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Math Help - Expected Value and Variance

  1. #1
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    Expected Value and Variance

    I need some help on this:

    E(X)=(1/(sigma(sqrt (2pi))) ∫(-infinity to infinity) xexp(-(x-m)^2/(2sigma^2)dx

    Show E(X)=m.

    I know to add and subtract m to the first x, but I don't know where to go from there.

    Thanks for any help.
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  2. #2
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    Quote Originally Posted by taypez View Post
    I need some help on this:

    E(X)=(1/(sigma(sqrt (2pi))) ∫(-infinity to infinity) xexp(-(x-m)^2/(2sigma^2)dx

    Show E(X)=m.

    I know to add and subtract m to the first x, but I don't know where to go from there.

    Thanks for any help.
    Make the substitution u = \frac{x-m}{\sigma \sqrt{2}}.

    After the appropriate substitutions and simplifying you should get:

    E(X) = \frac{1}{\sqrt{pi}} \int_{-\infty}^{+\infty} \sigma \, \sqrt{2} \, u \, e^{-u^2} + m e^{-u^2} \, du.

    The integral of the first bit is zero because u \, e^{-u^2} is odd.

    So E(X) = \frac{m}{\sqrt{\pi}} \int_{-\infty}^{+\infty} e^{-u^2} \, du = m.
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