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tristabo · 2024年03月29日

c选项不太理解

NO.PZ2016062402000005

问题如下：

Given that x and y are random variables and a, b, c and d are constants, which one of the following definitions is wrong?

选项：

A.

$E{(ax+by+c)}=aE{(x)}+bE{(y)}+c$ ，

if x and y are correlated.

B.

$V{(ax+by+c)}=V{(ax+by)}+c$ ,

if x and y are correlated.

C.

$Cov{(ax+by,cx+dy)}=acV{(x)}+bdV{(y)}+{(ad+bc)}Cov{(x,y)}$ ,

if x and y are correlated.

D.

$V{(x-y)}=V{(x+y)}=V{(x)}+V{(y)}$ ,

if x and y are uncorrelated.

解释：

Statement , as it is a linear operation. Statement C is correct, as in Equation:

$V(Y)=\sigma_p^2$

$=\sum_{i=1}^n\omega_i^2\sigma_i^2+\sum_{i=1}^N\sum_{j=1,j\neq i}^N\omega_i\omega_j\sigma_{i,j}$

$=\sum_{i=1}^N\omega_i^2\sigma_i^2+2\sum_{i=1}^N\sum_{j$

Statement D is correct, as the covariance term is zero if the variables are uncorrelated. Statement B is false, as adding a constant c to a variable cannot change the variance. The constant drops out because it is also in the expectation.

老师好，请问c选项是怎么推导出来的

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李坏_品职助教 · 2024年03月29日

嗨，从没放弃的小努力你好：

这道题C选项需要一点拓展思维，基础公式如下：

协方差（Covariance）是只针对变量进行计算（X和Y才是变量，如果只有a,b那就是常数，不需要参与计算）。

C选项的意思是针对ax+by和cx+dy计算协方差，由于两边都包含x和y两个变量，所以需要进行交叉计算：

cov(ax+by, cx+dy) = a*c*cov(x,x) + a*d*cov(x,y) + b*c*cov(x,y) + b*d*cov(y,y)，由于cov(x,x) = V(x)，cov(y,y)=V(y)，也就是截图里面第一个公式，所以：

cov(ax+by, cx+dy) = ac*V(x) + bd*V(y) + (ad+bc)*Cov(x,y)

----------------------------------------------
加油吧，让我们一起遇见更好的自己！

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