Law of total expectation

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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, or perhaps by any of a variety of 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

E(X) = E( E( X\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 nomenclature used here parallels the phrase law of total probability. See also law of total variance.

(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). )

[edit] Proof in the discrete case

\begin{align} \operatorname{E} \left( \operatorname{E}(X|Y) \right) &{} = \sum\limits_y \operatorname{E}(X|Y=y) \cdot \operatorname{P}(Y=y) \\ &{}=\sum\limits_y \left( \sum\limits_x x \cdot \operatorname{P}(X=x|Y=y) \right) \cdot \operatorname{P}(Y=y) \\ &{}=\sum\limits_y \sum\limits_x x \cdot \operatorname{P}(X=x|Y=y) \cdot \operatorname{P}(Y=y) \\ &{}=\sum\limits_y \sum\limits_x x \cdot \operatorname{P}(Y=y|X=x) \cdot \operatorname{P}(X=x) \\ &{}=\sum\limits_x x \cdot \operatorname{P}(X=x) \cdot \left( \sum\limits_y \operatorname{P}(Y=y|X=x) \right) \\ &{}=\sum\limits_x x \cdot \operatorname{P}(X=x) \\ &{}=\operatorname{E}(X). \end{align}
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