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:

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.