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相关系数边界：已知 Corr(x,y)=Corr(x,z

Correlation Bounds

专题
General / 综合
难度
L4

题目详情

随机变量 x,y,zx,y,z 满足 Corr(x,y)=0.8(x,y)=0.8，Corr(x,z)=0.8(x,z)=0.8。

问 Corr(y,z)(y,z) 最大可能是多少、最小可能是多少？

We have random variables x,y,zx,y,z with Corr(x,yx,y)=0.8, Corr(x,zx,z)=0.8. The correlation between yy and zz can be at most _______ and at least _______?

解析

相关矩阵必须半正定：

P=[10.80.80.81ρ0.8ρ1]⪰0.P=\begin{bmatrix} 1&0.8&0.8\\ 0.8&1&\rho\\ 0.8&\rho&1 \end{bmatrix}\succeq 0.

等价于 det⁡(P)≥0\det(P)\ge 0，解得

0.28≤ρ≤1.0.28\le \rho\le 1.

Original Explanation

Interpreting correlation = 0.8 as cos⁡θ=0.8.\cos\theta = 0.8. Then x,yx,y share angle θ\theta and x,zx,z also share angle θ.\theta. The angle between yy and zz is between 00 (cosine=1) and 2θ2\theta (cosine=cos⁡(2θ)=0.82−(0.6)2=0.28\cos(2\theta)=0.8^2 - (0.6)^2=0.28). So min=0.28, max=1.