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Math Help - Joint Density expected values

  1. #1
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    Joint Density expected values

    Let X and Y be random variables with p(x,y)=e^ {-y} for  0<x<y<+ \infty

    Compute  E[X | Y ] and  E[Y | X ]

    My solution:

    For  E[X | Y ] , I have  \int x p(x | y) dx = \int x \frac {p(x,y)}{p(y)} dx

    = \int x \frac {e^{-y}}{p(y)} dx

    But I'm stuck here because I don't know what p(y) is.

    Any help is appreciated, thank you!
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  2. #2
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    Re: Joint Density expected values

    Quote Originally Posted by tttcomrader View Post
    Let X and Y be random variables with p(x,y)=e^ {-y} for  0<x<y<+ \infty
    Is that even a density?

    CB
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    Re: Joint Density expected values

     p(x,y)=e^ {-y} for 0<x<y<+ \infty is the joint pdf, nothing else is given. Is  p(y) = \int ^y _0 p(x,y) dy then?
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    Re: Joint Density expected values

    Quote Originally Posted by tttcomrader View Post
     p(x,y)=e^ {-y} for 0<x<y<+ \infty is the joint pdf, nothing else is given. Is  p(y) = \int ^y _0 p(x,y) dy then?
    I meant is the integral over the 1st quadrant 1? Checking it now the answer appears to be yes.

    p(y) is a marginal and so:

    p(y)=\int_0^y e^{-y} \; dx=ye^{-y}


    CB
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    Re: Joint Density expected values

    Thank you! Now to find p(x), should I do this:

     \int ^ x _ 0 e^{-y} dy = -e^{-x}-1 ?
    Last edited by tttcomrader; September 3rd 2011 at 12:37 PM.
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    Re: Joint Density expected values

    Quote Originally Posted by tttcomrader View Post
    Thank you! Now to find p(x), should I do this:

     \int ^ x _ 0 e^{-y} dy = -e^{-x}-1 ?
    Except:

     \int ^ x _ 0 e^{-y}\; dy = \biggl[-e^{-x}\biggr]_0^x=1-e^{-x}

    CB
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    Re: Joint Density expected values

    f_X(x)=e^{-x} an exponential

    f_Y(y)=ye^{-y} a gamma

    f_{X|Y}(x,y)=1/y but where 0<x<y, which is a uniform

    E(X|Y)=\int_0^y{x\over y}dx={y\over 2} the center of that uniform

    E(Y|X)=\int_x^{\infty}ye^{-y}e^xdy=e^x\int_x^{\infty}ye^{-y}dy
    Last edited by matheagle; September 5th 2011 at 06:46 AM.
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