# 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

[itex]E( (X)_n )[itex]

where

[itex](x)_n=x(x-1)(x-2)\cdots(x-n+1)[itex]

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

[itex]E( (X)_n )=\lambda^n.[itex]

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