兩隨機變數乘積的期望值
以下推導參考Distribution of the product of two random variables - Expectation of product of random variables。
E ( X Y ) = E ( E ( X Y ∣ Y ) ) law of total expectation = E ( Y ⋅ E [ X ∣ Y ] ) 外層給定Y=y,所以Y對內層期望值來說為常數 \begin{aligned} \operatorname{E}(XY) &= \operatorname{E} ( \operatorname{E} (X Y \mid Y)) && \text{law of total expectation}\\&= \operatorname{E} ( Y\cdot \operatorname{E}[X\mid Y]) && \text{外層給定Y=y,所以Y對內層期望值來說為常數}\end{aligned} E(XY)=E(E(XY∣Y))=E(Y⋅E[X∣Y])law of total expectation外層給定Y=y,所以Y對內層期望值來說為常數
這條式子不論 X X X和 Y Y Y是否獨立都成立。
其中law of total expectation的介紹和證明可以參考law of total expectation。
在
X
X
X和
Y
Y
Y獨立的情況下,有:
E
[
X
∣
Y
]
=
E
[
X
]
\operatorname{E}[X \mid Y] =\operatorname{E}[X]
E[X∣Y]=E[X]
將它代入上式得:
E
(
X
Y
)
=
E
(
Y
⋅
E
[
X
∣
Y
]
)
=
E
(
Y
⋅
E
[
X
]
)
X
的期望值並不受
Y
影響
=
E
(
X
)
E
(
Y
)
把
E
(
X
)
這個常數從
E
裡搬出來
\begin{aligned}\operatorname{E}(XY) &= \operatorname{E} ( Y\cdot \operatorname{E}[X\mid Y]) \\& = \operatorname{E} ( Y\cdot \operatorname{E}[X])&& X\text{的期望值並不受}Y\text{影響} \\&= \operatorname{E}(X) \operatorname{E}(Y) && \text{把}\operatorname{E}(X)\text{這個常數從}\operatorname{E}\text{裡搬出來} \end{aligned}
E(XY)=E(Y⋅E[X∣Y])=E(Y⋅E[X])=E(X)E(Y)X的期望值並不受Y影響把E(X)這個常數從E裡搬出來