The factorial moment-generating function of a discrete random variable X is given by
![](https://test.transtutors.com/qimg/c0dc4e2e-9dd8-48c8-9258-6391cff8745a.png)
Show that the rth derivative of FX(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
![](https://test.transtutors.com/qimg/131431a3-0373-4524-a088-43368facf390.png)
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
![](https://test.transtutors.com/qimg/7a89cbf3-1bea-4029-9b15-a9b33969a47e.png)