Factorial moment generating function

In probability theory and statistics, the factorial moment generating function of a random variable X is

${\displaystyle M_{X}(t)=E\left(t^{X}\right),\quad t\in \mathbb {R} ,}$

wherever this expectation exists. The factorial moment generating function generates the factorial moments of the probability distribution.

Provided the factorial moment generating function exists in an interval around t = 1, the nth moment is given by

${\displaystyle E\left((x)_{n}\right)=M_{X}^{(n)}(1)=\left.{\frac {\mathrm {d} ^{n}}{\mathrm {d} t^{n}}}\right|_{t=1}M_{X}(t).}$

Example, Suppose X has a Poisson distribution with expected value λ, then the factorial moment generating function of X is

${\displaystyle M_{X}(t)=\sum _{0}^{\infty }{\frac {\lambda ^{x}e^{-\lambda }}{x!}}\,}$

${\displaystyle =e^{-\lambda (1-t)}}$

and thus we have

${\displaystyle E((X)_{n})=\lambda ^{n}.}$