Problem 150E

Refer to Exercise 3.147. Use the uniqueness of moment-generating functions to give the distribution

of a random variable with moment-generating function

Reference

If Y has a geometric distribution with probability of success p, show that the moment-generating function for Y is

Solution:

Step 1 of 2:

Let Y has a Geometric distribution with probability of success p, and q be the probability of failure

The moment generating function is

m(t) = , where q = 1 - p

We have, m(t) = - (1)

The claim is to give the distribution