雙重期望值定理

雙重期望値定理(Double expectation theorem)，亦稱重疊期望値定理(Iterated expectation theorem)、全期望値定理(Law of total expectation)，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

运算过程 
<math>
\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) \\

参考 
 (Theorem 34.4)
http://sims.princeton.edu/yftp/Bubbles2007/ProbNotes.pdf, especially equations (16) through (18)

數學公式
概率论