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Thread: How to integrate a Function of Random Variables using joint probability density?

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    Red face How to integrate a Function of Random Variables using joint probability density?

    How to integrate a Function of Random Variables using joint probability density?-randommath2.png
    Last edited by littlejon; Jan 21st 2017 at 12:44 PM. Reason: clearer image
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    Re: How to integrate a Function of Random Variables using joint probability density?

    integration by parts will take care of this easily enough
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    Re: How to integrate a Function of Random Variables using joint probability density?

    @romsek yes that what I was thinking yet the answer the book gets is very baffling indeed.



    I am sorry to trouble you with my trifles but would it be possible if you could show me what is u and what is dv?
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    Re: How to integrate a Function of Random Variables using joint probability density?

    Quote Originally Posted by littlejon View Post
    @romsek yes that what I was thinking yet the answer the book gets is very baffling indeed.



    I am sorry to trouble you with my trifles but would it be possible if you could show me what is u and what is dv?
    $\displaystyle{\int_0^\infty}~u e^{-u}~du$

    let's rewrite this in $x$ to avoid confusion

    $\displaystyle{\int_0^\infty}~x e^{-x}~dx$

    $u=x,~du=dx$

    $dv = e^{-x}~dx,~v = -e^{-x}$

    $\left . -x e^{-x}\right |_0^\infty + \displaystyle{\int_0^\infty}~e^{-x}~dx=$

    $\displaystyle{\int_0^\infty}~e^{-x}~dx=1$
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    Re: How to integrate a Function of Random Variables using joint probability density?

    Quote Originally Posted by romsek View Post
    $\displaystyle{\int_0^\infty}~u e^{-u}~du$

    let's rewrite this in $x$ to avoid confusion

    $\displaystyle{\int_0^\infty}~x e^{-x}~dx$

    $u=x,~du=dx$

    $dv = e^{-x}~dx,~v = -e^{-x}$

    $\left . -x e^{-x}\right |_0^\infty + \displaystyle{\int_0^\infty}~e^{-x}~dx=$

    $\displaystyle{\int_0^\infty}~e^{-x}~dx=1$

    What does the Gamma mean?
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    Re: How to integrate a Function of Random Variables using joint probability density?

    Quote Originally Posted by littlejon View Post
    What does the Gamma mean?
    wat?

    I don't see any Gamma anywhere
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    Re: How to integrate a Function of Random Variables using joint probability density?

    Sorry for the late reply in the book the answer was 2Γ =1. I was wondering where the gamma came from. I guess it is not so important if you can solve it without it.
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