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 |t|=1, see characteristic function. If X is a discrete random variable taking values only in the set {0,1, ...} of non-negative integers, then M_X is also called probability-generating function (PGF) of X and M_X(t) is well-defined at least for all t on the closed unit disk |t|\le1. The factorial moment generating function generates the factorial moments of the probability distribution. Provided M_X 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