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Q.22

Expert-verifiedFound in: Page 360

Book edition
9th

Author(s)
Sheldon M. Ross

Pages
432 pages

ISBN
9780321794772

Show that $Y=a+bX$, then

$\rho (X,Y)=\left\{\begin{array}{l}+1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}b>0\\ -1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}b<0\end{array}\right.$

We prove that

$Y=a+bX$

Given in the question that, We have to find

Show that $Y=a+bX$.

If $Y=a+bX$ with parameters $a$ and $b$

$\rho (X,Y)=\frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}\left(X\right)}\sqrt{\mathrm{Var}\left(Y\right)}}$

$Cov(X,Y)=Cov(X,a+bX)$

$Var\left(Y\right)=Var(a+bX)$

$=o+{b}^{2}Var\left(X\right)$

Now,

$\rho (X,Y)=\frac{b\mathrm{Var}\left(X\right)}{\sqrt{\mathrm{Var}\left(X\right)}\sqrt{{b}^{2}\mathrm{Var}\left(X\right)}}$

Finally,

$\rho (X,Y)=\left\{\begin{array}{l}+1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}\mathrm{b}>0\\ -1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}\mathbf{b}<0\end{array}\right.$

We prove that

$Y=a+bX$

and

$\rho (X,Y)=\left\{\begin{array}{l}+1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}\mathrm{b}>0\\ -1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}\mathrm{b}<0\end{array}\right.$

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