The lifetime of a certain component, in years, has probability density function

\(f(x)= \begin{cases}e^{-x} & x>0 \\ 0 & x \leq 0\end{cases}\)

Two such components, whose lifetimes are independent, are available. As soon as the first component fails, it is replaced with the second component. Let denote the lifetime of the first component, and let denote the lifetime of the second component.

a. Find the joint probability density function of and .

b. Find \(P(X \leq 1 \text { and } Y>1)\).

c. Find \(\mu_{X}\).

d. Find \(\mu_{X+Y}\).

e. Find \(P(X+Y \leq 2)\). (Hint: Sketch the region of the plane where \(x+y\leq2\), and then integrate the joint probability density function over that region.)

Equation Transcription:

Text Transcription:

f(x)={_0 x{</=}0 ^e^-x x>0

P(X{</=}1 and Y>1)

mu_X

mu_X+Y

P(X+Y2)

x+y{</=}2

Answer :

Step 1 of 6:

Given,

The probability density function of the lifetime of a certain component, in years.

We have two components, whose lifetimes are independent. As soon as the first component fails, it replaces with second component.

Let, X = lifetime of the first component. And Y = lifetime of the second component.