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Show that the result given in Exercise 3.158 also holds

Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer ISBN: 9780495110811 47

Solution for problem 137E Chapter 4

Mathematical Statistics with Applications | 7th Edition

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Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer

Mathematical Statistics with Applications | 7th Edition

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Problem 137E

Problem 137E

Show that the result given in Exercise 3.158 also holds for continuous random variables. That is, show that, if Y is a random variable with moment-generating function m(t) and U is given by U = aY + b, the moment-generating function of U is etbm(at). If Y has mean μ and variance σ 2, use the moment-generating function of U to derive the mean and variance of U .

Reference

If Y is a random variable with moment-generating function m(t) and if W is given by W = aY + b, show that the moment-generating function of W is etbm(at).

Step-by-Step Solution:
Step 1 of 3

Solution 137E

Step1 of 2:

Let us consider a random variable Y it has mean and variance . If Y is a random variable with moment-generating function m(t).

Also we have U = aY + b.

We need to find the moment generating function of U and also we need to derive mean and variance of U.


Step2 of 3:

Let,

Substitute U value in above equation we get,

               

               

             

Therefore moment generating function of U is .


Step3 of 3:

Consider,

             

                   

                               

                       

     

Hence, .

Now,

         

                   

                   

                               

                   

                   

Hence, .

Therefore, the variance of U is:

                                   

       

                                               

Hence,


Step 2 of 3

Chapter 4, Problem 137E is Solved
Step 3 of 3

Textbook: Mathematical Statistics with Applications
Edition: 7
Author: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer
ISBN: 9780495110811

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Show that the result given in Exercise 3.158 also holds