STAT 230 Final, Fall 1996

AMERICAN UNIVERSITY OF BEIRUT MATH 233 FINAL EXAMINATION Time = 1 hour 30 minutes February 1, 1997 . Let A and B be two events defined on the same sample space. (a) If P(A) =.3, P(B) = .4, and P(A|B) = .5, what is the conditional probability P(A|B*)? (5pts) (b) Prove or disprove: If P(A) = 3/4 and P(B) = 3/8 then 1/8 < P(AN B) < 3/8. (10pts) . A fair die is tossed 12 times. (a) Let X be the number of the face 6 and Y be the number of the face 1. Find the joint probability distribution of (X,Y). (10pts) (b) Find COV(X,Y). (Spts) . The truncated Poisson distribution is defined as the following: ua . (e4 — 1)a!’ Find E(X) and Var(X). (10pts) E P(X =2)= z=1,2,... . Given that the joint density of (X,Y) is k zit > Oy >0 fw = Gays (a) Find the constant k. (5pts) (b) Find the marginal density f,(z). (Spts) (c) Find the conditional density fij2(zly). (Spts) . Suppose that X and Y are independent random variables with the same p.d.f. f(x) = 6e-%; z > 0. Show that X+Y and X/Y are independent and compute E(X/Y). (15pts) . Let X and Y be independent random variable with densities f(z) = (1 — 2?)-¥/?/a;-1 < < 1 and g(y) = ye~*/*/4;y > 0 respectively. Find the distribution of the product random variable XY. (10pts)