Factorial moment
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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
- E((X)n)
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
- E((X)n) = λn.
