into a pile, and whenever we draw a letter we already have, we put it back in the bag and draw again. how many different piles of 5 blocks could result? possible? piles will contain at least one repeated letter? (c) if we draw out 5 blocks wit hout looking at them, how many different piles are (d) if we draw out 5 blocks without looking at them, how many of the possible 2. (4) consider the following formula. 12 give two different proofs, one using the factorial formulas and the other combina torial.