With reference to exercise 12 find the factorial

Question

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

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