Factorial moment generating function: Difference between revisions

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In [[probability theory]] and [[statistics]], the '''factorial moment generating function''' of the [[probability distribution]] of a [[real number|real-valued]] [[random variable]] ''X'' is defined as

:<math>M_X(t)=\operatorname{E}\bigl[t^{X}\bigr], \quad t \in \mathbb{R},</math>

for all [[complex number]]s ''t'' for which this [[expected value]] exists. This is the case at least for all ''t'' on the [[unit circle]] <math>|t|=1</math>, see [[characteristic function (probability theory)|characteristic function]]. If&nbsp;''X'' is a discrete random variable taking values only in the set {0,1, ...} of non-negative [[integer|integers]], then <math>M_X</math> is also called [[probability-generating function]] of ''X'' and <math>M_X(t)</math> is well-defined at least for all ''t'' on the [[closed set|closed]] [[unit disk]] <math>|t|\le1</math>.

:<math>M_X(t)=\operatorname{E}\bigl[t^{X}\bigr], \quad t \in \mathbb{R},</math>

wherever this expectation exists. The factorial moment generating function generates the [[factorial moment]]s of the [[probability distribution]].

Provided the factorial moment generating function exists in an interval around ''t''&nbsp;=&nbsp;1, the ''n''th factorial moment is given by

wherever this expectation exists. The factorial moment generating function generates the [[factorial moment]]s of the [[probability distribution]].

Provided the factorial moment generating function<math>M_X</math> exists in ana [[neighbourhood interval(mathematics)|neighbourhood]] aroundof ''t''&nbsp;=&nbsp;1, the ''n''th factorial moment is given by

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

where the [[Pochhammer symbol]] (''x'')_''n'' is the [[falling factorial]]

:<math>(x)_n = x(x-1)(x-2)\cdots(x-n+1).\,</math>

(Confusingly, some mathematicians, especially in the field of [[special function]]s, use the same notation to represent the [[rising factorial]].)

==Example==

Suppose ''X'' has a [[Poisson distribution]] with [[expected value]] &lambda;, then its factorial moment generating function is

:<math>M_X(t)

:<math>M_X(t) = \sum_{k=0}^\infty \frac{(t\lambda)^k e^\underbrace{-\lambdaoperatorname{P}(X=k)}_{k!} = e\,\lambda^ke^{-\lambda(1-t)},\qquad t\in\mathbb{R/k!},</math>

=e^{-\lambda}\sum_{k=0}^\infty \frac{(t\lambda)^k}{k!} = e^{\lambda(t-1)},\qquad t\in\mathbb{C},

</math>

(use the [[Exponential_function#Formal_definition|definition of the exponential function]]) and thus we have

:<math>\operatorname{E}[(X)_n]=\lambda^n.</math>

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* [[Moment-generating function]]

* [[Cumulant-generating function]]

[[Category:Probability theory]]