全期望公式

全期望公式，即设X,Y,Z为随机变量，g(·)和h(·)为连续函数，下列期望和条件期望均存在，则

1. $\operatorname {E} (X) = \operatorname{E} ( \operatorname{E} ( X \mid Y));$

运算过程

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

参考

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