STAT 233 Final, Spring 2003

I ft: ” > JAMERICAS UNIVER! Y ft 1 BR ARY _ - ani OF BEIRUT 6F BEIROT AMERICAN UNIVERSITY OF BEIRUT arama PAE 933 FINAL EXAMINA FION Time = 1 hour 30 minutes June 12, 2003 . 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*)? (Spts) ; (b) Prove or disprove: If P(A|B) = P(A|B*) then A and B are inde- pendent. (5 pts) . A fair die is tossed 12 times. (a) Let X be the number of dice that show the face 6 and Y be the number of dice that show the face 1. Find the joint probability distribution of (X,Y). (5 pts) (b) Find E(XY). (5pts) . The truncated Geometric distribution is defined as the following: P(X =n) =k(1—p)*"'p; if = 2,3... Find constant k, E(X), and Var(X). (10pts) . Given that the joint density of (X,Y) is k = —_ 0: f(z, y) (Gaapyet 7 yr? (a) Find the constant &. (Spts) (b) Find the marginal density fi(z). (Spts) (c) Find the conditional density f,)2(zly). (5pts) . Suppose that X and Y are independent random variables with the same p.d.f. f(x) = be; 2 > 0. Show that X+Y and X/Y are independent and compute E(X/Y). (15pts) _ Let X and Y be independent and identically distributed random vari- ables with common density function, f(z) = ze-® /? for z > 0. Find the distribution of the random variable R = VX? + Y? (10pts)