Factorial moment generating function

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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 {0,1, ...} 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 [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.)

Example[edit]

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[edit]