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Math Help - Probability, Expectation, Cov

  1. #1
    Member Maccaman's Avatar
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    Probability, Expectation, Cov

    Let X and Y be 2 random variables with |\mathbb{E}[X]|, |\mathbb{E}[Y]|, and |\mathbb{E}[\frac{X}{Y}]| all finite, and with  \mathbb{P}(Y = 0) = 0 and  \mathbb{E}[Y] \ne 0 .
    Prove that \mathbb{E}[\frac{X}{Y}] = \frac{\mathbb{E}[X]}{\mathbb{E}[Y]} if and only if Cov (Y,\frac{X}{Y}) = 0
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  2. #2
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    solution

    Use Cov(Y,X/Y)=E(Y*X/Y)-E(Y)*E(X/Y)=0. Because random variable Y*X/Y=X for all points except those where Y=0 the condition P(Y=0)=0
    gives that E(X)=E(Y*X/Y).
    Last edited by kobylkinks; August 14th 2009 at 05:58 AM.
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  3. #3
    Member Maccaman's Avatar
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    wow, that was surprisingly easy. Thanks for your help.
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