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| | 58 years old Secondary College Teacher Dorothy from Baie-d'Urfe, likes to spend some time amateur radio, [http://ganhardinheironainternet.comoganhardinheiro101.com como ganhar dinheiro] na internet and operating on cars. Has travelled since childhood and has visited many destinations, like Hanseatic Town of Visby. |
| 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
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| :<math>M_X(t)=\operatorname{E}\bigl[t^{X}\bigr]</math>
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| 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 ''X'' is a discrete random variable taking values only in the set {0,1, ...} of non-negative [[integer]]s, 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>.
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| The factorial moment generating function generates the [[factorial moment]]s of the [[probability distribution]].
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| Provided <math>M_X</math> exists in a [[neighbourhood (mathematics)|neighbourhood]] of ''t'' = 1, the ''n''th factorial moment is given by <ref>http://homepages.nyu.edu/~bpn207/Teaching/2005/Stat/Generating_Functions.pdf</ref>
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| :<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>
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| where the [[Pochhammer symbol]] (''x'')<sub>''n''</sub> is the [[falling factorial]]
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| :<math>(x)_n = x(x-1)(x-2)\cdots(x-n+1).\,</math>
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| (Confusingly, some mathematicians, especially in the field of [[special function]]s, use the same notation to represent the [[rising factorial]].)
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| ==Example==
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| Suppose ''X'' has a [[Poisson distribution]] with [[expected value]] λ, then its factorial moment generating function is
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| :<math>M_X(t)
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| =\sum_{k=0}^\infty t^k\underbrace{\operatorname{P}(X=k)}_{=\,\lambda^ke^{-\lambda}/k!}
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| =e^{-\lambda}\sum_{k=0}^\infty \frac{(t\lambda)^k}{k!} = e^{\lambda(t-1)},\qquad t\in\mathbb{C},
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| </math>
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| (use the [[Exponential_function#Formal_definition|definition of the exponential function]]) and thus we have
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| :<math>\operatorname{E}[(X)_n]=\lambda^n.</math>
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| ==See also==
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| * [[Moment (mathematics)]]
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| * [[Moment-generating function]]
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| * [[Cumulant-generating function]]
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| {{Reflist}}
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| {{DEFAULTSORT:Factorial Moment Generating Function}}
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| [[Category:Factorial and binomial topics]]
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| [[Category:Theory of probability distributions]]
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| [[Category:Generating functions]]
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58 years old Secondary College Teacher Dorothy from Baie-d'Urfe, likes to spend some time amateur radio, como ganhar dinheiro na internet and operating on cars. Has travelled since childhood and has visited many destinations, like Hanseatic Town of Visby.