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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 : 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 is a partition of the whole outcome space, i.e. these events are mutually exclusive and exhaustive, then : ==Example== Suppose that two factories supply light bulbs to the market. Factory ''X'' Applying the law of total expectation, we have: where * is the expected life of the bulb; * is the probability that the purchased bulb was manufactured by factory ''X''; * is the probability that the purchased bulb was manufactured by factory ''Y''; * is the expected lifetime of a bulb manufactured by ''X''; * 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
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