Show that the result given in Exercise 3.158 also holds for continuous random variables

Chapter 4, Problem 4.137

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QUESTION:

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 and variance 2, use the moment-generating function of U to derive the mean and variance of U.

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QUESTION:

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 and variance 2, use the moment-generating function of U to derive the mean and variance of U.

ANSWER:

Step 1 of 5

Given that,

If Y is a random variable with a moment-generating function  and U is given by .

 

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