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


 * $$E( (X)_n )$$

where


 * $$(x)_n=x(x-1)(x-2)\cdots(x-n+1)$$

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 &minus; 1); the present notation is used by combinatorialists).

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


 * $$E( (X)_n )=\lambda^n.$$