Solution Found!
Let X represent the lifetime of a component, in weeks. Let
Chapter 4, Problem 27SE(choose chapter or problem)
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.