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).

Asystem of linear equations with more equations than unknowns is sometimes called an overdetermined system. can such a system be consistent? illustrate your answer with a specific system of three equations in two unknowns. choose the correct answer below. a. yes, overdetermined systems can be consistent. for example, the system of equations below is consistent because it has the solution nothing. (type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 6 b. no, overdetermined systems cannot be consistent because there are fewer free variables than equations. for example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 12 c. yes, overdetermined systems can be consistent. for example, the system of equations below is consistent because it has the solution nothing. (type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 8 d. no, overdetermined systems cannot be consistent because there are no free variables. for example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 24