双重期望值定理 双重期望値定理(Double expectation theorem)，亦称重叠期望値定理(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 E ( X ) = E ( E ( X ∣ Y ) ) ; {\displaystyle \operatorname {E} (X)=\operatorname {E} (\operatorname {E} (X\mid Y));} 运算过程 E ( E ( X | Y ) ) = ∑ y E ( X | Y = y ) ⋅ P ( Y = y ) = ∑ y ( ∑ x x ⋅ P ( X = x | Y = y ) ) ⋅ P ( Y = y ) = ∑ y ∑ x x ⋅ P ( X = x | Y = y ) ⋅ P ( Y = y ) = ∑ y ∑ x x ⋅ P ( Y = y | X = x ) ⋅ P ( X = x ) = ∑ x ∑ y x ⋅ P ( Y = y | X = x ) ⋅ P ( X = x ) = ∑ x x ⋅ P ( X = x ) ⋅ ( ∑ y P ( Y = y | X = x ) ) = ∑ x x ⋅ P ( X = x ) = E ( X ) . {\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)
