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Get Full Access to Mathematical Statistics With Applications - 7 Edition - Chapter 4 - Problem 137e
Get Full Access to Mathematical Statistics With Applications - 7 Edition - Chapter 4 - Problem 137e

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# Show that the result given in Exercise 3.158 also holds ISBN: 9780495110811 47

## Solution for problem 137E Chapter 4

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 .

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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

Step 3 of 3

##### ISBN: 9780495110811

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