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Math Help - Expected value of random variable

  1. #1
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    Expected value of random variable

    It is known that for any non-negative Y random variable:

    E(Y)= \int_0^\infty P\{Y>t\}\,\mathrm{d}t

    Show that for any non-negative X random variable:

    E(X^n)= \int_0^\infty nx^{n-1}P\{X>x\}\,\mathrm{d}x

    Thank you very much in advance!
    Last edited by doug; October 22nd 2011 at 10:48 AM. Reason: typo
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  2. #2
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    Re: Expected value of random variable

    As it is written, I'm not sure it exists.

    Anyway, it looks like a simple job for an Integration by Parts. Let's see what you get.
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  3. #3
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    Re: Expected value of random variable

    Quote Originally Posted by TKHunny View Post
    As it is written, I'm not sure it exists.

    Anyway, it looks like a simple job for an Integration by Parts. Let's see what you get.
    You are right there was typo in the text, I corrected the variable t to x in the second integral.

    I suppose that E(X^n)=\int_0^\infty P(X^n>t)\,\mathrm{d}t and then changing the variable: t:=x^n. But to tell the truth I don't know how to continue.
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    Re: Expected value of random variable

    Hello,

    Since X is almost surely positive, you can say that \{X^n>t\}\Leftrightarrow \{X>t^{1/n}\}

    and then change x=t^{1/n}
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