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

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

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

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

[edit] Iterated expectations with nested conditioning sets

The following formulation of the law of iterated expectations plays an important role in many macroeconomic and finance models:

$\operatorname{E} (X \mid I1) = \operatorname{E} ( \operatorname{E} ( X \mid I2) \mid I1),$

where I1 is a subset of I2. To build intuition, imagine an investor who forecasts a random stock price (X) based on the limited information set I1. The law of iterated expectations says that the investor can never gain a more precise forecast of X by conditioning on more specific information (I2), if the more specific forecast must itself be forecast with the original information (I1).

This formulation is often applied in a time series context, where E_t denotes expectations with respect to information observed through time period t. In typical models the information set t+1 contains all information available through time t, plus additional information revealed at time t+1. One can then write:

$\operatorname{E_t} (X) = \operatorname{E_t} ( \operatorname{E_{t+1}} ( X )),$