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Math Help - Converse to variable independence implying covariance is zero

  1. #1
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    Converse to variable independence implying covariance is zero

    If X and Y are two independent random variables, then E[X]E[Y] = E[XY]. The converse is not true in general, because there are examples of two random variables where this equality holds but they are dependent, however, in all the examples I've seen, E[XY] = 0, so I feel like this is kind of a "fluke". Can anyone give an example of two dependent random variables such that E[X]E[Y] = E[XY] \neq 0?
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  2. #2
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    Re: Converse to variable independence implying covariance is zero

    Hey SworD.

    Instead of an example, I'd recommend a much more general idea to show you how you can construct random variables systematically that follow this property.

    Define two random variables U and V where V = U^2. Now perform a principal component analysis on them to make them un-correlated and the solution to these un-correlated variables will always have the property you have provided where E[XY] = E[X]E[Y] for the transformed random variables X and Y.

    You can make U and V have any relationship you want and you will still get the property to be true for the transformed random variables.

    Principal Component Analysis does the equivalent of what Gram-Schmidt does to vectors: it creates an orthogonal basis (note - not orthonormal) which is ordered by the amount of variation that a particular variable contributes to the variability of the data.

    It might be useful for you to look at a good book on the subject if you are keen enough.
    Thanks from SworD
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