Factorial moment generating function

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

$M_X(t)=\mathrm{E}\left[t^X\right]$

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 $|t|\le 1$.

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

$\mathrm{E}[(X{)}_n]=M_X^{(n)}(1)={\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}|}_{t=1}{M}_X(t),$

where the Pochhammer symbol (x)_n is the falling factorial

$(x{)}_n=x(x-1)(x-2)\cdots (x-n+1).$

(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

$M_X(t)=\sum _{k=0}^{\mathrm{\infty }}{t}^k\underset{={\lambda }^k{e}^{-\lambda }/k!}{\underbrace{\mathrm{P}(X=k)}}=e^{-\lambda }\sum _{k=0}^{\mathrm{\infty }}\frac{(t\lambda {)}^k}{k!}=e^{\lambda (t-1)},t\in \mathbb{ℂ},$

(use the definition of the exponential function) and thus we have

$\mathrm{E}[(X{)}_n]={\lambda }^n.$

See also

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