Factorial Moment

In probability theory, the nth factorial moment of a probability distribution, also called the nth factorial moment of any random variable X with that probability distribution, is

where

is the falling factorial. (Confusingly, this same notation, the Pochhammer symbol (x)_n, is used by some mathematicians, especially in the theory of special functions, to denote the rising factorial x(x + 1)(x + 2) ... (x + n − 1); the present notation is used by combinatorialists.)

For example, if X has a Poisson distribution with expected value λ, then the nth factorial moment of X is

One context in which factorial moments occur naturally is at an initial stage in the use of probability-generating functions to derive the moments of discrete distributions.