Hello everyone,

I have two normal random variables $\displaystyle X$ and $\displaystyle Y$ such that both have zero mean and same standard deviation $\displaystyle \sigma$ and I want to find out the variance of the random variable $\displaystyle Z$ such that

$\displaystyle Z = aX + bY + X^2 - Y^2$

$\displaystyle Z = aX + bY + (X+Y)(X-Y)$

where a and b are constants. I understand that $\displaystyle aX +bY$ is a normal random variable. So are $\displaystyle X+Y$ and $\displaystyle X-Y$ and that their product has a PDF given by modified bessel function of second kind. But how can I determine the variance of $\displaystyle Z$ in terms of $\displaystyle a, b$ and $\displaystyle \sigma$.

Any help or pointers on this will be greatly appreciated. Thank you for your time.