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Expert-verified Found in: Page 360 ### A First Course in Probability

Book edition 9th
Author(s) Sheldon M. Ross
Pages 432 pages
ISBN 9780321794772 # Show that $Y=a+bX$, then$\rho \left(X,Y\right)=\left\{\begin{array}{l}+1 \text{if}b>0\\ -1 \text{if}b<0\end{array}\right\$

We prove that

$Y=a+bX$

See the step by step solution

## Step 1: Given information

Given in the question that, We have to find

Show that $Y=a+bX$.

## Step 2: Explanation

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

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

$Cov\left(X,Y\right)=Cov\left(X,a+bX\right)$

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

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

Now,

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

Finally,

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

We prove that

$Y=a+bX$

and

$\rho \left(X,Y\right)=\left\{\begin{array}{l}+1 \text{if}\mathrm{b}>0\\ -1 \text{if}\mathrm{b}<0\end{array}\right\$ ### Want to see more solutions like these? 