查看“︁雙重期望值定理”︁的源代码

{{NoteTA |G1=Math}}
'''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
#:<math>\operatorname {E} (X) = \operatorname{E} ( \operatorname{E} ( X \mid Y));</math>

== 运算过程 ==
:<math>
\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}
</math>

== 参考 ==
*{{cite book | last=Billingsley | first=Patrick | title=Probability and measure | url=https://archive.org/details/probabilitymeasu0000bill | publisher=John Wiley & Sons, Inc | location=New York, NY | year=1995 | isbn=0-471-00710-2}} (Theorem 34.4)
*http://sims.princeton.edu/yftp/Bubbles2007/ProbNotes.pdf{{Wayback|url=http://sims.princeton.edu/yftp/Bubbles2007/ProbNotes.pdf |date=20110720030539 }}, especially equations (16) through (18)

[[Category:數學公式]]
[[Category:概率论]]