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Law of Iterated Expectations

It is also know as LIE. X is a random variable. Y is another variable. We have:

\operatorname {E} (X)=\operatorname {E} (\operatorname {E} (X\mid Y))

In \operatorname {E} (X\mid Y), Y is a specific value. In \operatorname {E} (\operatorname {E} (X\mid Y)), we need to enumerate all the possible value of Y and multiply \operatorname {E} (X\mid Y) with the corresponding probability of that Y and sum together. Therefore, they are equal.