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).

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,