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 $|t|=1$, 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 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 "n"th factorial moment is given by:$operatorname\{E\}\; [(X)\_n]\; =M\_X^\{(n)\}(1)=left.frac\{mathrm\{d\}^n\}\{mathrm\{d\}t^n\}\; ight|\_\{t=1\}\; M\_X(t),$where the Pochhammer symbol ("x")_"n" is the falling factorial:$(x)\_n\; =\; x(x-1)(x-2)cdots(x-n+1).,$ (Confusingly, some 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:$M\_X(t)=sum\_\{k=0\}^infty\; t^kunderbrace\{operatorname\{P\}(X=k)\}\_\{=,lambda^ke^\{-lambda\}/k!\}=e^\{-lambda\}sum\_\{k=0\}^infty\; frac\{(tlambda)^k\}\{k!\}\; =\; e^\{lambda(t-1)\},qquad\; tinmathbb\{C\},$(use the definition of the exponential function) and thus we have:$operatorname\{E\}\; [(X)\_n]\; =lambda^n.$

ee also

* Moment (mathematics)
* Moment-generating function
* Cumulant-generating function