Factorial moment generating function


In probability theory and statistics, the factorial moment generating function 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. If X is a discrete random variable taking values only in the set of non-negative integers, then is also called probability-generating function of X and 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.
Provided exists in a neighbourhood of t = 1, the nth factorial moment is given by
where the Pochhammer symbol n is the falling factorial

Example

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