Question

Two variates, x and y, are uncorrelated and have standard deviations ${\sigma }_x$ and ${\sigma }_y$ respectively. What is the correlation coefficient between x + y and x - y?

• A. $\frac{{\sigma }_x{\sigma }_y}{{\sigma }_x^2+{\sigma }_y^2}$
• B. $\frac{{\sigma }_x+{\sigma }_y}{2{\sigma }_x{\sigma }_y}$
• C. $\frac{{\sigma }_x^2-{\sigma }_y^2}{{\sigma }_x^2+{\sigma }_y^2}$
• D. $\frac{{\sigma }_y-{\sigma }_x}{{\sigma }_x{\sigma }_y}$

Answer 1

Answer: (C)

$\frac{{\sigma }_x^2-{\sigma }_y^2}{{\sigma }_x^2+{\sigma }_y^2}$

Let $u=(x+y);v=(x-y)$

$\overline{u}=(\overline{x}+\overline{y});\overline{v}=(\overline{x}-\overline{y})$

$cov(u,v)=E(u-\overline{u})(v-\overline{u})=E\left\{\right(x-\overline{x})+(y-\overline{y}\left)\right\}\times \left\{\right(x-\overline{x})-(y-\overline{y}\left)\right\}$

= $E\left\{\right(x-\overline{x}{)}^2-(y-\overline{y}{)}^2\}={\sigma }_x^2-{\sigma }_y^2$

Also, $var\left(u\right)=E\left\{\right(u-\overline{u}\}=E(x-\overline{x})+(y-\overline{y}){\}}^2={\sigma }_x^2+{\sigma }_y^2$

Also, $var\left(u\right)=E\left\{\right(v-\overline{v}\}=E(x-\overline{x})+(y-\overline{y}){\}}^2={\sigma }_x^2+{\sigma }_y^2$

Thus, $\rho =\frac{cov(u,v)}{{\sigma }_u\times {\sigma }_v}=\frac{{\sigma }_x^2-{\sigma }_y^2}{{\sigma }_x^2+{\sigma }_y^2}$