advertisement

Texas A&M University Department of Mathematics Volodymyr Nekrashevych Summer 2010 MATH 411 Homework 4 4.1. Suppose that X and Y have joint density f (x, y) = c(x + y) for 0 < x, y < 1. (a) What is c? (b) What is P (X < 1/2)? 4.2. Suppose X and Y have joint density f (x, y) = 1 for 0 < x, y < 1. Find P (XY ≤ z). 4.3. Suppose X is uniform on (0, 1) and Y = X. Find the joint distribution function of X and Y . 4.4. Suppose a point (X, Y ) is chosen at random from the disk x2 + y 2 ≤ 1. Find (a) the marginal density of X and (b) the conditional density of Y given X = x. 4.5. Suppose X and Y have joint density f (x, y) = 6y when x > 0, y > 0, and x + y < 1. (a) Find the marginal densities of X and Y and (b) the conditional density of X given Y = y. 4.6. Suppose we draw 13 cards out of a deck of 52. What is the expected value of the number of aces we get? 4.7. Ten people get on an elevator on the first floor of a seven-story building. Each gets off at one of the six higher floors chosen at random. What is the expected number of stops the elevator makes? 4.8. Roll two dice and let Z = XY be the product of the two numbers obtained. What is the mean and variance of Z? 4.9. Suppose X takes on the values −2, −1, 0, 1, 2 with probability 1/5 each, and let Y = X 2 . (a) Find cov(X, Y ). (b) Are X and Y independent? 4.10. Suppose that it is known that the number of items produced at a factory per week is a random variable X with mean 50 and variance 25. What can we say about P (40 < X < 60)?