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[Exer16-2] Exercise 87 of Book {Devore2017B} P.247
Posted by haifeng on 2020-06-22 10:36:36 last update 2020-06-22 10:36:36 | Answers (1) | 收藏
- 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 $Y=aX+b$.
