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Math Help - I don't even know the subject of this question, possibly Bivariate Normal

  1. #1
    Senior Member chella182's Avatar
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    I don't even know the subject of this question, possibly Bivariate Normal

    Let X and Y be random variables with zero means, variances \sigma_{x}^{2}, \sigma_{y}^{2} and correlation \rho. Find the value of a which minimises Var(Y-aX) and evaluate this minimum value. Deduce the minimum value Var(X-bY) and the value of b at which the minimum is attained. Investigate the cases \rho=0,\pm1 explicitly. Comment generally.

    Literally no idea on this thanks for any help in advance.
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  2. #2
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    Quote Originally Posted by chella182 View Post
    Let X and Y be random variables with zero means, variances \sigma_{x}^{2}, \sigma_{y}^{2} and correlation \rho. Find the value of a which minimises Var(Y-aX) and evaluate this minimum value. Deduce the minimum value Var(X-bY) and the value of b at which the minimum is attained. Investigate the cases \rho=0,\pm1 explicitly. Comment generally.

    Literally no idea on this thanks for any help in advance.
    Why not explicitate {\rm Var}(Y-aX)? You have {\rm Var}(X+Y)={\rm Var}(X)+2{\rm Cov}(X,Y)+{\rm Var}(Y) and {\rm Var}(\lambda X)=\lambda^2 {\rm Var}(X), hence you should be able to write {\rm Var}(Y-aX) as a polynomial of degree 2 in the variable a.
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  3. #3
    Senior Member chella182's Avatar
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    Quote Originally Posted by Laurent View Post
    Why not explicitate {\rm Var}(Y-aX)? You have {\rm Var}(X+Y)={\rm Var}(X)+2{\rm Cov}(X,Y)+{\rm Var}(Y) and {\rm Var}(\lambda X)=\lambda^2 {\rm Var}(X), hence you should be able to write {\rm Var}(Y-aX) as a polynomial of degree 2 in the variable a.
    Alright, so I'm at a^2Var(Y)-2aCov(X,Y)+Var(x). Obviously Var(Y)=\sigma_{y}^2 and Var(X)=\sigma_{x}^2, but is there another way to express the Cov(X,Y)? Like \sigma_{x}\sigma_{y}\rho or something? And even after that am I differentiating or factorising or something? I hate questions like this.
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  4. #4
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    Quote Originally Posted by chella182 View Post
    Alright, so I'm at a^2Var(Y)-2aCov(X,Y)+Var(x). Obviously Var(Y)=\sigma_{y}^2 and Var(X)=\sigma_{x}^2, but is there another way to express the Cov(X,Y)? Like \sigma_{x}\sigma_{y}\rho or something? And even after that am I differentiating or factorising or something? I hate questions like this.
    Since you want to find a which minimizes Var(Y-aX), you should take the first derivative of Var(Y-aX) and set it to 0 and solve for a.
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  5. #5
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    Rewrite the covariance the way you suggested, take the derivative, and set her to zero. Bling Blang Blaow.
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  6. #6
    Senior Member chella182's Avatar
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    Yeah, I got it in the end, cheers
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