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Expectations on the product of two dependent random variables

  1. Dec 1, 2008 #1
    I am studying for the FRM and there is a question concerning the captioned. I try to start off by following the standard Expectation calculation and breakdown the pdf into Bayesian Conditional Probability function. Then i got stuck there. Anyone can help me to find a proof on it? Many thanks.
  2. jcsd
  3. Dec 4, 2008 #2


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    Homework Helper

    If the random variables [tex] X, Y [/tex] are independent then

    E[X \cdot Y] = E[X] \cdot E[Y]

    I sense from the tone of your question something more is involved?
  4. Dec 4, 2008 #3
    Thanks Statdad.

    But I wanna work out a proof of Expectation that involves two dependent variables, i.e. X and Y, such that the final expression would involve the E(X), E(Y) and Cov(X,Y).

    I suspect it has to do with the Joint Probability distribution function and somehow I need to separate this function into a composite one that invovles two single-variate Probability distribution and one that involves correlation coefficient.

    I just can't get beyond that step.
  5. Dec 5, 2008 #4


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    Sorry - I'm not sure how I did it, but when I first read your message I apparently saw, or thought I saw, a reference to independence.
  6. Dec 5, 2008 #5
    No need to look at conditional PDFs. We have that:

    Cov[X,Y] = E[(X-E[X])\cdot(Y-E[Y])] [/tex]
    = E[X \cdot Y] - E[X \cdot E[Y]] - E[E[X] \cdot Y] + E[E[X] \cdot E[Y]] [/tex]
    [tex] = E[X \cdot Y] - E[X] \cdot E[Y] [/tex]


    [tex]E[X \cdot Y] = Cov[X,Y] + E[X] \cdot E[Y] [/tex]


  7. Dec 10, 2009 #6
    Is anybody familiar with how this problem generalizes to multiple random variables? As a steppingstone, is there a formula for three random variables X, Y, and Z such that:

    E[XYZ] = E[X] * E[Y] * E[Z] + [term involving covariances]

    Thanks for your help!
  8. Dec 13, 2009 #7
    There is no such formula involving just covariances, you have to include higher order moments such as [itex]E[(X-E[X]) \cdot (Y-E[Y]) \cdot (Z-E[Z])] [/itex] for a 3-variable case.
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