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

${\displaystyle M_{X}(t)=\operatorname {E} {\bigl [}t^{X}{\bigr ]}}${\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 ${\displaystyle |t|=1}${\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 ${\displaystyle M_{X}}${\displaystyle M_{X}} is also called probability-generating function (PGF) of X and ${\displaystyle M_{X}(t)}${\displaystyle M_{X}(t)} is well-defined at least for all t on the closed unit disk ${\displaystyle |t|\leq 1}${\displaystyle |t|\leq 1}.

The factorial moment generating function generates the factorial moments of the probability distribution. Provided ${\displaystyle M_{X}}${\displaystyle M_{X}} exists in a neighbourhood of t = 1, the nth factorial moment is given by ^[1]

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

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

(Many mathematicians, especially in the field of special functions, use the same notation to represent the rising factorial.)

Suppose X has a Poisson distribution with expected value λ, then its factorial moment generating function is

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

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

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

• Factorial and binomial topics
• Moments (mathematics)
• Generating functions

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