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In probability theory, the conditional expectation of a random variable is another random variable equal to the average of the former over each possible "condition". In the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space. This definition is then generalized to any probability space using measure theory. Conditional expectation is also known as conditional expected value or conditional mean. In modern probability theory the concept of conditional probability is defined in terms of conditional expectation. == Concept == The concept of conditional expectation can be nicely illustrated through the following example. Suppose we have daily rainfall data (mm of rain each day) collected by a weather station on every day of the ten year period from Jan 1, 1990 to Dec 31, 1999. The conditional expectation of daily rainfall knowing the month of the year is the average of daily rainfall over all days of the ten year period that fall in a given month. These data then may be considered either as a function of each day (so for example its value for Mar 3, 1992, would be the sum of daily rainfalls on all days that are in the month of March during the ten years, divided by the number of these days, which is 310) or as a function of just the month (so for example the value for March would be equal to the value of the previous example). It is important to note the following. * The conditional expectation of daily rainfall knowing that we are in a month of March of the given ten years is not a monthly rainfall data, that is it is not the average of the ten monthly total March rainfalls. That number would be 31 times higher. * The average daily rainfall in March 1992 is not equal to the conditional expectation of daily rainfall knowing that we are in a month of March of the given ten years, because we have restricted ourselves to 1992, that is we have more conditions than just that of being in March. This shows that reasoning as "we are in March 1992, so I know we are in March, so the average daily rainfall is the March average daily rainfall" is incorrect. Stated differently, although we use the expression "conditional expectation knowing that we are in March" this really means "conditional expectation knowing nothing other than that we are in March". 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「Conditional expectation」の詳細全文を読む スポンサード リンク
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