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If X is a hypergeometric rv, show directly from the

Probability and Statistics for Engineers and the Scientists | 9th Edition | ISBN: 9780321629111 | Authors: Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying E. Ye ISBN: 9780321629111 32

Solution for problem 118E Chapter 3

Probability and Statistics for Engineers and the Scientists | 9th Edition

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Probability and Statistics for Engineers and the Scientists | 9th Edition | ISBN: 9780321629111 | Authors: Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying E. Ye

Probability and Statistics for Engineers and the Scientists | 9th Edition

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Problem 118E

If X is a hypergeometric rv, show directly from the definition that E(X) = nM/N ( consider only the case n < M). [Hint: Factor nM/N out of the sum for E(X), and show that the terms inside the sum are of the form h( y; n – 1, M – 1, N -1), where y – x -1.]

Step-by-Step Solution:

Answer : Step 1 : Given If X is a hypergeometric rv. Now we have to show that directly from the definition that E(X) = N Here X:hypergeometric row variable. P(X = x)=h(x;n,m,N) Proposition : If X is the number of S’s in a completely random sample of size n drawn from a population consisting of M S’s and (N-M) F’s , then the probability distribution of X, called the hypergeometric distribution, is given by M NM (x)( nx ) P(X = x) = h(x; n, M, N) = N (n) X= the number of successes from a sample size of n, from a population of size N, with M successes in it X~hypergeom( n, M, N ). Where n= number of successes among the population M= sample size N= total population size The expected value is n E(X) = P(X = x) x, x integer x=0 We know that. (x)( nx ) P(X = x) = ( n n M NM (x )(nx ) E(X) = N x x=0 ( n n E(X) = M ! (NM) ! n!(Nn) !x x=1(x!) (Mx)! (nx)! (N+Mn)(N)! n NM nM (M1)! (nx ) (n1)(N1(n1)! E(X) = N (x1)! (M1(x1)! (nx)! (N+Mn)! (N1)! x=1 n (M1)( nx ) E(X) = nM x1 N x=1 (N1) Let y = x-1 Then, n1 M1 NM nM ( y )( n1y E(X) = N (N1) y=0 nM n1 E(X) = N h(y;n 1,M 1,N 1) y=0 nM E(X) = N nM HenceE(X) = N

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Chapter 3, Problem 118E is Solved
Textbook: Probability and Statistics for Engineers and the Scientists
Edition: 9
Author: Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying E. Ye
ISBN: 9780321629111

Probability and Statistics for Engineers and the Scientists was written by and is associated to the ISBN: 9780321629111. The answer to “If X is a hypergeometric rv, show directly from the definition that E(X) = nM/N ( consider only the case n < M). [Hint: Factor nM/N out of the sum for E(X), and show that the terms inside the sum are of the form h( y; n – 1, M – 1, N -1), where y – x -1.]” is broken down into a number of easy to follow steps, and 59 words. This textbook survival guide was created for the textbook: Probability and Statistics for Engineers and the Scientists, edition: 9. Since the solution to 118E from 3 chapter was answered, more than 360 students have viewed the full step-by-step answer. This full solution covers the following key subjects: show, sum, hint, directly, factor. This expansive textbook survival guide covers 18 chapters, and 1582 solutions. The full step-by-step solution to problem: 118E from chapter: 3 was answered by , our top Statistics solution expert on 05/06/17, 06:21PM.

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