MTH 201 - Assignment 9 - written component Assignment 9 is due at the beginning of class on Wednesday April 29th, 2015. 1. Let X be an exponential random variable with parameter λ = 1. Exponential random variables are very successful in describing times until the next failure of an item in a large population of items. For the lifetime of an individual item however, the exponential is often a reasonable fit but sometimes some modifications are required to better fit the data. You will explore one such modification in this problem. 1 (a) For fixed a, b, c > 0, let Y = a + bX c where X is exponential with λ = 1. Find an explicit formula for the cumulative distribution function FY (y). Show your work. (b) Use your answer in (a) to find the density function fY (y) for Y . Show your work. Your answer should be a Weibull distribution discussed in section 5.6 of the book. Find formulas for the parameters ν, α, β in the Weibull distribution you get in terms of a, b, c. 24xy if 0 < x < 1, 0 < y < 1, and 0 < x + y < 1, 2. Let f (x, y) = 0 otherwise. (a) Show that f (x, y) is a joint probability density function for some continuous random variables X and Y . (b) Find E[X]. (c) Are X and Y independent? Justify your answer. x+y if 0 < x < 1, 0 < y < 1 3. The joint density function of X and Y is f (x, y) = 0 otherwise. (a) Find the density function of X. (b) Find P (X + Y < 1). (c) Are X and Y independent? Justify your answer. 4. Let X and Y be two independent random variables. Often one has to consider a new random variable W = min(X, Y ) which gives the smaller of the values of X and Y . (a) Find the cumulative distribution function FW (a) of W in terms of FX and FY . (Hint: Notice that min(X, Y ) > a if and only if both X > a and Y > a. Now relate the probability of the above event to FW (a)). (b) Now assume further that X and Y are continuous random variables. Find a formula for the density fW of W in terms of FX , fX , FY and fY . 5. Suppose X and Y are independent normal random variables with parameters µ1 , σ12 and µ2 , σ22 , respectively. Prove that X + Y is a normal random variable, and find its parameter values. 6. A deck of n cards numbered 1 through n is thoroughly shuffled so that all possible n! orderings can be assumed to be equally likely. Suppose you are to guess how the cards are ordered as follows: you make n guesses sequentially, where your ith guess is a guess of which card is in position i. Let N denote the number of correct guesses. (a) If, while you are guessing, you are not given any information about your earlier guesses, show that, for any strategy, E[N ] = 1. (b) Suppose that after each guess you are shown the card that was in the position in question. What do you think is the best strategy? Show that, under this strategy, E[N ] = 1 1 1 1 + + + · · · + + 1, n n−1 n−2 2 and conclude that E[N ] ≈ log(n). (c) Suppose that you are told after each guess whether you are right or wrong. In this case, it can be shown that the strategy which maximizes E[N ] is one that keeps on guessing the same card until you are told you are correct and then changes to a new card. For this strategy, show that E[N ] = 1 1 1 1 + + + · · · + + 1, n! (n − 1)! (n − 2)! 2! and conclude that E[N ] ≈ e − 1. Hint: For all parts, express N as the sum of indicator (that is, Bernoulli) random variables. 7. An insurance company has 10,000 automobile policyholders. The expected yearly claim per policyholder is $240, with a standard deviation of $800. Approximate the probability that the total yearly claim exceeds $2.7 million. Explain any assumptions you make about the underlying distributions. 8. Explain why a gamma random variable with parameters (α, λ) has an approximately normal distribution when α is large. Hint: Consider α as a large integer, and recall how to interpret a gamma distributed random variable in terms of waiting times for a Poisson distributed event. Then invoke the central limit theorem.