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

M_X(t)=\operatorname{E}\bigl[t^{X}\bigr]

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 nth factorial moment is given by ^[1]

\operatorname{E}[(X)_n]=M_X^{(n)}(1)=\left.\frac{\mathrm{d}^n}{\mathrm{d}t^n}\right|_{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.)

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^k\underbrace{\operatorname{P}(X=k)}_{=\,\lambda^ke^{-\lambda}/k!} =e^{-\lambda}\sum_{k=0}^\infty \frac{(t\lambda)^k}{k!} = e^{\lambda(t-1)},\qquad t\in\mathbb{C},

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