Refer to Exercises 4.141 and 4.137. Suppose that Y is

QUESTION:

Problem 142E

Refer to Exercises 4.141 and 4.137. Suppose that Y is uniformly distributed on the interval (0, 1) and that a > 0 is a constant.

a Give the moment-generating function for Y .

b Derive the moment-generating function of W = aY. What is the distribution of W? Why?

c Derive the moment-generating function of X = −aY. What is the distribution of X? Why?

d If b is a fixed constant, derive the moment-generating function of V = aY + b. What is the distribution of V? Why?

Reference

4.141 If θ1 < θ2, derive the moment-generating function of a random variable that has a uniform distribution on the interval (θ1, θ2).

4.137 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 σ2, use the moment-generating function of U to derive the mean and variance of U .