Question

Show that $Y=a+bX$, then

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

Short answer

We prove that

$Y=a+bX$

Step-by-step solution

Given information

Given in the question that, We have to find

Show that $Y=a+bX$.

Explanation

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,

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

Final answer

We prove that

$Y=a+bX$

and

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