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Mathematical Statistics with Applications, Seventh Edition

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www.downloadslide.com Exercises 253 5.52 In Exercise 5.8, we derived the fact that { 4y1 y 2 , 0 ≤ y 1 ≤ 1, 0 ≤ y 2 ≤ 1, f (y 1 , y 2 ) = 0, elsewhere is a valid joint probability density function. Are Y 1 and Y 2 independent? 5.53 In Exercise 5.9, we determined that { 6(1 − y2 ), 0 ≤ y 1 ≤ y 2 ≤ 1, f (y 1 , y 2 ) = 0, elsewhere is a valid joint probability density function. Are Y 1 and Y 2 independent? 5.54 In Exercise 5.10, we proved that { 1, 0 ≤ y1 ≤ 2, 0 ≤ y 2 ≤ 1, 2y 2 ≤ y 1 , f (y 1 , y 2 ) = 0, elsewhere is a valid joint probability density function for Y 1 , the amount of pollutant per sample collected above the stack without the cleaning device, and Y 2 , the amount collected above the stack with the cleaner. Are the amounts of pollutants per sample collected with and without the cleaning device independent? 5.55 Suppose that, as in Exercise 5.11, Y 1 and Y 2 are uniformly distributed over the triangle shaded in the accompanying diagram. Are Y 1 and Y 2 independent? 5.56 In Exercise 5.12, we were given the following joint probability density function for the random variables Y 1 and Y 2 , which were the proportions of two components in a sample from a mixture of insecticide: { 2, 0 ≤ y1 ≤ 1, 0 ≤ y 2 ≤ 1, 0 ≤ y 1 + y 2 ≤ 1, f (y 1 , y 2 ) = 0, elsewhere. Are Y 1 and Y 2 independent? 5.57 In Exercises 5.13 and 5.31, the joint density function of Y 1 and Y 2 was given by { 30y1 y2 2 f (y 1 , y 2 ) = , y 1 − 1 ≤ y 2 ≤ 1 − y 1 , 0 ≤ y 1 ≤ 1, 0, elsewhere. Are the random variables Y 1 and Y 2 independent? 5.58 Suppose that the random variables Y 1 and Y 2 have joint probability density function, f (y 1 , y 2 ), given by (see Exercises 5.14 and 5.32) { 6y 2 1 y 2 , 0 ≤ y 1 ≤ y 2 , y 1 + y 2 ≤ 2, f (y 1 , y 2 ) = 0, elsewhere. Show that Y 1 and Y 2 are dependent random variables. 5.59 If Y 1 is the total time between a customer’s arrival in the store and leaving the service window and if Y 2 is the time spent in line before reaching the window, the joint density of these variables, according to Exercise 5.15, is { e −y 1 , 0 ≤ y 2 ≤ y 1 ≤∞ f (y 1 , y 2 ) = 0, elsewhere. Are Y 1 and Y 2 independent?

www.downloadslide.com 254 Chapter 5 Multivariate Probability Distributions 5.60 In Exercise 5.16, Y 1 and Y 2 denoted the proportions of time that employees I and II actually spent working on their assigned tasks during a workday. The joint density of Y 1 and Y 2 is given by { y1 + y 2 , 0 ≤ y 1 ≤ 1, 0 ≤ y 2 ≤ 1, f (y 1 , y 2 ) = 0, elsewhere. Are Y 1 and Y 2 independent? 5.61 In Exercise 5.18, Y 1 and Y 2 denoted the lengths of life, in hundreds of hours, for components of types I and II, respectively, in an electronic system. The joint density of Y 1 and Y 2 is { (1/8)y1 e −(y 1+y 2 )/2 , y 1 > 0, y 2 > 0, f (y 1 , y 2 ) = 0, elsewhere. Are Y 1 and Y 2 independent? 5.62 Suppose that the probability that a head appears when a coin is tossed is p and the probability that a tail occurs is q = 1 − p. Person A tosses the coin until the first head appears and stops. Person B does likewise. The results obtained by persons A and B are assumed to be independent. What is the probability that A and B stop on exactly the same number toss? 5.63 Let Y 1 and Y 2 be independent exponentially distributed random variables, each with mean 1. Find P( Y 1 > Y 2 | Y 1 < 2Y 2 ). 5.64 Let Y 1 and Y 2 be independent random variables that are both uniformly distributed on the interval (0, 1). Find P( Y 1 < 2Y 2 | Y 1 < 3Y 2 ). *5.65 Suppose that, for −1 ≤ α ≤ 1, the probability density function of (Y 1 , Y 2 ) is given by { [1 − α{(1 − 2e −y 1 )(1 − 2e −y 2)}]e −y 1−y 2 , 0 ≤ y 1 , 0 ≤ y 2 , f (y 1 , y 2 ) = 0, elsewhere. a Show that the marginal distribution of Y 1 is exponential with mean 1. b What is the marginal distribution of Y 2 ? c Show that Y 1 and Y 2 are independent if and only if α = 0. Notice that these results imply that there are infinitely many joint densities such that both marginals are exponential with mean 1. *5.66 Let F 1 (y 1 ) and F 2 (y 2 ) be two distribution functions. For any α, −1 ≤ α ≤ 1, consider Y 1 and Y 2 with joint distribution function F(y 1 , y 2 ) = F 1 (y 1 )F 2 (y 2 )[1 − α{1 − F 1 (y 1 )}{1 − F 2 (y 2 )}]. a What is F(y 1 , ∞), the marginal distribution function of Y 1 ?[Hint: What is F 2 (∞)?] b What is the marginal distribution function of Y 2 ? c If α = 0 why are Y 1 and Y 2 independent? d Are Y 1 and Y 2 independent if α ≠ 0? Why? Notice that this construction can be used to produce an infinite number of joint distribution functions that have the same marginal distribution functions. 5.67 In Section 5.2, we argued that if Y 1 and Y 2 have joint cumulative distribution function F(y 1 , y 2 ) then for any a < b and c < d P(a < Y 1 ≤ b, c < Y 2 ≤ d) = F(b, d) − F(b, c) − F(a, d) + F(a, c).

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