STAT 230 Final, Spring 2008

AMERICAN UNIVERSITY OF BEIRUT MATH 233 FINAL EXAMINATION Time = 2 hours 1. Six cards are drawn at random and without replacement from an ordi- nary deck of 52 cards. (a) Find the joint density of the number, of aces x. and the number of kings Y. (b) Find the conditional density of X given Y. - Hint: use the fact ; k . ny Ne of Mt Ne (5) (2) ("2") 2. Let X and Y be independent and identically distribiited random vari- ables with common density - f(x) = O(1 — 6)? for z= 0, 1,2, fae Find P(X = Y). - 3, The discrete density of . the random:variable xX is. given by f(r) = a/8 for « = 1,2 and fy\x(y|z) is binomial with:parameters # and 1/2; that is e frix(y|z) = ( ? ) (=)* for y =0,1,...,2 and « = 1,2 y Nie (a) Find £(X) and Var(X). (b) Find E(Y). ; (c) Find the joint distribution of X and Y. aN ee SOONIVERS Ey, RARY . eis shes boca alls | Sarde tiie Se, Rie AEE AG eke iecs eae eran 1 Rey oe. j 4 4 a Sabir emis: oa? ; a 4. Let X1,X2,...,X_ be n independent and identically distributed random x variables with common density function © : f(z) =27 fore > 1. Let ¥, = min{X1, X2,..., Xn}. te # (a) Does E(X;) exist? If so, Find it. : (b) Does E(Y,) exist? If so, Find it. 4 5. Let X; and X, be a random sample of size 2 from the density : f(z) = ser for > “ Find the distribution of the random variable Y = X, 1X3. ; 6. Let X, and X2 be a random sample of'size 2 from the density 4 f(z) = 1 for 0 <2 <1 Sok yes 5 Find the distribution of Y = (X; + X,)/2. 7. Let X and X, be a random sample from N (0, 1); that is the common 4 pd.f.is f(z) = (1/V2zr) exp(—1/227) for all « S (a) Show that (Xi +X2)/V/2 and (K+ X2)/v2 are 2 independent and 3 N(0,1) random variables. che 8 (b) Find the distribution of (X; +: XX ~ ¥”3) i} : 4 2 4 . 3 t ; one