Temple University AS 2101 Actuarial Probability and Statistics II, Fall 2016 Assignment 3 Note: Assignment 3 is due on Friday, October 21, at the beginning of the class. You are encouraged to work on these problems in groups of no more than 3 or 4. Each student must hand in her/his own answer sheet. Throughout this assignment, fY |X (y|x) denotes the conditional density of Y given that X = x. 1. You are given that the joint density function of Y1 and Y2 , f (y1 , y2 ) is a constant over the region C = {(y1 , y2 ) : 0 < y1 < 6, 0 < y2 < 6, 0 < y1 + y2 < 10}, and zero otherwise. (a) Determine the joint density function f (y1 , y2 ). (b) Calculate Pr(Y1 < 2Y2 ). (c) Calculate Pr(Y1 < Y2 |Y2 < 5). (d) Find the marginal densities of Y1 and Y2 , respectively. (e) Find the conditional density of Y1 , given that Y2 = 2, i.e. fY1 |Y2 (y1 |2). (f) What is the probability that Y1 is larger than 3, given that Y2 = 2. (g) Calculate E[Y1 Y2 ]. 2. An insurance company insures a large number of drivers. Let X be the random variable representing the company's losses under collision insurance, and let Y represent the company's losses under liability insurance. X and Y have joint density function ( k(x + 1 0.5y), 0 < x < 1 and 0 < y < 2, f (x, y) = 0, otherwise. (a) Determine k. (b) Find the joint (cumulative) distribution function of X and Y . (c) Calculate the probability that the total company loss is at least 1. (d) Find the marginal density of Y . (e) Find the conditional density of Y given that X = 0.5. (f) Calculate the expected value of the total company loss. 3. Let X and Y denote the values of two stocks at the end of a five-year period. X is uniformly distributed on the interval (0, 12). Given X = x, Y is uniformly distributed on the interval (0, x). (a) Find the joint density function of X and Y , f (x, y). (Hint: f (x, y) = fY |X (y|x)fX (x)). (b) Find the marginal density of Y . (c) Calculate E[Y ]. (d) Calculate E[XY ]. 1 4. Tom has two bulbs. The second bulb is a backup for the first, so the second is used immediately (and only) when the first has failed. Let X and Y be the times (in years) at which the first and second bulb fail, respectively. X and Y have joint probability density function ( kex e2y , 0 < x < y < , f (x, y) = 0, otherwise (a) Determine k. (b) Calculate the probability that Y < 2X. (c) Find the marginal density function of Y . (d) Calculate E[Y ]. (e) Determine whether X and Y are independent. (f) Given that the first bulb fails at X = 0.5 years, find the probability that the second bulb lasts longer than 1 year. 5. The joint density function of (X, Y ) is given by ( 4ye(x+2y) , 0 < x < , 0 < y < f (x, y) = 0, otherwise (a) Determine whether X and Y are independent. (b) Calculate the probability that Y < X 1. (c) Calculate E[X + Y ]. (d) Calculate E[XY 2 ]. 2