Reference no: EM131129214
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 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
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
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