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Math Help - discrete random variable prove var[x]=E[x^2]- (E[x])^2

  1. #1
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    Exclamation discrete random variable prove var[x]=E[x^2]- (E[x])^2

    If X is a discrete random variable and E[X] exists and the function given by f(x) for each x in the domain of the function is the probability mass function at x, then
    VAR[x]=E[x^2] - (E[x])^2

    Im confused on what im suppose to satisfy for the hypothesis. i know the claim is true by trial and error, but can i just pick any discrete function and show it (solved it using a bernoulli) ? or do i have to show it for all functions which i don't know how to do.
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  2. #2
    MHF Contributor harish21's Avatar
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    The expected value of a random variable is its mean \mu

    The variance of a random variable is its second central moment.

    V(X) = E(X - \mu)^2 = E(X - E(X))^2 = E [ X^2] -2XE[X] +(E(X))^2] = E(X^2) -2E(X \; E(X)) +E(X)^2

    since, E(X \; E(X)) = E(X)E(X) =E(X)^2

    what do you get???
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  3. #3
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    This is a standard proof. Here it is on wiki
    Variance - Wikipedia, the free encyclopedia
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  4. #4
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    thakn you both!! i knew the answer but just needed direction on how to get there!
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