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Math Help - Joint density function

  1. #1
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    Joint density function

    I'm a little confused about this problem:

    Let X and Y be independent, iid uniform random variables on [0,1]. Compute the joint density of:
    a) U = X + Y, V = X/Y
    b) U = X, V = X/Y
    c) U = X + Y, V = X/(X+Y)

    I know how to approach this problem, but I am having trouble with some of the details. I rearranged U and V in part (a) to get X=UV/(V+1) and Y=U/(V+1). I know you are supposed to integrate the function and somehow use the jacobian determinant, which I calculated to be (-X/Y^2) - (1/Y). However I am not sure how to actually calculate the joint distribution function.
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  2. #2
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    I'm interested in your problem. Please, give me a simple example as to understand the details. As a first step, I would infere the support for U and V. For a, 0<=U<=1 and 0<Z<Inf. Then, I would construct the marginal for U and V. At last, construct f(u,v). But, I don't exactly know how.

    Do you have any web page were the issue is discussed?

    Thank you.


    Federico.
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  3. #3
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    Quote Originally Posted by almono View Post
    I'm a little confused about this problem:

    Let X and Y be independent, iid uniform random variables on [0,1]. Compute the joint density of:
    a) U = X + Y, V = X/Y
    [snip]
    X = UV/(V+1) \text{ and } Y = U/(V+1)

    so the Jacobian matrix is

    J = <br />
\begin{pmatrix}<br />
V/V+1 & U/(V+1)^2\\<br />
1/(V+1) & -U/(V+1)^2<br />
\end{pmatrix}

    and |\det(J)| = \frac{U}{(V+1)^2}

    So by the change-of-variables formula, the joint pdf of U and V is
    <br />
f(U,V) =<br />
\begin{cases}<br />
\frac{U}{(V+1)^2}   &\text{ if } 0 \leq U/(V+1) \leq 1 \text{ and } 0 \leq UV/(V+1) \leq 1\\<br />
0                            &\text{ otherwise}<br />
\end{cases}

    The only part that is a little tricky is sketching the region where the pdf is non-zero, and I'm not good at posting images, but it's the region of the first quadrant in the U-V plane bounded by the V axis, the U axis, the line V = U-1 and the curve V = 1/(U-1). The region stretches off to infinity in the direction of positive V.

    jw
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