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

<math>E( (X)_n )</math>

where

<math>(x)_n=x(x-1)(x-2)\cdots(x-n+1)</math>

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

<math>E( (X)_n )=\lambda^n.</math>

See also

• moment (mathematics)
• cumulant
• Factorial moment generating function