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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 Law of the unconscious statistician ： ウィキペディア英語版 Law of the unconscious statistician In probability theory and statistics, the law of the unconscious statistician (sometimes abbreviated LOTUS) is a theorem used to calculate the expected value of a function ''g''(''X'') of a random variable ''X'' when one knows the probability distribution of ''X'' but one does not explicitly know the distribution of ''g''(''X''). The form of the law can depend on the form in which one states the probability distribution of the random variable ''X''. If it is a discrete distribution and one knows its probability mass function ''ƒX'' (but not ''ƒg(X)''), then the expected value of ''g''(''X'') is : $\backslash operatorname()\; =\; \backslash sum\_x\; g(x)\; f\_X(x),\; \backslash ,$ where the sum is over all possible values ''x'' of ''X''. If it is a continuous distribution and one knows its probability density function ''ƒX'' (but not ''ƒg(X)''), then the expected value of ''g''(''X'') is : $\backslash operatorname()\; =\; \backslash int\_^\backslash infty\; g(x)\; f\_X(x)\; \backslash ,\; dx$ (provided the values of ''X'' are real numbers as opposed to vectors, complex numbers, etc.). Regardless of continuity-versus-discreteness and related issues, if one knows the cumulative probability distribution function ''FX'' (but not ''Fg(X)''), then the expected value of ''g''(''X'') is given by a Riemann–Stieltjes integral : $\backslash operatorname()\; =\; \backslash int\_^\backslash infty\; g(x)\; \backslash ,\; dF\_X(x)$ (again assuming ''X'' is real-valued).〔Eric Key (1998) ( Lecture 6: Random variables ), Lecture notes, University of Leeds〕〔Bengt Ringner (2009) ("Law of the unconscious statistician" ), unpublished note, Centre for Mathematical Sciences, Lund University〕 However, the result is so well known that it is usually used without stating a name for it: the name is not extensively used. For justifications of the result for discrete and continuous random variables see.〔(Virtual Laboratories in Probability and Statistics ), Sect. 3.1 "Expected Value: Definition and Properties", item "Basic Results: Change of Variables Theorem".〕 ==From the perspective of measure== A technically complete derivation of the result is available using arguments in measure theory, in which the probability space of a transformed random variable ''g''(''X'') is related to that of the original random variable ''X''. The steps here involve defining a pushforward measure for the transformed space, and the result is then an example of a change of variables formula.〔S.R.Srinivasa Varadhan (2002) (Lecture notes on Limit Theorems ), NYU (Section 1.4)〕 : $\backslash int\_\backslash Omega\; g\; \backslash circ\; X\; dP\; =\; \backslash int\_\backslash mathbb\; g\; d(X\_$ * P) We say $X$ has a density if $d(X\_$ * P) is absolutely continuous with respect to the Lebesgue measure $\backslash mu$. In that case : $d(X\_$ *P) = f d_\mu where $f\; :\; \backslash mathbb\; \backslash to\; \backslash mathbb$ is the density (see Radon-Nikodym derivative). So the above can be rewritten as the more familiar : $\backslash operatorname()\; =\; \backslash int\_\backslash Omega\; g\; \backslash circ\; X\; dP\; =\; \backslash int\_\backslash mathbb\; g(x)\; f(x)\; dx$ 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「Law of the unconscious statistician」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.