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Thread: [SOLVED] urgent... i have a question about expectation value.. i need help.

  1. #1
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    [SOLVED] urgent... i have a question about expectation value.. i need help.

    Consider a random variable X that takes only nonnegative integer values. Show that the following
    is a way of computing the expectation of X:
    E(X) = (sum of this function from x=0 to x= infinity) (sum symbol) Pr(X > x)
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  2. #2
    Super Member Anonymous1's Avatar
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    Well, what is $\displaystyle P(X>x)?$

    It is the cumulative distribution function. What do you get when you sum over the distribution function for all $\displaystyle x?$ What do you get when you sum over the distribution function to some $\displaystyle x?$
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  3. #3
    Super Member Anonymous1's Avatar
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    I'll just make your life easy.

    $\displaystyle \sum_{x=0}^{\infty} P(X>x) = \sum_{j=0}^{\infty}\sum_{x=0}^{j} P(X=j) = \sum_{j=0}^{\infty}jP(X=j) = E[X].$

    Note this only necessarily holds for integer-valued non-negative random variates.
    Last edited by mr fantastic; Mar 27th 2010 at 04:49 PM. Reason: Merged posts
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    thanks a lot

    it was really important
    Quote Originally Posted by Anonymous1 View Post
    I'll just make your life easy.

    $\displaystyle \sum_{x=0}^{\infty} P(X>x) = \sum_{j=0}^{\infty}\sum_{x=0}^{j} P(X=j) = \sum_{j=0}^{\infty}jP(X=j) = E[X].$

    Note this only necessarily holds for integer-valued non-negative random variates.
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