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nanaluo · 2021年09月20日

请老师解释下这道题的题目和答案,谢谢

NO.PZ2020010801000024

问题如下:

Suppose Yi and Xi followed a bivariate normal so that:

[YiXi]iidN([00],[10.90.93])\begin{bmatrix}Y_i\\X_i\end{bmatrix}\overset{iid}\sim N\left(\begin{bmatrix}0\\0\end{bmatrix},\begin{bmatrix}1&0.9\\0.9&3\end{bmatrix}\right)

What is the R2R^2 in a model that regresses Y on X? What would the R2R^2 be in a model that regressed X on Y?

选项:

解释:

Because this is a regression with a single explanatory variable, the R2R^2 is the squared correlation. The correlation is 0.9/3=.5190.9/\sqrt 3 = .519 and so the R2=0.27R^2 = 0.27. The R2R^2 is the same in both regressions because it only depends on the correlation between the variables.

请老师解释下这道题的题目和答案,题目的矩阵应该怎么看?谢谢!

1 个答案
已采纳答案

李坏_品职助教 · 2021年09月20日

嗨,爱思考的PZer你好:


矩阵的意思是:

左边的[0, 0]表示X和Y的均值都是0。

右边矩阵的左上角的1表示Y的方差是1,右下角的3表示X的方差是3。所以Y标准差是1,X的标准差是根号3。左下角和右上角的0.9是X和Y的协方差cov.


所以X和Y的相关系数=cov/(1*根号3) = 0.9/根号3。 一元线性回归里面的R^2 = 相关系数的平方,也就是0.519^2 = 0.27


这个R^2对于X on Y 和Y on X 都是一样的。


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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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