Solutions - 5240-06

(1) Recall the formula for $\rho [X,Y]$:

$$\rho [X,Y]=\frac{\text{Cov}[X,Y]}{{\sigma }_X{\sigma }_Y}$$

Therefore, if $\rho =1$, then $\text{Cov}[X,Y]={\sigma }_X{\sigma }_Y$, and note that $\rho [X,Y]=\text{Cov}[\tilde{X},\tilde{Y}]=1$.

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(2) Compute $E\left[{(\tilde{X}-\tilde{Y})}^2\right]$:

$$E\left[{(\tilde{X}-\tilde{Y})}^2\right]=E\left[{\tilde{X}}^2\right]-2E\left[\tilde{X}\tilde{Y}\right]+E\left[{\tilde{Y}}^2\right]=1-2+1=0$$

Thus, $\frac{X-{\mu }_X}{{\sigma }_X}=\frac{Y-{\mu }_Y}{{\sigma }_Y}$.

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(3) Isolate $Y$ in the above equation:

$$Y=\frac{{\sigma }_Y}{{\sigma }_X}X-\frac{{\sigma }_Y}{{\sigma }_X}{\mu }_X+{\mu }_Y$$

Thus, $Y=aX+b$, where $a=\frac{{\sigma }_Y}{{\sigma }_X}$ and $b=-\frac{{\sigma }_Y}{{\sigma }_X}{\mu }_X+{\mu }_Y$.