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Math Help - Joint PDF

  1. #1
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    Joint PDF

    Any help appreciated! Thanks in advance!

    Suppose that X and Y are independent uniform U(0, 1) random variables.

    (i) Find the joint PDF of U and V , fU,V (u, v), defined by U = X + Y and
    V = X Y .

    (ii) Obtain the marginal PDF of U.
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  2. #2
    MHF Contributor matheagle's Avatar
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    The marginal pdf of U is the classic triangular density...

    f_U(u)=u on 0<u<1

    f_U(u)=2-u on 1<u<2

    and zero elsewhere.
    This can be done directly via CDFs.

    As for the joint density of U and V, you need to use calc 3 and jacobians.
    Let's see your work..............

    f_{XY}(x,y)=1 on 0<x<1, 0<y<1.
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  3. #3
    Senior Member chella182's Avatar
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    I'm not sure what you mean by "calc 3" and Jacobians

    This question isn't remotely similar to any examples we were given when we were learning the stuff with Jacobians
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  4. #4
    MHF Contributor matheagle's Avatar
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    You can get the marginals with the CDF technique but I think you have to do a two to two change of variables like you do in calculus with polar coordinates to get the joint density of U and V.
    Do recall in polar, that dxdy=rdrd\theta. The r is the jacobian, it's the determinant of the 2 by 2 matrix of partial derivatives.
    Last edited by matheagle; December 10th 2009 at 12:49 AM.
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  5. #5
    Senior Member chella182's Avatar
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    Still don't know what calc 3 is... an American maths class, maybe?

    This is seeming a little more complicated than necessary... darn lecturer's not here until Friday either. Oh well.
    Last edited by mr fantastic; December 9th 2009 at 06:17 PM. Reason: m --> r
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  6. #6
    MHF Contributor matheagle's Avatar
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    I would use calculus, but I also figured you could use the joint CDF.
    I've never done that before...

    P(U \le z, V\le b) = P(X+Y \le a, X-Y \le b)=

    the density of X,Y is one on the unit square.
    You will need to draw and consider the various cases of a and b.
    Last edited by mr fantastic; December 10th 2009 at 02:37 PM. Reason: Fixed a latex tag and some code
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