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 $|t|\leq 1$ .

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 ^[1]

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

(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 ^{k}e^{{-\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}.