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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 total covariance ： ウィキペディア英語版 Law of total covariance In probability theory, the law of total covariance,〔Matthew R. Rudary, ''On Predictive Linear Gaussian Models'', ProQuest, 2009, page 121.〕 covariance decomposition formula, or ECCE states that if ''X'', ''Y'', and ''Z'' are random variables on the same probability space, and the covariance of ''X'' and ''Y'' is finite, then :$\backslash operatorname(X,Y)=\backslash operatorname(\backslash operatorname(X,Y\; \backslash mid\; Z))+\backslash operatorname(\backslash operatorname(X\backslash mid\; Z),\backslash operatorname(Y\backslash mid\; Z)).\backslash ,$ The nomenclature in this article's title parallels the phrase ''law of total variance''. Some writers on probability call this the "conditional covariance formula"〔Sheldon M. Ross, ''A First Course in Probability'', sixth edition, Prentice Hall, 2002, page 392.〕 or use other names. (The conditional expected values E( ''X'' | ''Z'' ) and E( ''Y'' | ''Z'' ) are random variables in their own right, whose values depends on the value of ''Z''. Notice that the conditional expected value of ''X'' given the ''event'' ''Z'' = ''z'' is a function of ''z'' (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!). If we write E( ''X'' | ''Z'' = ''z'') = ''g''(''z'') then the random variable E( ''X'' | ''Z'' ) is just ''g''(''Z''). Similar comments apply to the conditional covariance.) ==Proof== The law of total covariance can be proved using the law of total expectation: First, :$\backslash operatorname()\; =\; \backslash operatorname()\; -\; \backslash operatorname()\backslash operatorname()$ from the definition of covariance. Then we apply the law of total expectation by conditioning on the random variable ''Z'': ::$=\; \backslash operatorname]\backslash operatorname\; +\; \backslash operatorname(Z)\backslash operatorname(Z)\backslash right]\; -\; \backslash operatorname]$ Since expectation of a sum is the sum of expectations, we can regroup the terms: ::$=\; \backslash operatorname\backslash !\backslash left\backslash operatorname(Z)\backslash right]\; -\; \backslash operatorname]$ Finally, we recognize the final two terms as the covariance of the conditional expectations E() and E(): ::$=\; \backslash operatorname(\backslash operatorname(X,Y\; \backslash mid\; Z))+\backslash operatorname(\backslash operatorname(X\backslash mid\; Z),\backslash operatorname(Y\backslash mid\; Z))$ 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 total covariance」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.