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Math Help - Expected value of a product of RV's

  1. #1
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    Expected value of a product of RV's

    Hi. I need some help verifying something.

    Let X, Y be discrete random variables such that 1<=X,Y<=10. Then:

     \mathbb{E}[XY]=\sum_{1\leq x,y\leq10}xyP(X=x\cap Y=y)

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by JoachimAgrell View Post
    Hi. I need some help verifying something.

    Let X, Y be discrete random variables such that 1<=X,Y<=10. Then:

     \mathbb{E}[XY]=\sum_{1\leq x,y\leq10}xyP(X=x\cap Y=y)

    Thanks in advance.
    This is correct; it follows from \displaystyle \mathbb{E}[X]=\sum xP(X=x) and the fact that XY is a discrete random variable.

    Note that P(XY=2) = P(X=1\cap Y=2) + P(X=2\cap Y=1) and that 2P(XY=2) = 2P(X=1\cap Y=2) + 2P(X=2\cap Y=1), illustrating that the sum you wrote down is in fact equal to \sum zP(XY=z) where I wrote z to avoid confusion; z ranges over the values that XY can take on.
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    Quote Originally Posted by undefined View Post
    This is correct; it follows from \displaystyle \mathbb{E}[X]=\sum xP(X=x) and the fact that XY is a discrete random variable.

    Note that P(XY=2) = P(X=1\cap Y=2) + P(X=2\cap Y=1) and that 2P(XY=2) = 2P(X=1\cap Y=2) + 2P(X=2\cap Y=1), illustrating that the sum you wrote down is in fact equal to \sum zP(XY=z) where I wrote z to avoid confusion; z ranges over the values that XY can take on.
    That makes sense. Thank you.

    But generally is it true that given a discrete random vector X and the set S of its values, \mathbb{E}[f(\mathbf{X})]=\sum_{\mathbf{x}\in S}f(\mathbf{x})P(\mathbf{X}=\mathbf{x})?
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    Quote Originally Posted by JoachimAgrell View Post
    That makes sense. Thank you.

    But generally is it true that given a discrete random vector X and the set S of its values, \mathbb{E}[f(\mathbf{X})]=\sum_{\mathbf{x}\in S}f(\mathbf{x})P(\mathbf{X}=\mathbf{x})?
    Coincidentally, I just came across the formal statement of this recently in another thread on interestingly named math concepts. It is true, and sometimes known as the law of the unconscious statistician.

    This isn't your question, but a word of caution on something semi-related: It is not correct to do things like E(|X-Y|) = |E(X)-E(Y)|.
    Last edited by undefined; July 9th 2010 at 08:03 AM.
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