雙重期望值定理

雙重期望值値定理(Double expectation theorem)，亦稱重疊期望值値定理(Iterated expectation theorem)、全期望值値定理(Law of total expectation)，即設X,Y,Z為隨機變數，g(·)和h(·)為連續函數，下列期望值和條件期望值均存在，則

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

運算過程

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

參考

• 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（頁面存檔備份，存於網際網路檔案館）, especially equations (16) through (18)