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相关系数在平移/缩放下是否改变

The correlation

专题
Statistics / 统计
难度
L4

题目详情

The correlation between XX and YY is ρ\rho . What is the correlation between X+5X + 5 and YY ? What is the correlation between 5X5X and YY ?

解析

相关系数

corr(U,V)=Cov(U,V)Var(U)Var(V).\mathrm{corr}(U,V)=\frac{\mathrm{Cov}(U,V)}{\sqrt{\mathrm{Var}(U)\mathrm{Var}(V)}}.
  • corr(X+5,Y)=ρ\mathrm{corr}(X+5,Y)=\rho：因为加常数不改变协方差与方差。
  • corr(5X,Y)=ρ\mathrm{corr}(5X,Y)=\rho：因为协方差与标准差都会被同一倍数缩放并相互抵消：
Cov(5X,Y)Var(5X)Var(Y)=5Cov(X,Y)25Var(X)Var(Y)=ρ.\frac{\mathrm{Cov}(5X,Y)}{\sqrt{\mathrm{Var}(5X)\mathrm{Var}(Y)}}=\frac{5\mathrm{Cov}(X,Y)}{\sqrt{25\mathrm{Var}(X)\mathrm{Var}(Y)}}=\rho.