AMS 311, Spring 2016 Homework 8 Due at the beginning of class, April 19, 2016 1. (20 points) Two friends A and B go on a date and agree to meet at 7pm. Friend A is an impatient person and will scold b if A arrives before B and B arrives later than 7.15pm. Let X denote the number of minutes after 7pm that A arrives and Y denote the number of minutes after 7pm that B arrives. Assume that X and Y are independent and that they are both uniformly distributed on the interval (30, 30). (a) (10 points) Draw a graph with X and Y as the two axes and shade the region which represents the event that B will be scolded. (b) (10 points) Hence, or otherwise, nd the probability that B will be scolded. 2. (20 points) Let X and Y be two independent random variables with means 1 and 2 and 2 2 variances 1 and 2 , respectively. 2 2 (a) (6 points) Show that X Y and X + Y are uncorrelated if and only if 1 = 2 . 2 (b) (7 points) Show that Cov(X, XY ) = 2 1 . 2 2 2 2 (c) (7 points) Show that Var(XY ) = 1 2 + 2 2 + 2 1 . 2 1 3. (30 points) Suppose the joint probability density function of X and Y is given by f (x, y) = x+y 50 if x = 1, 2, . . . , y and y = 1, 2, 3, 4. (a) (10 points) Find the following probabilities: i. P (X 2, Y 3). ii. P (X + Y 4). (b) (10 points) Find the marginal probability density function X and Y . (c) (10 points) Find the conditional probability density function of Y given X = x and use it to nd E(Y |X = 2). 4. (40 points) If X and Y have a joint probability density function f (x, y) = 1, if 0 < y < 1, x + y < 1, x y > 1 0 otherwise (a) (5 points) Find P (Y < 0.5|X < 0.5). (b) (10 points) Find the marginal probability density functions X and Y . (c) (10 points) Find the conditional probability density functions of Y given X = x and X given Y = y. (d) (15 points) Using the density function found in (c), evaluate c PF.Kuan i. P (Y < 0.3|X = 0.5) ii. P (0.5 < X < 0.5|Y = 0.5) iii. P (X > 0|Y = 0.2) 5. (20 points) Suppose the conditional distribution of Y given X = x is Gamma with = 4 and = 1/x and the marginal distribution of X is exponential with mean 2. (a) (8 points) Find the conditional mean and variance of Y given X = x. (b) (8 points) Using the results from (a), nd the mean and variance of Y . (c) (4 points) Find the correlation coecient of X and Y . c PF.Kuan