Question

The coefficient of correlation between two variable $x$ and $y$ is given by

• A. $r=[{{\sigma }_x}^2+{{\sigma }_y}^2+{{\sigma }_{x-y}}^2]/[2{\sigma }_x{\sigma }_Y]$
• B. $r=[{{\sigma }_x}^2+{{\sigma }_y}^2-{{\sigma }_{x-y}}^2]/[2{\sigma }_x{\sigma }_Y]$
• C. $r=[{{\sigma }_x}^2+{{\sigma }_y}^2+{{\sigma }_{x-y}}^2]/[{\sigma }_x{\sigma }_Y]$
• D. $r=[{{\sigma }_x}^2+{{\sigma }_y}^2-{{\sigma }_{x-y}}^2]/[{\sigma }_x{\sigma }_Y]$

Answer 1

The correct option is B

$r=[{{\sigma }_x}^2+{{\sigma }_y}^2-{{\sigma }_{x-y}}^2]/[2{\sigma }_x{\sigma }_Y]$

Explanation for the correct option :

Finding relation between $r$ and ${{\sigma }_{x-y}}^2$:

$$\begin{gathered} {{\sigma }_{x-y}}^2=\frac{{\left[\right(x-y)-(\overline{x}-\overline{y}\left)\right]}^2}{n} \\ =\frac{{\left[\right(x-\overline{x})-(y-\overline{y}\left)\right]}^2}{n} \\ =\frac{{(x-\overline{x})}^2+{(y-\overline{y})}^2-2(x-\overline{x})(y-\overline{y})}{n} \\ =\frac{{(x-\overline{x})}^2}{n}+\frac{{(y-\overline{y})}^2}{n}-2\frac{(x-\overline{x})(y-\overline{y})}{n} \\ ={{\sigma }_x}^2+{{\sigma }_y}^2-2\mathrm{Cov}(\mathrm{x},\mathrm{y}) \\ ={{\mathrm{\sigma }}_{\mathrm{x}}}^2+{{\mathrm{\sigma }}_{\mathrm{y}}}^2-2{\mathrm{r\sigma }}_{\mathrm{x}}{\mathrm{\sigma }}_{\mathrm{y}} \end{gathered}$$

Rearrange above equation.

$$\begin{gathered} \therefore 2r{\sigma }_x{\sigma }_y={{\sigma }_x}^2+{{\sigma }_y}^2-{{\sigma }_{x-y}}^2 \\ \Rightarrow r=\frac{{{\sigma }_x}^2+{{\sigma }_y}^2-{{\sigma }_{x-y}}^2}{2{\sigma }_x{\sigma }_y} \end{gathered}$$

Therefore, option (B) is correct.