Posted by haifeng on 2020-06-22 10:36:36 last update 2020-06-22 10:36:36 | Answers (1)
Use the general formula for the variance of a linear combination to write an expression for $V(aX+Y)$. Then let $a=\frac{\sigma_Y}{\sigma_X}$ and show that $\rho\geqslant -1$. [{\it Hint:} Variance is always $\geqslant 0$, and $\mathrm{Cov}(X,Y)=\sigma_X\cdot\sigma_Y\cdot\rho$.]
By considering $V(aX-Y)$, conclude that $\rho\leqslant 1$.
Use the fact that $V(W)=0$ only if $W$ is a constant to show that $\rho=1$ only if $Y=aX+b$.
