Variance of a combination of normal random variables

**siffi28** · October 6th 2009, 12:20 PM

Hello everyone,

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

$Z = aX + bY + X^2 - Y^2$
$Z = aX + bY + (X+Y)(X-Y)$

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

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

**matheagle** · October 6th 2009, 09:47 PM

Are X and Y independent?

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

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

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

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

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

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

If $E(X)=0$ , then for a normal I believe $E(X^3)=0$ too.

**siffi28** · October 8th 2009, 01:18 AM

Thank you very much matheagle! Really appreciate the guidance.

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