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)