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Let X represent the lifetime of a component, in weeks. Let

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 27SE Chapter 4

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 27SE

Let X represent the lifetime of a component, in weeks. Let Y represent the lifetime of the component in days, so \(Y=7 X\). Suppose \(X \sim \operatorname{Exp}(\lambda)\)

a. Let \(F_{Y}\) be the cumulative distribution function of Y and let \(F_{X}\) be the cumulative distribution function of X.Show that \(F_{Y}(y)=1-e^{-\lambda y / 7}\). [Hint:\(F_{Y}(y)=P(Y \leq y)=P(7 X \leq y)=P(X \leq y / 7)\)

b. Show that \(Y \sim \operatorname{Exp}(\lambda / 7)\).[Hint  Find the probability density function of \(Y\) by differentiating \(F_{Y}(y)\).]

Equation Transcription:

Text Transcription:

Y = 7X

X  Tilde Exp(lambda)

F_Y

F_X

F_Y (y) = 1-e^-lambda y/7

F_Y (y) = P(Y less than or equal to y) = P(X less than or equal to y/7)

Y Tilde Exp(lambda/7)

F_Y(y)

Step-by-Step Solution:

Answer:

Step 1 of 2:

(a)

In this question, we are asked to prove the following equation.

Where represent the lifetime of the component in days and .

 represent the lifetime of a component in weeks.

Suppose

 is a parameter.

 be the cumulative distribution function of

be the cumulative distribution function of

Since  is a random variable whose distribution is exponential with parameter , we can therefore express the CDF as:

 =        

                                  =   0                    

Given

Therefore

 =

 =

 =  …………..(2)

We know the CDF of  

 = ……….(3)

Compare equation (2) and (3) and substitute the value accordingly.

 =

 =

Hence  and is equal when

 =

Substituein above equation

We have

  =

Hence proved.

Step 2 of 2

Chapter 4, Problem 27SE is Solved
Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

Statistics for Engineers and Scientists was written by and is associated to the ISBN: 9780073401331. The full step-by-step solution to problem: 27SE from chapter: 4 was answered by , our top Statistics solution expert on 06/28/17, 11:15AM. Since the solution to 27SE from 4 chapter was answered, more than 347 students have viewed the full step-by-step answer. The answer to “?Let X represent the lifetime of a component, in weeks. Let Y represent the lifetime of the component in days, so \(Y=7 X\). Suppose \(X \sim \operatorname{Exp}(\lambda)\)a. Let \(F_{Y}\) be the cumulative distribution function of Y and let \(F_{X}\) be the cumulative distribution function of X.Show that \(F_{Y}(y)=1-e^{-\lambda y / 7}\). [Hint:\(F_{Y}(y)=P(Y \leq y)=P(7 X \leq y)=P(X \leq y / 7)\)b. Show that \(Y \sim \operatorname{Exp}(\lambda / 7)\).[Hint Find the probability density function of \(Y\) by differentiating \(F_{Y}(y)\).]Equation Transcription:Text Transcription: Y = 7XX Tilde Exp(lambda)F_YF_XF_Y (y) = 1-e^-lambda y/7F_Y (y) = P(Y less than or equal to y) = P(X less than or equal to y/7)Y Tilde Exp(lambda/7)F_Y(y)” is broken down into a number of easy to follow steps, and 108 words. This full solution covers the following key subjects: let, function, component, cumulative, differentiating. This expansive textbook survival guide covers 153 chapters, and 2440 solutions. This textbook survival guide was created for the textbook: Statistics for Engineers and Scientists , edition: 4.

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Let X represent the lifetime of a component, in weeks. Let