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Thread: Covariance question

  1. #1
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    Covariance question

    Hi,
    I just want you expert to confirm something for me.

    Suppose you have two random variables X and Y. Further suppose that fXY(x,y) = g(x) where x and y are bounded, fXY(x,y)= 0 elsewhere. That is fXY(x,y) is just a function of x (or y). Does it follow that Cov (X,Y) = 0????
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  2. #2
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    Re: Covariance question

    If $f_{XY}(x,y) = f_X(x)$ then $f_Y(y)=1$

    $E[XY] = \displaystyle{\int_{-\infty}^\infty\int_{-\infty}^\infty} ~x y f(x,y)~dx~dy =$

    $\displaystyle{\int_{-\infty}^\infty y \int_{-\infty}^\infty} ~x f(x)~dx~dy =$

    $\displaystyle{\int_{-\infty}^\infty y f_Y(y) \int_{-\infty}^\infty} ~x f(x)~dx~dy =$

    $\left(\displaystyle{\int_{-\infty}^\infty} y f_Y(y)~dy \right) \left(\displaystyle{\int_{-\infty}^\infty} ~x f(x)~dx\right) = E[y]E[x]$

    $E[XY] -E[x]E[y]=0$
    Last edited by romsek; Jan 31st 2017 at 07:55 PM.
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  3. #3
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    Re: Covariance question

    Quote Originally Posted by romsek View Post
    If $f_{XY}(x,y) = f_X(x)$ then $f_Y(y)=1$

    $E[XY] = \displaystyle{\int_{-\infty}^\infty\int_{-\infty}^\infty} ~x y f(x,y)~dx~dy =$

    $\displaystyle{\int_{-\infty}^\infty y \int_{-\infty}^\infty} ~x f(x)~dx~dy =$

    $\displaystyle{\int_{-\infty}^\infty y f_Y(y) \int_{-\infty}^\infty} ~x f(x)~dx~dy =$

    $\left(\displaystyle{\int_{-\infty}^\infty} y f_Y(y)~dy \right) \left(\displaystyle{\int_{-\infty}^\infty} ~x f(x)~dx\right) = $

    $E[y]E[x]$

    $E[XY] = E[x]E[y]=0$
    This is exactly what I did. I think that the last line should be E[XY] - E[X][Y] = 0 and not E[XY] = E[X][Y] = 0. Am I correct about that?
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    Re: Covariance question

    Quote Originally Posted by JaguarXJS View Post
    This is exactly what I did. I think that the last line should be E[XY] - E[X][Y] = 0 and not E[XY] = E[X][Y] = 0. Am I correct about that?
    yes
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