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Math Help - Marginals and conditional of a uniform distribution

  1. #1
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    Marginals and conditional of a uniform distribution

    Hello, I have been going over this problem for a while. I might have a correct solution but I'm not sure about it. Can anyone verify if it is correct? if is not, can you point me in the right direction. Thanks in advanced.

    Suppose that (X,Y) are uniformily distributed over the interval:

    0 \leq y \leq 1-x^2 , y-1 \leq x \leq 1

    I'm unsure about the limits of integration, This I think is the graph



    a) Find the marginals of X and Y:



    To find  f_{x}(x) I think I have to use:

    \int_0^{1-x^2}\frac{1}{b-a}dy

    and obtain

     \frac{1-x^2}{b-a}

    for  f_{y}(y)

    \int_{y-1}^{1}\frac{1}{b-a}dx

    and obtain

     \frac{2-1}{b-a}


    Again I'm unsure about the limits of integration

    For the second part

    b)Find both conditional density functions


    f(x|y)= \frac{\int_{-1}^{0}\int_{x-1}^{1-x^2} \frac{1}{b-a} dy dx}{\frac{2-1}{b-a}}

    All I have left to do is integrate and simplify to obtain the conditional density . Can someone verify the limits of integration again? I will not keep posting the details because if the analysis is right I know the rest and if the analysis is wrong , well, I don't have much time. Any help will be greatly appreciated.
    Last edited by gablar; May 14th 2009 at 04:51 AM. Reason: fixed typo
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  2. #2
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    Joined
    Jan 2009
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    Quote Originally Posted by gablar View Post
    Hello, I have been going over this problem for a while. I might have a correct solution but I'm not sure about it. Can anyone verify if it is correct? if is not, can you point me in the right direction. Thanks in advanced.

    Suppose that (X,Y) are uniformily distributed over the interval:

    0 \leq Y \leq 1-x^2 , y-1 \leq X \leq 1

    I'm unsure about the limits of integration, This I think is the graph



    a) Find the marginals of X and Y:



    To find  f_{x}(x) I think I have to use:

    \int_0^{1-x^2}\frac{1}{b-a}dy

    and obtain

     \frac{1-x^2}{b-a}

    for  f_{y}(y)

    \int_{y-1}^{1}\frac{1}{b-a}dx

    and obtain

     \frac{2-1}{b-a}


    Again I'm unsure about the limits of integration

    For the second part

    b)Find both conditional density functions


    f(x|y)= \frac{\int_{-1}^{0}\int_{x-1}^{1-x^2} \frac{1}{b-a} dy dx}{\frac{2-1}{b-a}}

    All I have left to do is integrate and simplify to obtain the conditional density . Can someone verify the limits of integration again? I will not keep posting the details because if the analysis is right I know the rest and if the analysis is wrong , well, I don't have much time. Any help will be greatly appreciated.
    The bounds for the region over which the pdf f(x, y) is uniform do not make sense to me. Are you sure they are correct?
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  3. #3
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    May 2008
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    I changed X and Y to small case as the original problem states. Othewise the bounds are the ones given.

    I also found the boundaries a little weird. Is it correct that the area of integration is in the 4th cuadrant?
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