×
Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 2.6 - Problem 21e
Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 2.6 - Problem 21e

×

# The lifetime of a certain component, in years, has

ISBN: 9780073401331 38

## Solution for problem 21E Chapter 2.6

Statistics for Engineers and Scientists | 4th Edition

• Textbook Solutions
• 2901 Step-by-step solutions solved by professors and subject experts
• Get 24/7 help from StudySoup virtual teaching assistants

Statistics for Engineers and Scientists | 4th Edition

4 5 1 354 Reviews
17
4
Problem 21E

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

Step-by-Step Solution:

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.

Step 2 of 6

Step 3 of 6

#### Related chapters

Unlock Textbook Solution