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a. Show that b(x; n, 1 - p) = b(n - x; n, p). b. Show that

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 63E 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 63E

a.? ?Show that b(x; n, 1 - p) = b(n - x; n, p). b. ?Show that B(x; n, 1 – p) = 1 – B ( n – x – 1; n, p). [Hint: At most x S’s is equivalent to at least ( n – x) F’s.] c. ?What do parts (a) and (b) imply about the necessity of including values of p greater than .5 in Appendix Table A.1?

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Problem 63E Answer: Step1: We need to show that, a. Show that b(x; n, 1 - p) = b(n - x; n, p). b. Show that B(x; n, 1 – p) = 1 – B ( n – x – 1; n, p). [Hint: At most x S’s is equivalent to at least ( n – x) F’s.] c. What do parts (a) and (b) imply about the necessity of including values of p greater than .5 in Appendix Table A.1 Step2: The probability mass function of binomial distribution is given by p(x) = C p q n-, x = 0,1,2,...,n x With mean E(x) = np and Var(x) = np(1 - p) a). 1).Consider, n x nx b(x;n, 1 p)= ( )(1x p) [1 (1 p)] . = n! (1 p) (p) nx x!(nx)! = n! (p)nx (1 p) x x!(nx)! Hence, b(x;n, 1 p) = n! (p)nx (1 p) x ……..(1) x!(nx)! 2).Again consider, n nx n(nx) b(n x;n,p) = ( nx)(p) (1 p) n! nx x = (nx)![n(nx)]!) (1 p) n! nx x = (nx)!x!p) (1 p) n! nx x Hence, b(n x;n,p)= (nx)!x!p) (1 p) ……..(2) Therefore, From equation (1) and (2) we have b(x;n, 1 p) = b(n x;n,p) n! nx x n! nx x x!(nx)!p) (1 p) = (nx)!x!p) (1 p) Hence the proof Step3: b). We have random variable X and it follows binomial distribution with parameter n and (1 - p) i.e, X ~B(n, 1- p) Let us consider the other random variable Y and it follows binomial distribution with parameter n and p. i.e, Y ~B(n, p) b(x;n, 1 p)= P(Xx) = 1 - P(X > x) = 1 - P(n-Y > x) = 1 - P(Y < n - x) = 1 - P(Y n - x -1) = 1 - B(n - x - 1;n, p) Therefore, b(x;n, 1 p) = [1 - B(n - x - 1;n, p)] Hence the proof. c). The above part(a) and part(b) imply that in the binomial probability tables in the back of the textbook, it is not necessary to publish the value for the success probability p > 0.5 because we can find the needed probabilities using the failure probability (1-p) which will be less than 0.5 whenever p > 0.5.

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Chapter 3, Problem 63E is Solved
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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 full step-by-step solution to problem: 63E from chapter: 3 was answered by , our top Statistics solution expert on 05/06/17, 06:21PM. This textbook survival guide was created for the textbook: Probability and Statistics for Engineers and the Scientists, edition: 9. The answer to “a.? ?Show that b(x; n, 1 - p) = b(n - x; n, p). b. ?Show that B(x; n, 1 – p) = 1 – B ( n – x – 1; n, p). [Hint: At most x S’s is equivalent to at least ( n – x) F’s.] c. ?What do parts (a) and (b) imply about the necessity of including values of p greater than .5 in Appendix Table A.1?” is broken down into a number of easy to follow steps, and 72 words. Since the solution to 63E from 3 chapter was answered, more than 464 students have viewed the full step-by-step answer. This full solution covers the following key subjects: show, least, Greater, hint, imply. This expansive textbook survival guide covers 18 chapters, and 1582 solutions.

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a. Show that b(x; n, 1 - p) = b(n - x; n, p). b. Show that