# Variance of a combination of normal random variables

• Oct 6th 2009, 12:20 PM
siffi28
Variance of a combination of normal random variables
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.
• Oct 6th 2009, 09:47 PM
matheagle
Are X and Y independent?

If so you need $\displaystyle V(aX+X^2)+V(bY-Y^2)$

You can obtain $\displaystyle V(aX+X^2)$

via $\displaystyle V(aX+X^2)=a^2V(X)+V(X^2)+2aCov(X,X^2)$

Now $\displaystyle V(X)=\sigma^2$

$\displaystyle V(X^2)=E(X^4)-(E(X^2))^2$

and $\displaystyle Cov(X,X^2)=E(X^3)-E(X)E(X^2)=E(X^3)$

If $\displaystyle E(X)=0$, then for a normal I believe $\displaystyle E(X^3)=0$ too.
• Oct 8th 2009, 01:18 AM
siffi28
Thank you very much matheagle! Really appreciate the guidance.