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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 variance ： ウィキペディア英語版 Law of total variance In probability theory, the law of total variance〔Neil A. Weiss, ''A Course in Probability'', Addison–Wesley, 2005, pages 385–386.〕 or variance decomposition formula, also known as Eve's law, states that if ''X'' and ''Y'' are random variables on the same probability space, and the variance of ''Y'' is finite, then :$\backslash operatorname()=\backslash operatorname\_X(\backslash operatorname(X))+\backslash operatorname\_X(\backslash operatorname(X)).\backslash ,$ Some writers on probability call this the "conditional variance formula". In language perhaps better known to statisticians than to probabilists, the two terms are the "unexplained" and the "explained" components of the variance respecively (cf. fraction of variance unexplained, explained variation). In actuarial science, specifically credibility theory, the first component is called the expected value of the process variance (EVPV) and the second is called the variance of the hypothetical means (VHM). There is a general variance decomposition formula for ''c'' ≥ 2 components (see below).〔Bowsher, C.G. and P.S. Swain, Proc Natl Acad Sci USA, 2012: 109, E1320–29.〕 For example, with two conditioning random variables: :$$ \begin \operatorname() & = & \operatorname(\operatorname(X_1 , X_2 )) + \operatorname(\operatorname) \\ & & + \operatorname(\operatorname(X_1 )),\, \end which follows from the law of total conditional variance:〔 :$\backslash operatorname(X\_1)=\backslash operatorname(\backslash operatorname(X\_1,X\_2)\backslash mid\; X\_1)+\backslash operatorname(\backslash operatorname(X\_1,X\_2)\backslash mid\; X\_1).\backslash ,$ Note that the conditional expected value is a random variable in its own right, whose value depends on the value of ''X''. Notice that the conditional expected value of ''Y'' given the ''event'' ''X'' = ''x'' is a function of ''x'' (this is where adherence to the conventional and rigidly case-sensitive notation of probability theory becomes important!). If we write E( ''Y'' | ''X'' = ''x'' ) = ''g''(''x'') then the random variable is just ''g''(''X''). Similar comments apply to the conditional variance. One special case, (similar to the Law of total expectation) states that if $A\_1,\; A\_2,\; \backslash ldots,\; A\_n$ is a partition of the whole outcome space, i.e. these events are mutually exclusive and exhaustive, then ^(A_i)} |- | | $+\; \backslash sum\_^n\; (A\_i))\backslash operatorname(A\_i)\}$ |- | | $-\; 2\backslash sum\_^\backslash sum\_^\backslash operatorname(X\; \backslash mid\; A\_i)\; \backslash operatorname(A\_i)\backslash operatorname(X\; \backslash mid\; A\_j)\; \backslash operatorname(A\_j).$ |} In this formula, the first component is the expectation of the conditional variance; the other two rows are the variance of the conditional expectation. ==Proof== The law of total variance can be proved using the law of total expectation.〔Neil A. Weiss, ''A Course in Probability'', Addison–Wesley, 2005, pages 380–383.〕 First, :$\backslash operatorname()\; =\; \backslash operatorname()\; -\; \backslash right]\; -\; ^2$ Now we rewrite the conditional second moment of Y in terms of its variance and first moment: ::$=\; \backslash operatorname\; \backslash left]^2\backslash right]\; -\; ^2$ Since the expectation of a sum is the sum of expectations, the terms can now be regrouped: ::$=\; \backslash operatorname\; -\; ^2\backslash right)$ Finally, we recognize the terms in parentheses as the variance of the conditional expectation E(): ::$=\; \backslash operatorname\; ]$ 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 variance」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.