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Math Help - 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 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.
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  2. #2
    MHF Contributor matheagle's Avatar
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    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.
    Last edited by matheagle; October 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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