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.
If X is a discrete random variable taking values only in the set {0,1, ...} of non-negative integers, then
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 [1] 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.)
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