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.
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“Every moment instructs, and every object: for wisdom is infused into every form. It has been poured into us as blood; it convulsed us as pain; it slid into us as pleasure; it enveloped us in dull, melancholy days, or in days of cheerful labor; we did not guess its essence, until after a long time.”
—Ralph Waldo Emerson (1803–1882)