## Moment-Generating Function

Given a Random Variable , if there exists an such that

 (1)

for , then
 (2)

is the moment-generating function.
 (3)

where is the th Moment about zero. The moment-generating function satisfies
 (4)

If is differentiable at zero, then the th Moments about the Origin are given by
 (5)

 (6)

 (7)

 (8)

The Mean and Variance are therefore
 (9) (10)

It is also true that
 (11)

where and is the th moment about the origin.

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

 (12) (13) (14)

But , so
 (15) (16)

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.