With reference to Exercise 12, find the factorial moment-generating function of
(a) the Bernoulli distribution and show that μ'(1) = θ and μ'(r) = 0 for r > 1;
(b) the binomial distribution and use it to find μ and σ2.
Exercise 12
The factorial moment-generating function of a discrete random variable X is given by
![](https://test.transtutors.com/qimg/b1bc29ae-91c1-41d7-b4e2-21fb103b6daf.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/554add76-2b64-405f-91b5-6bca9c31f74f.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/3370ae5d-388f-45b4-a9e8-fc5f43a57205.png)