Question

Let X₁,X₂ and X₃ be random variables with equal variances but with correlation coefficients ρ₁₂=0.3,ρ₁₃=0.5,ρ₂₃=0.2 find the correlation coefficient of the linear function Y=X₁+X₂ and Y=X₂+X₃

Answer 1

the correlation coefficient of the linear function Y=X₁+X₂ and Y=X₂+X₃:

$r_{X_1+X_2,X_2+X_3}=\frac{0.3\sigma^2+0.5\sigma^2+0.2\sigma^2}{\sigma_{X_1+X_2}\sigma_{X_2+X_3}}$

$Var(X_1+X_2)=Var(X_1)+2cov(X_1,X_2)+Var(X_2)=\sigma^2+2\cdot0.3\sigma^2+\sigma^2=$

$=2.6\sigma^2$

similarly:

$Var(X_2+X_3)=\sigma^2+2\cdot0.2\sigma^2+\sigma^2=2.4\sigma^2$

$r_{X_1+X_2,X_2+X_3}=\frac{0.3\sigma^2+0.5\sigma^2+0.2\sigma^2}{\sqrt{2.6\sigma^2\cdot2.4\sigma^2}}=\frac{1}{\sqrt{2.6\cdot2.4}}=0.4003$