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claireteng · 2022年06月21日

B项展开是怎么样的？

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.

B项展开的公式是什么？这部分讲义讲的比较简单，何老师没有展开讲，做题时感觉都不会

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李坏_品职助教 · 2022年06月22日

嗨，爱思考的PZer你好：

看一下基础班讲义P49;

V(ax+by) = A^2 * V(x) + b^2 * V(Y) + 2ab * Cov(x,y)

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努力的时光都是限量版，加油！

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李坏_品职助教 · 2022年06月21日

嗨，从没放弃的小努力你好：

计算方差的时候，常数c可以直接忽略，因为常数的方差是0，并且常数项和随机变量x、y之间都是不相关的。

所以B项的V(ax+by+c) = V(ax + by)，不需要做什么复杂的变换。参考基础班讲义P20：

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就算太阳没有迎着我们而来，我们正在朝着它而去，加油！

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claireteng · 2022年06月22日

不会考对v（ax+by）的展开吗

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2024-04-07 16:05 1 · 回答

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2021-12-25 15:23 1 · 回答

B的正确版本是不是Var（ax+by+c）=Var（ax+by）？

2020-03-17 14:54 1 · 回答