AMS 311, Spring 2017 Homework 8 Due at the beginning of class, April 20, 2017 1. 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 between 6.30 and 7.30pm that A arrives and Y denote the number of minutes 6.30 and 7.30pm that B arrives. Assume that X and Y are independent and that they are both uniformly distributed on the interval (30, 30). (a) 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) Hence, or otherwise, find the probability that B will be scolded. 2. Let X and Y be two independent random variables with means 1 and 2 and variances 12 and 22 , respectively. (a) Show that X Y and X + Y are uncorrelated if and only if 12 = 22 . (b) Show that Cov(X, XY ) = 2 12 . (c) Show that Var(XY ) = 12 22 + 21 22 + 22 12 . 3. 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) Find the following probabilities: i. P (X 2, Y 3). ii. P (X + Y 4). (b) Find the marginal probability density function X and Y . (c) Find the conditional probability density function of Y given X = x and use it to find E(Y |X = 2). 4. If X and Y have a joint probability density function ( 1, if 0 < y < 1, x + y < 1, x y > 1 f (x, y) = 0 otherwise (a) Find P (Y < 0.5|X < 0.5). (b) Find the marginal probability density functions X and Y . (c) Find the conditional probability density functions of Y given X = x and X given Y = y. (d) 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. 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) Find the conditional mean and variance of Y given X = x. (b) Using the results from (a), find the mean and variance of Y . (c) Find the correlation coefficient of X and Y . c PF.Kuan