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Use the definition in Expression (3.13) to prove that V(aX

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 41E 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 41E

Use the definition in Expression (3.13) to prove that V(aX + b) = ?2. ?2x [ Hint: With h(X) = aX + b, E[h(X)] = aµ = b where µ = E(X).] Reference Expression (3.13 The variance of h(X) is the expected value of the squared difference between h(X) and its expected value: When h(X) = aX + b, a linear function, Substituting this into (3.13) gives a simple relationship between V[h(X)] and V(X):

Step-by-Step Solution:
Step 1 of 3

Problem 41E Answer: Step1: We have the variance of h(X) is the expected value of the squared difference between h(X) and its expected value: V [h(x)]= 2 = {h(x) E[h(x)]} • P(x) ……….(1) h(x) x When h(X) = aX + b, a linear function, h(x) E[h(x)] = ax + b (a + b) = a(x ) Substituting this into (1) gives a simple relationship between V[h(X)] and V(X): We need to prove V (aX + b) = a • 2 x Step2: Let us assume that Y = aX + b W.k.t, V [h(x)]= 2 = {h(x) E[h(x)]} • P(x) h(x) x Now, V ar(Y ) = 2 = {y E[y]} • P(Y = y) h(x) y 2 = {y } y P(Y = y) y Substitute y = aX + b in above equation we get 2 V ar(Y ) = {(ax + b) ax+b} • P(X = x) x = {(ax + b) a + b}x• P(X = x) x = (ax a ) • P(X = x) x x 2 2 = {a (x ) }x• P(X = x) x 2 2 = a {(x ) } x P(X = x) x = a V ar(x) [from equ (1)] From the given information we have V [h(x)]= 2 h(x) 2 Hence, V ar(x) = x Therefore, V ar(Y ) = a V ar(x) = a • x V ar(Y ) = a • 2 x Hence the proof.

Step 2 of 3

Chapter 3, Problem 41E is Solved
Step 3 of 3

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. This full solution covers the following key subjects: Expected, Expression, prove, difference, function. This expansive textbook survival guide covers 18 chapters, and 1582 solutions. Since the solution to 41E from 3 chapter was answered, more than 451 students have viewed the full step-by-step answer. The answer to “Use the definition in Expression (3.13) to prove that V(aX + b) = ?2. ?2x [ Hint: With h(X) = aX + b, E[h(X)] = aµ = b where µ = E(X).] Reference Expression (3.13 The variance of h(X) is the expected value of the squared difference between h(X) and its expected value: When h(X) = aX + b, a linear function, Substituting this into (3.13) gives a simple relationship between V[h(X)] and V(X):” is broken down into a number of easy to follow steps, and 74 words. The full step-by-step solution to problem: 41E 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.

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Use the definition in Expression (3.13) to prove that V(aX