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Lede: removed an unnecessary statement: the previous paragraph applies to any probability spaces *already*, so there is no need to generalize anything |
The two equalities are false if the distribution of X is not uniform |
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=== Conditional expectation with respect to an event ===
Let X be a random variable with a uniform distribution. In [[Classical definition of probability|classical probability theory]] the '''conditional expectation''' of <math>X</math> given an event <math>H</math> (which may be the event <math>Y=y</math> for a random variable <math>Y</math>) is the average of <math>X</math> over all outcomes in <math>H</math>, that is
:<math>\operatorname{E} (X \mid H ) = \frac{\sum_{\omega \in H} X(\omega)}{|H|},</math>
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