Moment-Generating Function

Given a random variable  and a probability density function , if there exists an  such that

(1)

for , where  denotes the expectation value of , then  is called the moment-generating function.

For a continuous distribution,

			(2)
			(3)
			(4)

where  is the th raw moment.

For independent  and , the moment-generating function satisfies

			(5)
			(6)
			(7)
			(8)

If  is differentiable at zero, then the th moments about the origin are given by 

			(9)
			(10)
			(11)
			(12)

The mean and variance are therefore

			(13)
			(14)
			(15)
			(16)

It is also true that

(17)

where  and  is the th raw moment.

It is sometimes simpler to work with the logarithm of the moment-generating function, which is also called the cumulant-generating function, and is defined by

			(18)
			(19)
			(20)

But , so

			(21)
			(22)

REFERENCES:

Kenney, J. F. and Keeping, E. S. "Moment-Generating and Characteristic Functions," "Some Examples of Moment-Generating Functions," and "Uniqueness Theorem for Characteristic Functions." §4.6-4.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 72-77, 1951.