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Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 4 - Problem 27se
Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 4 - Problem 27se

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

ISBN: 9780073401331 38

## Solution for problem 27SE Chapter 4

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:

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

##### 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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