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# Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com 5.9 The Multinomial Probability Distribution 281 classes. Then we wish to find n! p(y 1 , y 2 , y 3 , y 4 , y 5 ) = y 1 ! y 2 ! y 3 ! y 4 ! y 5 ! py 1 1 py 2 2 py 3 3 py 4 4 py 5 5 , for n = 5 and y 1 = 1, y 2 = 2, y 3 = 0, y 4 = 2, and y 5 = 0. Substituting these values into the formula for the joint probability function, we obtain p(1, 2, 0, 2, 0) = 5! 1! 2! 0! 2! 0! (.18)1 (.23) 2 (.16) 0 (.27) 2 (.16) 0 = 30(.18)(.23) 2 (.27) 2 = .0208. THEOREM 5.13 If Y 1 , Y 2 ,...,Y k have a multinomial distribution with parameters n and p 1 , p 2 ,...,p k , then 1. E(Y i ) = np i , V (Y i ) = np i q i . 2. Cov(Y s , Y t ) =−np s p t , ifs ≠ t. Proof The marginal distribution of Y i can be used to derive the mean and variance. Recall that Y i may be interpreted as the number of trials falling into cell i. Imagine all of the cells, excluding cell i, combined into a single large cell. Then every trial will result in cell i or in a cell other than cell i, with probabilities p i and 1 − p i , respectively. Thus, Y i possesses a binomial marginal probability distribution. Consequently, E(Y i ) = np i and V (Y i ) = np i q i , where q i = 1 − p i . The same results can be obtained by setting up the expectations and evaluating. For example, E(Y 1 ) = ∑ ∑ n! ···∑ y 1 y 1 y 2 y k y 1 !y 2 ! ···y k ! py 1 1 py 2 2 ···py k k . Because we have already derived the expected value and variance of Y i ,we leave the summation of this expectation to the interested reader. The proof of part 2 uses Theorem 5.12. Think of the multinomial experiment as a sequence of n independent trials and define, for s ≠ t, { 1, if trial i results in class s, U i = 0, otherwise, and { 1, if trial i results in class t, W i = 0, otherwise. Then n∑ n∑ Y s = U i and Y t = W j . i=1 j=1

www.downloadslide.com 282 Chapter 5 Multivariate Probability Distributions (Because U i = 1 or 0 depending upon whether the ith trial resulted in class s, Y s is simply the sum of a series of 0s and 1s. A 1 occurs in the sum everytime we observe an item from class s, and a 0 occurs everytime we observe any other class. Thus, Y s is simply the number of times class s is observed. A similar interpretation applies to Y t .) Notice that U i and W i cannot both equal 1 (the ith item cannot simultaneously be in classes s and t). Thus, the product U i W i always equals zero, and E(U i W i ) = 0. The following results allow us to evaluate Cov(Y s , Y t ): E(U i ) = p s E(W j ) = p t Cov(U i , W j ) = 0, if i ≠ j because the trials are independent Cov(U i , W i ) = E(U i W i ) − E(U i )E(W i ) = 0 − p s p t From Theorem 5.12, we then have n∑ n∑ Cov(Y s , Y t ) = Cov(U i , W j ) = = i=1 j=1 n∑ Cov(U i , W i ) + ∑∑ Cov(U i , W j ) i=1 i ≠ j n∑ (−p s p t ) + ∑∑ 0 =−np s p t . i=1 The covariance here is negative, which is to be expected because a large number of outcomes in cell s would force the number in cell t to be small. Inferential problems associated with the multinomial experiment will be discussed later. i ≠ j Exercises 5.119 A learning experiment requires a rat to run a maze (a network of pathways) until it locates one of three possible exits. Exit 1 presents a reward of food, but exits 2 and 3 do not. (If the rat eventually selects exit 1 almost every time, learning may have taken place.) Let Y i denote the number of times exit i is chosen in successive runnings. For the following, assume that the rat chooses an exit at random on each run. a Find the probability that n = 6 runs result in Y 1 = 3, Y 2 = 1, and Y 3 = 2. b For general n, find E(Y 1 ) and V (Y 1 ). c Find Cov(Y 2 , Y 3 ) for general n. d To check for the rat’s preference between exits 2 and 3, we may look at Y 2 − Y 3 . Find E(Y 2 − Y 3 ) and V (Y 2 − Y 3 ) for general n. 5.120 A sample of size n is selected from a large lot of items in which a proportion p 1 contains exactly one defect and a proportion p 2 contains more than one defect (with p 1 + p 2 < 1). The cost of repairing the defective items in the sample is C = Y 1 + 3Y 2 , where Y 1 denotes the number of

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