Factorial moment - Citizendia

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) . Probability theory is the branch of Mathematics concerned with analysis of random phenomena In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way In Mathematics, the Pochhammer symbol (x_{n}\ introduced by Leo August Pochhammer, represents either the rising or the falling In Mathematics, the Pochhammer symbol (x_{n}\ introduced by Leo August Pochhammer, represents either the rising or the falling Special functions are particular mathematical functions which have more or less established names and notations due to their importance for the Mathematical analysis In Mathematics, the Pochhammer symbol (x_{n}\ introduced by Leo August Pochhammer, represents either the rising or the falling . . (x + n − 1); the present notation is used by combinatorialists). Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects

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

E ((X) n) = λ n . In Probability theory and Statistics, the Poisson distribution is a Discrete probability distribution that expresses the probability of a number of events

See also