STAT 230 Final, Spring 2007

American University of Beirut STAT 230 Introduction to Probability and Random Variables Spring 2008 Final Exam Name: ID #: circle your section please: 1(TTH 8am) 2 (TTH 2 pm) Exercise 1 (20 points) Let f(a) = 2c, if -1 < z < 2, be the pdf of a continuous random variable X. Find the pdf of Y = X?. Exercise 2 (20 points) The jomt pdf of two random variables X and ¥ is f(zyl=ky O<a<y<l a. find the value of the constant k b. find the marginal pdf of X and Y. Are they independent? c. find P(XY < 1/2) d. find E(X?Y) Exercise 3 (20 points) Let X1,X2 be two independent chi-square random variables with ri and rz degrees of freedom, respectively. Let ¥; = (X1/ri)/(Xe/r2) and Yo = Xp. Find the joint pdf of Y; xthen find the marginal pdf of Yj. ots Exercise 4 (25 points) Let X,, X2, X3 denote a random sample of size n = 3 from a distribution 3\ (1\F pc=ny=(2) (2) =a... a. find P(X, = 1, Xo = 3,X3 = 1) with the geometric pdf b. determine P(X, + X2 + X3 = 5) c. find the moment generating function of Z = X, + X2 + Xz, then give the distribution of Z d. let Y = max(X 1, X2,X3) i. find P(Y <2) ii. find P(Y <n), then deduce P(Y =n) Exercise 5 (15 points) In the casino game roulette, the probability of winning with a bet on red is p = 18/38. Let Y equal the number of winning bets out of 1000 independent bets that are placed. Approximate P(Y > 500). (hint: consider Y = X,+...+X1o00 where X; is ‘the winning at the ith bet, then use the central limit theorem) good luck