STAT 230 Final, Fall 2007

American University of Beirut STAT 230 Introduction to Probability and Random Variables Fall 2007-2008 Final Exam Name: ............................... ID #: ................................. Exercise 1 Let X1, X2, X3 be three independent random variables with binomial distributions b(4, 1/2), b(6, 1/3), and b(12, 1/6) respectively. a. find P(X1 = 2, X2 = 2, X3 = 5) b. find E(X1X2X3) c. find the mean and the variance of Y = X1 + X2 Exercise 2 The joint pdf of two random variables X and Y is f(x, y) = kx 0 < x < y < 1 a. find the value of the constant k b. find the marginal pdf of X and Y . Are they independent? c. find P(X + Y < 1/2) d. find E(X2Y ) Exercise 3 A man and a woman decide to meet at a certain location. If each person indepen- dently arrives at a time uniformly distributed between 12 noon and 1 PM, find the probability that the first to arrive has to wait no longer than 10 minutes. Exercise 4 Let X and Y be a random sample of from the exponential distribution with pdf f(x) = e−x 0 < x < ∞ Let U = X/Y and V = X + Y . Find the joint pdf of the couple (U, V ). Are U and V independent? Exercise 5 Let X1, X2, . . . , Xn be a random sample of size n with uniform distribution over (0, 1). Find the pdf of Y = max(X1, X2, . . . , Xn). Find E(Y ), V ar(Y ) and MY (t), the moment generating function of Y . good luck