The factorial moment-generating function of a discrete random variable X is given by

Show that the rth derivative of F_X(t) with respect to t at t = 1 is μ'_(r), the rth factorial moment defined in Exercise 11.

In the proof of Theorem 2 we determined the quantity E[X(X - 1)], called the second factorial moment. In general, the rth factorial moment of X is given by

Express μ'₂, μ'₃, and μ'₄ in terms of factorial moments. 1

Theorem 2

The mean and the variance of the binomial distribution are