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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 Moment-generating function ： ウィキペディア英語版 Moment-generating function In probability theory and statistics, the moment-generating function of a random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. Note, however, that not all random variables have moment-generating functions. In addition to univariate distributions, moment-generating functions can be defined for vector- or matrix-valued random variables, and can even be extended to more general cases. The moment-generating function does not always exist even for real-valued arguments, unlike the characteristic function. There are relations between the behavior of the moment-generating function of a distribution and properties of the distribution, such as the existence of moments. ==Definition== In probability theory and statistics, the moment-generating function of a random variable ''X'' is :$M\_X(t)\; :=\; \backslash mathbb\backslash !\backslash left(),\; \backslash quad\; t\; \backslash in\; \backslash mathbb,$ wherever this expectation exists. $M\_X(0)$ always exists and is equal to 1. A key problem with moment-generating functions is that moments and the moment-generating function may not exist, as the integrals need not converge absolutely. By contrast, the characteristic function always exists (because it is the integral of a bounded function on a space of finite measure), and thus may be used instead. More generally, where $\backslash mathbf\; X\; =\; (\; X\_1,\; \backslash ldots,\; X\_n)$T, an ''n''-dimensional random vector, one uses $\backslash mathbf\; t\; \backslash cdot\; \backslash mathbf\; X\; =\; \backslash mathbf\; t^\backslash mathrm\; T\backslash mathbf\; X$ instead of ''tX'': :$M\_(\backslash mathbf\; t)\; :=\; \backslash mathbb\backslash !\backslash left(e^\backslash right).$ The reason for defining this function is that it can be used to find all the moments of the distribution.〔Bulmer, M.G., Principles of Statistics, Dover, 1979, pp. 75–79〕 The series expansion of ''e''''tX'' is: : $$ e^ = 1 + t\,X + \frac + \frac + \cdots +\frac + \cdots. Hence: : $$ \begin M_X(t) = \mathbb(e^) &= 1 + t \,\mathbb(X) + \frac + \frac+\cdots + \frac+\cdots \\ & = 1 + tm_1 + \frac + \frac+\cdots + \frac+\cdots, \end where ''m''''n'' is the ''n''th moment. Differentiating ''M''''X''(t) ''i'' times with respect to ''t'' and setting ''t'' = 0 we hence obtain the ''i''th moment about the origin, ''m''''i'', see Calculations of moments below. 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「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.