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Math Help - Covariance equal to variance

  1. #1
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    Covariance equal to variance

    If covariance(X,X)=variance(X), it can be possible that covariance(X,Y)=variance(X) if X not equal to Y?
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  2. #2
    MHF Contributor matheagle's Avatar
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    By definition you have covariance(X,X)=variance(X) always,
    while covariance(X,Y)=variance(X) can be true since two numbers can be equal

    Covariance(X,Y)=E(XY)-X(X)E(Y) while variance(X)=E(X^2)-(EX)^2
    let E(X)=0, then E(XY) could be the same as E(X^2) depending on the distribution.

    The restriction comes from the Cauchy-Schwartz inequality, staying with mean zero for X and Y, to keep this simple...

    -1\le {E(XY)\over \sqrt{E(X^2)E(Y^2)}}\le 1
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