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

Chapter 4, Problem 27SE

(choose chapter or problem)

Get Unlimited Answers
QUESTION:

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)

Questions & Answers

QUESTION:

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)

ANSWER:

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.

Add to cart


Study Tools You Might Need

Not The Solution You Need? Search for Your Answer Here:

×

Login

Login or Sign up for access to all of our study tools and educational content!

Forgot password?
Register Now

×

Register

Sign up for access to all content on our site!

Or login if you already have an account

×

Reset password

If you have an active account we’ll send you an e-mail for password recovery

Or login if you have your password back