Math 520, Fall 2016 --- Homework 1 Due Thursday, Sept. 15, 2016 A subset of these problems will be graded - 20 points total 1. Problem 1.7 (i.e. Exercise 7 from Chapter 1) 2. Problem 1.9 3. Problem 1.13 4. Calculate the mean and variance of the claim random variable X, where the probability of a claim in one year is q = 0.10 and the claim amount random variable B is uniformly distributed between 0 and 600. Assume that there is never more than one claim in a year. 5. Consider the automobile collision example on page 29 of the textbook and in the class notes. As in the class notes, let D be the actual damage to the car from a collision, and let B be the claim amount as defined in the example. In this case define the indicator I by I = 1 if a collision occurred. You are given: Pr[I = 0] = 0.75 0.9 D | I = 1 has density for 0 x 2250 , with Pr[D 2250 | I = 1]=0.9. 2250 Let X = IB. Calculate EX and Var(X). 6. The number of claims N in a year has a discrete distribution: Pr[ N k ] 0.1(4 k ) for k 0,1,2,3 The amount for each claim has mean 300 and variance 30000. The distribution of amount per claim is independent of N Calculate the mean and variance of the total claim amount. Hint: Use formulas (2.2.10) and (2.2.11). 7. Problem 2.3 continued next page HW1_graded.docx Last saved 9/7/2016 8:00 AM 8. Problem 2.7 except replace the distributions of the Xi with the following: x Pr[X1=x] Pr[X2=x] Pr[X3=x] Pr[X4=x] 0 0.6 0.5 0.5 0.8 1 0.0 0.25 0.5 0.0 2 0.4 0.25 0.0 0.2 Note that you need to specify FS(x) for x = 0, 1, ... 7. Partial answer: Pr[S=3]=0.225, and FS(3)= 0.755 9. Let X be a random variable with distribution function: FX (x) = x2 for 0 x 1. Define g(y) on the support of FX as g(y) = 1 - 3y for 0 y 1. Define Y = g(X). Calculate the distribution function FY(y) and density function fY(y) of Y, specifying the values of y for which the formulas hold. Hint: The Background material for the course discusses the change of variable formula. 1 10. Let X = ln ln(U ) , where U has the uniform distribution on (0,1). U Calculate the distribution function FX(x) for x > 0. 11. X is a random variable representing the life of a piece of machinery. X is x exponentially distributed with FX(x) = 1 exp for x 0. You place a bet with a 2 colleague on how long the machine will last. Your winnings W are: 100 if X 1 W = 0 if 1 X 3 400 if X 3 Find E(W). Note: W is a discrete random variable. HW1_graded.docx (Answer: 49.91) Last saved 9/7/2016 8:00 AM