PSTAT 120A: Discussion Session 8 (May 16 - May 20) Practice problems: Problem 1. Select a point (X, Y ) at random (i.e. uniformly) from the unit circle with center (0, 0) and radius 1, and let R be the distance of the point from the origin. (a) Find the probability density function (PDF) of R. Hint: One approach is to first find the cumulative distribution function (CDF) of R. (b) Compute P (a < R < b) for 0 < a < b < 1. Homework problems (the quiz in Week 9 will be based on the following problems): Problem 1. The time it takes for an express train to arrive at your station is an exponential random variable with rate of 5 trains per hour. The time it takes for a local train to arrive at your station follows an exponential distribution with rate of 15 trains per hour. An express train gets you to work in 16 minutes and a local train gets you there in 28 minutes. You always take the first local train to arrive, but your co-worker always takes the first express train. You are both waiting at the same station. What is the probability that you will get to work before your co-worker? Problem 2. John wants to purchase a bond which will pay him X thousand dollars after two years, where X is equally likely to be any of the numbers in the set {0, 1, 2, 3, 4, 5}. John believes that the continuously compounded rate of interest, R, is independent of X and has a uniform distribution on the interval (0.04, 0.08). The present value of this bond is given by V = Xe2R . Compute the mean of the present value of the bond. Problem 3. Let X be a uniform random variable defined on the interval (0, 1). If Y = 6X 2 6X +1, compute the correlation of X and Y . (a) Are X and Y uncorrelated? (b) Can you use your answer in part (a) to conclude that X and Y are independent? Problem 4. Let (X, Y ) have a uniform distribution in the region D = {(x, y) : 0 < x < 2, 0 < y < 4, x < y}. (a) Compute P(X < 1). (b) Compute P(Y < X 2 ). Hint: The distribution of (X, Y ) is uniform in the region D, so probabilities can be computed by comparing sizes of areas. 1 Problem 5. The joint probability density function of (X, Y ) is given by ( c(x + y)2 , if 0 x 1, and 0 y 1 fX,Y (x, y) = 0, otherwise. (a) Find the value of c. (b) Compute the probability P(X + Y > 1). (c) Are X and Y independent. Justify your answer. 2