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

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

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

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

(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

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

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

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

See also

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