雙重期望值定理

Template:NoteTA 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

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

运算过程

$\begin{array}{cc}& \mathrm{E}(\mathrm{E}(X|Y))\\ & =\sum _y\mathrm{E}(X|Y=y)\cdot \mathrm{P}(Y=y)\\ & =\sum _y(\sum _{x}x\cdot \mathrm{P}(X=x|Y=y))\cdot \mathrm{P}(Y=y)\\ \end{array}$

参考

• Template:Cite book (Theorem 34.4)
• -{R|http://sims.princeton.edu/yftp/Bubbles2007/ProbNotes.pdf}-Template:Wayback, especially equations (16) through (18)