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^[1]

$\operatorname{E}\bigl[(X)_n\bigr]$

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 − 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

$\operatorname{E}\bigl[(X)_n\bigr] =\lambda^n.$

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.

See also

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

References

1. ^ http://www.staff.city.ac.uk/~ra359/X3MathFinance/Newby2006/mathematical%20modelling%20for%20finance%5Bpart%201%5D.pdf