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 ) = E [ t X ] {\displaystyle 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 {\displaystyle |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 {\displaystyle M_{X}} is also called probability-generating function (PGF) of X and M X ( t ) {\displaystyle M_{X}(t)} is well-defined at least for all t on the closed unit disk | t | ≤ 1 {\displaystyle |t|\leq 1} .
The factorial moment generating function generates the factorial moments of the probability distribution. Provided M X {\displaystyle M_{X}} exists in a neighbourhood of t = 1, the nth factorial moment is given by
E [ ( X ) n ] = M X ( n ) ( 1 ) = d n d t n | t = 1 M X ( t ) , {\displaystyle \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 ) ⋯ ( x − n + 1 ) . {\displaystyle (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 ) = ∑ k = 0 ∞ t k P ( X = k ) ⏟ = λ k e − λ / k ! = e − λ ∑ k = 0 ∞ ( t λ ) k k ! = e λ ( t − 1 ) , t ∈ C , {\displaystyle 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
E [ ( X ) n ] = λ n . {\displaystyle \operatorname {E} [(X)_{n}]=\lambda ^{n}.}See also
References
Néri, Breno de Andrade Pinheiro (2005-05-23). "Generating Functions" (PDF). nyu.edu. Archived from the original (PDF) on 2012-03-31. https://web.archive.org/web/20120331042031/https://files.nyu.edu/bpn207/public/Teaching/2005/Stat/Generating_Functions.pdf ↩