STAT 230 Final, Spring 2001

AMERICAN UNIVERSITY OF BEIRUT MATH 233. Final Exam June 9. 2001 Time = 1 Honr and 50 Mintes JH PCA). 1 and PCB) = t/docan cl and 2 be disjoint? Explain. (5 pts) 2. Let fer) = (1/3)|1 — |(r = a)/3\] ha - 8 <9 <a + 8 where —oo < a<tooand 3 > 0. (a) Show that f(r) is a probability density function. (5 pts) (b} Find the distribution fimetion that corresponds to f(r). (10 pts) (c) Pind the mean and variance of .X. (5 pts) 3. XN isa binomial raudom variable with = | and probability of success p. Further, the raudom variable ¥ has B(Y|.Y = 0) = Land B(Y|X = b) = 2, what is P(Y)? (5 pts) 4. Let X and Y have the joint probability density function [ny =kifO<4r<land0<y<l (1) (a) What is the value of & for which f(a.y) is a probability density function? (5 pts} (b) What is the distribution of Z = X +? (10 pts) 5. Let XN aud ¥ have the joint probability density function fly) =e" if 0< X< cand <y <0 Q2) What is the distribution of the random variable Z ~. Y/Y? (10 pts) )». Let Xy and No be a random sample of size 2 from a distribution 2 with a probability density fimetion f(r) ~ 2 2 aifa > 1. Let Y = min{ X,, Xo} (a) Does ECYX,). ¢ = 1,2. exist? Tf so. find it. (5 pts) (b) Does F(¥)} exist? Ifso. find it. (LO pts) ABRARS OF nana y i Lon