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 {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

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.