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Thread: Variance of a combination of normal random variables

  1. #1
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    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.
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  2. #2
    MHF Contributor matheagle's Avatar
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    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.
    Last edited by matheagle; Oct 7th 2009 at 08:28 PM.
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  3. #3
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    Thank you very much matheagle! Really appreciate the guidance.
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