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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Factorial moment generating function ： ウィキペディア英語版 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 :$M\_X(t)=\backslash operatorname\backslash bigl()$ 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 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 $|t|\backslash le1$. The factorial moment generating function generates the factorial moments of the probability distribution. Provided $M\_X$ exists in a neighbourhood of ''t'' = 1, the ''n''th factorial moment is given by 〔http://homepages.nyu.edu/~bpn207/Teaching/2005/Stat/Generating_Functions.pdf〕 :$\backslash operatorname()=M\_X^(1)=\backslash left.\backslash frac\backslash right|\_\; M\_X(t),$ where the Pochhammer symbol (''x'')''n'' is the falling factorial :$(x)\_n\; =\; x(x-1)(x-2)\backslash cdots(x-n+1).\backslash ,$ (Many 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_^\infty t^k\underbrace_ =e^\sum_^\infty \frac = e^,\qquad t\in\mathbb, (use the definition of the exponential function) and thus we have :$\backslash operatorname()=\backslash lambda^n.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Factorial moment generating function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.