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# Use the result in Exercise 3.158 to prove that, if W = aY

ISBN: 9780495110811 47

## Solution for problem 159E Chapter 3

Mathematical Statistics with Applications | 7th Edition

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Problem 159E

Problem 159E

Use the result in Exercise 3.158 to prove that, if W = aY + b, then E(W ) = aE(Y ) + b and V (W ) = a2 V (Y ).

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:

Solution :

Step 1 of 1:

Let Y denotes a random variable with moment-generating function m(t).

Our goal is:

If  W=aY+b. We need to prove that E(W)=aE(Y)+b and V(W)=.

Now we have to prove that E(W)=aE(Y)+b and V(W)=.

If W=aY+b we know that the moment generating function.

The moment generating function is .

The expected value is the first derivative of the moment generating function evaluated at t=0.

E(W)=

E(W)=

E(W)=

E(W)=

E(W)=

E(W)=

E(W)=

The variance is the second derivatives of the moment generating function evaluated at t=0.

V(W)=

V(W)=

V(W)=

V(W)=

Therefore, E(W)=and V(W)=.

Step 2 of 1

##### ISBN: 9780495110811

This textbook survival guide was created for the textbook: Mathematical Statistics with Applications , edition: 7. The answer to “Use the result in Exercise 3.158 to prove that, if W = aY + b, then E(W ) = aE(Y ) + b and V (W ) = a2 V (Y ).ReferenceIf 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).” is broken down into a number of easy to follow steps, and 61 words. The full step-by-step solution to problem: 159E from chapter: 3 was answered by , our top Statistics solution expert on 07/18/17, 08:07AM. Mathematical Statistics with Applications was written by and is associated to the ISBN: 9780495110811. This full solution covers the following key subjects: function, moment, generating, random, exercise. This expansive textbook survival guide covers 32 chapters, and 3350 solutions. Since the solution to 159E from 3 chapter was answered, more than 1205 students have viewed the full step-by-step answer.

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Use the result in Exercise 3.158 to prove that, if W = aY