Law of total expectation

The proposition in probability theory known as the law of total expectation , the law of iterated expectations, Adam's law, the tower rule, the smoothing theorem, among 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

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

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).

Proof in the discrete case

${\begin{aligned}\operatorname {E} _{y}\left(\operatorname {E} (X|Y)\right)&{}=\operatorname {E} _{y}{\Bigg [}\sum _{x}x\cdot \operatorname {P} (X=x|Y){\Bigg ]}\\[6pt]&{}=\sum _{y}{\Bigg [}\sum _{x}x\cdot \operatorname {P} (X=x|Y=y){\Bigg ]}\cdot \operatorname {P} (Y=y)\\[6pt]&{}=\sum _{y}\sum _{x}x\cdot \operatorname {P} (X=x|Y=y)\cdot \operatorname {P} (Y=y)\\[6pt]&{}=\sum _{x}x\sum _{y}\operatorname {P} (X=x|Y=y)\cdot \operatorname {P} (Y=y)\\[6pt]&{}=\sum _{x}x\sum _{y}\operatorname {P} (X=x,Y=y)\\[6pt]&{}=\sum _{x}x\cdot \operatorname {P} (X=x)\\[6pt]&{}=\operatorname {E} (X).\end{aligned}}$

Iterated expectations with nested conditioning sets

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

$\operatorname {E} (X\mid I_{1})=\operatorname {E} (\operatorname {E} (X\mid I_{2})\mid I_{1}),$

where the value of I₁ is determined by that 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 conditional on only the information observed up to and including 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:

$\operatorname {E} _{t}(X)=\operatorname {E} _{t}(\operatorname {E} _{t+1}(X)).$

References

• Billingsley, Patrick (1995). Probability and measure. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-00710-2. (Theorem 34.4)
• http://sims.princeton.edu/yftp/Bubbles2007/ProbNotes.pdf, especially equations (16) through (18)