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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 expectation ： ウィキペディア英語版 Law of total expectation The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, the smoothing theorem, and Adam's Law among other names, states that if ''X'' is an integrable random variable (i.e., a random variable satisfying E( | ''X'' | ) < ∞) and ''Y'' is any random variable, not necessarily integrable, on the same probability space, then :$\backslash operatorname\; (X)\; =\; \backslash operatorname\_Y\; (\; \backslash operatorname\_\; (\; X\; \backslash mid\; Y)),$ i.e., the expected value of the conditional expected value of ''X'' given ''Y'' is the same as the expected value of ''X''. The conditional expected value E( ''X'' | ''Y'' ) is a random variable in its own right, whose value depends on the value of ''Y''. Notice that the conditional expected value of ''X'' given the ''event'' ''Y'' = ''y'' is a function of ''y'' (this is where adherence to the conventional, rigidly case-sensitive notation of probability theory becomes important!). If we write E( ''X'' | ''Y'' = ''y'') = ''g''(''y'') then the random variable E( ''X'' | ''Y'' ) is just ''g''(''Y''). One special case 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 :$\backslash operatorname\; (X)\; =\; \backslash sum\_^(A\_i)\}.$ ==Example== Suppose that two factories supply light bulbs to the market. Factory ''X'''s bulbs work for an average of 5000 hours, whereas factory ''Y'''s bulbs work for an average of 4000 hours. It is known that factory ''X'' supplies 60% of the total bulbs available. What is the expected length of time that a purchased bulb will work for? Applying the law of total expectation, we have: $\backslash operatorname\; (L)\; =\; \backslash operatorname(L\; \backslash mid\; X)\; \backslash operatorname(X)+\backslash operatorname(L\; \backslash mid\; Y)\; \backslash operatorname(Y)\; =\; 5000(.6)+4000(.4)=4600$ where * $\backslash operatorname\; (L)$ is the expected life of the bulb; * $\backslash Pr(X)=$ is the probability that the purchased bulb was manufactured by factory ''X''; * $\backslash Pr(Y)=$ is the probability that the purchased bulb was manufactured by factory ''Y''; * $\backslash operatorname(L\; \backslash mid\; X)=5000$ is the expected lifetime of a bulb manufactured by ''X''; * $\backslash operatorname(L\; \backslash mid\; Y)=4000$ is the expected lifetime of a bulb manufactured by ''Y''. Thus each purchased light bulb has an expected lifetime of 4600 hours. 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 expectation」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.