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

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

运算过程

${\displaystyle {\begin{aligned}\operatorname {E} \left(\operatorname {E} (X|Y)\right)&{}=\sum \limits _{y}\operatorname {E} (X|Y=y)\cdot \operatorname {P} (Y=y)\\&{}=\sum \limits _{y}\left(\sum \limits _{x}x\cdot \operatorname {P} (X=x|Y=y)\right)\cdot \operatorname {P} (Y=y)\\&{}=\sum \limits _{y}\sum \limits _{x}x\cdot \operatorname {P} (X=x|Y=y)\cdot \operatorname {P} (Y=y)\\&{}=\sum \limits _{y}\sum \limits _{x}x\cdot \operatorname {P} (Y=y|X=x)\cdot \operatorname {P} (X=x)\\&{}=\sum \limits _{x}\sum \limits _{y}x\cdot \operatorname {P} (Y=y|X=x)\cdot \operatorname {P} (X=x)\\&{}=\sum \limits _{x}x\cdot \operatorname {P} (X=x)\cdot \left(\sum \limits _{y}\operatorname {P} (Y=y|X=x)\right)\\&{}=\sum \limits _{x}x\cdot \operatorname {P} (X=x)\\&{}=\operatorname {E} (X).\end{aligned}}}$ 

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（页面存档备份，存于互联网档案馆）, especially equations (16) through (18)