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
#:\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) \\
\end{align}

参考

• (Theorem 34.4)

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