$\displaystyle X$ and $\displaystyle Y$ are random variables with expectation equal to zero and variances $\displaystyle \sigma^2$ and $\displaystyle \omega^2\sigma^2$ respectively.

a) If $\displaystyle X$ and $\displaystyle Y$ are independent show that the correlation of $\displaystyle X+Y$ and $\displaystyle X-Y$ is

$\displaystyle \frac{1-\omega^2}{1+\omega^2}$.

b) Find the corresponding correlation if $\displaystyle X$ and $\displaystyle Y$ are not independent but have correlation $\displaystyle \rho$.