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Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com Supplementary Exercises 293 5.155 Suppose that Y 1 , Y 2 , and Y 3 are independent χ 2 -distributed random variables with ν 1 ,ν 2 , and ν 3 degrees of freedom, respectively, and that W 1 = Y 1 + Y 2 and W 2 = Y 1 + Y 3 . a In Exercise 5.87, you derived the mean and variance of W 1 . Find Cov(W 1 , W 2 ). b Explain why you expected the answer to part (a) to be positive. 5.156 Refer to Exercise 5.86. Suppose that Z is a standard normal random variable and that Y is an independent χ 2 random variable with ν degrees of freedom. a Define W = Z/ √ Y . Find Cov(Z, W ). What assumption do you need about the value of ν? b With Z, Y , and W as above, find Cov(Y, W ). c One of the covariances from parts (a) and (b) is positive, and the other is zero. Explain why. 5.157 A forester studying diseased pine trees models the number of diseased trees per acre, Y ,asa Poisson random variable with mean λ. However, λ changes from area to area, and its random behavior is modeled by a gamma distribution. That is, for some integer α, ⎧ ⎪⎨ 1 f (λ) = Ɣ(α)β α λα−1 e −λ/β , λ > 0, ⎪⎩ 0, elsewhere. Find the unconditional probability distribution for Y . 5.158 A coin has probability p of coming up heads when tossed. In n independent tosses of the coin, let X i = 1iftheith toss results in heads and X i = 0iftheith toss results in tails. Then Y , the number of heads in the n tosses, has a binomial distribution and can be represented as Y = ∑ n i=1 X i. Find E(Y ) and V (Y ), using Theorem 5.12. *5.159 The negative binomial random variable Y was defined in Section 3.6 as the number of the trial on which the rth success occurs, in a sequence of independent trials with constant probability p of success on each trial. Let X i denote a random variable defined as the number of the trial on which the ith success occurs, for i = 1, 2,...,r. Now define W i = X i − X i−1 , i = 1, 2,...,r, where X 0 is defined to be zero. Then we can write Y = ∑ r i=1 W i. Notice that the random variables W 1 , W 2 ,...,W r have identical geometric distributions and are mutually independent. Use Theorem 5.12 to show that E(Y ) = r/p and V (Y ) = r(1 − p)/p 2 . 5.160 A box contains four balls, numbered 1 through 4. One ball is selected at random from this box. Let X 1 = 1ifball1orball2isdrawn, X 2 = 1ifball1orball3isdrawn, X 3 = 1 if ball 1 or ball 4 is drawn. The X i values are zero otherwise. Show that any two of the random variables X 1 , X 2 , and X 3 are independent but that the three together are not. 5.161 Suppose that we are to observe two independent random samples: Y 1 , Y 2 ,...,Y n denoting a random sample from a normal distribution with mean µ 1 and variance σ 2 1 ; and X 1, X 2 ,...,X m denoting a random sample from another normal distribution with mean µ 2 and variance σ 2 2 . An approximation for µ 1 − µ 2 is given by Y − X, the difference between the sample means. Find E(Y − X) and V (Y − X).

www.downloadslide.com 294 Chapter 5 Multivariate Probability Distributions 5.162 In Exercise 5.65, you determined 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, and is such that the marginal distributions of Y 1 and Y 2 are both exponential with mean 1. You also showed that Y 1 and Y 2 are independent if and only if α = 0. Give two specific and different joint densities that yield marginal densities for Y 1 and Y 2 that are both exponential with mean 1. *5.163 Refer to Exercise 5.66. If F 1 (y 1 ) and F 2 (y 2 ) are two distribution functions then for any α, −1 ≤ α ≤ 1, 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 )}] is a joint distribution function such that Y 1 and Y 2 have marginal distribution functions F 1 (y 1 ) and F 2 (y 2 ), respectively. a b c d If F 1 (y 1 ) and F 2 (y 2 ) are both distribution functions associated with exponentially distributed random variables with mean 1, show that the joint density function of Y 1 and Y 2 is the one given in Exercise 5.162. If F 1 (y 1 ) and F 2 (y 2 ) are both distribution functions associated with uniform (0, 1) random variables, for any α, −1 ≤ α ≤ 1, evaluate F(y 1 , y 2 ). Find the joint density functions associated with the distribution functions that you found in part (b). Give two specific and different joint densities such that the marginal distributions of Y 1 and Y 2 are both uniform on the interval (0, 1). *5.164 Let X 1 , X 2 , and X 3 be random variables, either continuous or discrete. The joint momentgenerating function of X 1 , X 2 , and X 3 is defined by m(t 1 , t 2 , t 3 ) = E(e t 1 X 1 +t 2 X 2 +t 3 X 3 ). a Show that m(t, t, t) gives the moment-generating function of X 1 + X 2 + X 3 . b Show that m(t, t, 0) gives the moment-generating function of X 1 + X 2 . c Show that ] ∂ k 1+k 2 +k 3 m(t 1 , t 2 , t 3 ) ∂t k 1 1 ∂t k 2 2 ∂t k 3 3 t 1 =t 2 =t 3 =0 ( ) = E X k 1 1 X k 2 2 X k 3 3 . *5.165 Let X 1 , X 2 , and X 3 have a multinomial distribution with probability function n! n∑ p(x 1 , x 2 , x 3 ) = x 1 !x 2 !x 3 ! px 1 1 px 2 2 px 3 3 , x i = n. i=1 Use the results of Exercise 5.164 to do the following: a Find the joint moment-generating function of X 1 , X 2 , and X 3 . b Use the answer to part (a) to show that the marginal distribution of X 1 is binomial with parameter p 1 . c Use the joint moment-generating function to find Cov(X 1 , X 2 ). *5.166 A box contains N 1 white balls, N 2 black balls, and N 3 red balls (N 1 + N 2 + N 3 = N). A random sample of n balls is selected from the box (without replacement). Let Y 1 , Y 2 , and Y 3

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