In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable X is defined as

for all complex numbers t for which this expected value exists. This is the case at least for all t on the unit circle, see characteristic function. IfX is a discrete random variable taking values only in the set {0,1, ...} of non-negative integers is also called probability-generating function (PGF) of X is well-defined at least for all t on the closed unit disk.

The factorial moment generating function generates the factorial moments of the probability distribution exists in a neighbourhood of t=1, the nth factorial moment is given by

where the Pochhammer symbol (x)n is the falling factorial

(Many mathematicians, especially in the field of special functions, use the same notation to represent the rising factorial.)

Examples

Poisson distribution

Suppose X has a Poisson distribution with expected value λ, then its factorial moment generating function is

(use the definition of the exponential function) and thus we have

See also

• Moment (mathematics)

• Moment-generating function

• Cumulant-generating function

References