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www.downloadslide.com Exercises 289 of θ, the joint distribution of Y and θ is obtained by multiplying the conditional by the marginal. The marginal distribution of Y is then obtained from the joint distribution by integrating or summing over the possible values of θ. The results of this section can be used to find E(Y ) and V (Y ) **with**out finding this marginal distribution. Other examples of hierarchical models are contained in Exercises 5.136, 5.138, 5.141 and 5.142. Exercises 5.133 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. a Find E(Y 1 |Y 2 = y 2 ). b Use the answer derived in part (a) to find E(Y 1 ). (Compare this **with** the answer found in Exercise 5.77.) 5.134 In Examples 5.32 and 5.33, we determined that if Y is the number of defectives, E(Y ) = 1.25 and V (Y ) = 1.5625. Is it likely that, on any given day, Y will exceed 6? 5.135 In Exercise 5.41, we considered a quality control plan that calls for randomly selecting three items from the daily production (assumed large) of a certain machine and observing the number of defectives. The proportion p of defectives produced by the machine varies from day to day and has a uniform distribution on the interval (0, 1). Find the a b expected number of defectives observed among the three sampled items. variance of the number of defectives among the three sampled. 5.136 In Exercise 5.42, the number of defects per yard in a certain fabric, Y , was known to have a Poisson distribution **with** parameter λ. The parameter λ was assumed to be a random variable **with** a density function given by { e −λ , λ ≥ 0, f (λ) = 0, elsewhere. a Find the expected number of defects per yard by first finding the conditional expectation of Y for given λ. b Find the variance of Y . c Is it likely that Y exceeds 9? 5.137 In Exercise 5.38, we assumed that Y 1 , the weight of a bulk item stocked by a supplier, had a uniform distribution over the interval (0, 1). The random variable Y 2 denoted the weight of the item sold and was assumed to have a uniform distribution over the interval (0, y 1 ), where y 1 was a specific value of Y 1 . If the supplier stocked 3/4 ton, what amount could be expected to be sold during the week? 5.138 Assume that Y denotes the number of bacteria per cubic centimeter in a particular liquid and that Y has a Poisson distribution **with** parameter λ. Further assume that λ varies from location to location and has a gamma distribution **with** parameters α and β, where α is a positive integer. If we randomly select a location, what is the

www.downloadslide.com 290 Chapter 5 Multivariate Probability Distributions a b expected number of bacteria per cubic centimeter? standard deviation of the number of bacteria per cubic centimeter? 5.139 Suppose that a company has determined that the the number of jobs per week, N, varies from week to week and has a Poisson distribution **with** mean λ. The number of hours to complete each job, Y i , is gamma distributed **with** parameters α and β. The total time to complete all jobs in a week is T = ∑ N i=1 Y i. Note that T is the sum of a random number of random variables. What is a b E(T | N = n)? E(T ), the expected total time to complete all jobs? 5.140 Why is E[V (Y 1 |Y 2 )] ≤ V (Y 1 )? 5.141 Let Y 1 have an exponential distribution **with** mean λ and the conditional density of Y 2 given Y 1 = y 1 be { 1/y1 , 0 ≤ y 2 ≤ y 1 , f (y 2 | y 1 ) = 0, elsewhere. Find E(Y 2 ) and V (Y 2 ), the unconditional mean and variance of Y 2 . 5.142 Suppose that Y has a binomial distribution **with** parameters n and p but that p varies from day to day according to a beta distribution **with** parameters α and β. Show that a E(Y ) = nα/(α + β). nαβ(α + β + n) b V (Y ) = (α + β) 2 (α + β + 1) . *5.143 If Y 1 and Y 2 are independent random variables, each having a normal distribution **with** mean 0 and variance 1, find the moment-generating function of U = Y 1 Y 2 . Use this moment-generating function to find E(U) and V (U). Check the result by evaluating E(U) and V (U) directly from the density functions for Y 1 and Y 2 . 5.12 Summary The multinomial experiment (Section 5.9) and its associated multinomial probability distribution convey the theme of this chapter. Most experiments yield sample measurements, y 1 , y 2 ,...,y k , which may be regarded as observations on k random variables. Inferences about the underlying structure that generates the observations— the probabilities of falling into cells 1, 2,...,k—are based on knowledge of the probabilities associated **with** various samples (y 1 , y 2 ,...,y k ). Joint, marginal, and conditional distributions are essential concepts in finding the probabilities of various sample outcomes. Generally we draw from a population a sample of n observations, which are specific values of Y 1 , Y 2 ,...,Y n . Many times the random variables are independent and have the same probability distribution. As a consequence, the concept of independence is useful in finding the probability of observing the given sample. The objective of this chapter has been to convey the ideas contained in the two preceding paragraphs. The numerous details contained in the chapter are essential in providing a solid background for a study of inference. At the same time, you should be careful to avoid overemphasis on details; be sure to keep the broader inferential objectives in mind.

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