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The lifetimes, in months, of two components in a system,

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 20E Chapter 2.6

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 20E

Problem 20E

The lifetimes, in months, of two components in a system, denoted X and Y, have joint probability density function

a. What is the probability that both components last longer than one month?

b. Find the marginal probability density functions fX(0) and fY(y).

c. Are X and Y independent? Explain.

Step-by-Step Solution:
Step 1 of 3

Solution 20E

Step1 of 4:

Let us consider a random variables X and Y they presents the lifetimes, in months, of two components in a system and they have joint probability density function.

 

Here our goal is:

a).We need to find the probability that both components last longer than one month?

b).We need to find the marginal probability density functions

c).We need to check whether X and Y independent? Explain.


Step2 of 4:

a).

                                     

                                       

First integrate above equation with respect to “y” we get

                                       

Apply integration by parts we get

                                             

                                             

Apply integral substitution and let u = -2x-y

                                                      du = -1dy

                                   

                                    )

Substitute “u” value in above equation we get

                                         

                                   

                                   

Now,

Integrate above equation with respect x we get 

                                   

                                   

Apply integral by parts we get

                                         

                                   

Let us consider u = -2x-1

                        du = -2

                                           

                                     

                                     

Substitute “u” value in above equation we get

                                           

                                     = {}

                                     = 0 - (-)

                                     =

Hence, = .


Step3 of 4:

b).

The marginal probability density function is given by:

                               

                                     

Apply integral substitution and let u = -2x-y

                                                      du = -1dy

                                 

                                 )

Substitute “u” value in above equation we get

                                     

                               

                               

                               

                               

Therefore, marginal probability density function of is given below

Similarly,

The marginal probability density function is given by:

                               

Apply integration by parts we get

                               

Let u = -2x-y

      du = -2du

                               

                               

                                

Substitute “u” value in above equation we get

 

                                   

 =  

 Therefore, marginal probability density function of is given below


Step4 of 4:

c).

Consider,

=

Where, = 4xy                      [given]

           =  and                    [from part (b)]

           =                            [from part (b)]

Now,

  =

Step 2 of 3

Chapter 2.6, Problem 20E is Solved
Step 3 of 3

Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

The full step-by-step solution to problem: 20E from chapter: 2.6 was answered by , our top Statistics solution expert on 06/28/17, 11:15AM. Statistics for Engineers and Scientists was written by and is associated to the ISBN: 9780073401331. This textbook survival guide was created for the textbook: Statistics for Engineers and Scientists , edition: 4. The answer to “The lifetimes, in months, of two components in a system, denoted X and Y, have joint probability density function a. What is the probability that both components last longer than one month?________________b. Find the marginal probability density functions fX(0) and fY(y).________________c. Are X and Y independent? Explain.” is broken down into a number of easy to follow steps, and 47 words. Since the solution to 20E from 2.6 chapter was answered, more than 1364 students have viewed the full step-by-step answer. This full solution covers the following key subjects: Probability, components, Density, independent, explain. This expansive textbook survival guide covers 153 chapters, and 2440 solutions.

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