Moment-Generating Function

The moment-generating function of a real random variable is defined by

(1)

where denotes the expectation value of . Here the moment-generating function is said to exist when for with some . This definition applies to arbitrary probability distributions, including discrete distributions. Existence of a probability density function is not required.

If has a probability density function , then for ,

			(2)
			(3)
			(4)

where is the th raw moment. The power series can be integrated term by term because its absolute sum has finite expectation value under the stated hypothesis.

For independent random variables and , the moment-generating function of their sum satisfies

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

Finiteness on implies that is infinitely differentiable there and that every absolute moment of positive integer order is finite. Differentiation under the expectation value gives the following identities for and positive integers (Durrett 2019, pp. 106-107):

			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)

The mean and variance are therefore

			(17)
			(18)
			(19)
			(20)

It is also true that

(21)

where is the th central moment, , 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

			(22)
			(23)
			(24)

But , so

			(25)
			(26)