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Math Help - transformation of variable

  1. #1
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    transformation of variable

    i've been looking at examples and have a thorough idea of the process of transformation of variables using the jacobian method, but still cannot solve this question. after solving the determinant i resulted with a zero, which i am sure is incorrect?

    this is the question:

    "X and Y have joint pdf fX,Y(x,y)=2 for 0<x<y<1 and fX,Y(x,y)=0 elsewhere. Find the joint pdf for U=Y-X and V=X, (and hence find the marginal pdf for U.)"

    my main problem is finding fU,V(u,v). If anyone can show me how to solve this question, id highly appreciate it!! thanks!!! =P
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  2. #2
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    Quote Originally Posted by appleting View Post
    i've been looking at examples and have a thorough idea of the process of transformation of variables using the jacobian method, but still cannot solve this question. after solving the determinant i resulted with a zero, which i am sure is incorrect?

    this is the question:

    "X and Y have joint pdf fX,Y(x,y)=2 for 0<x<y<1 and fX,Y(x,y)=0 elsewhere. Find the joint pdf for U=Y-X and V=X, (and hence find the marginal pdf for U.)"

    my main problem is finding fU,V(u,v). If anyone can show me how to solve this question, id highly appreciate it!! thanks!!! =P

    hey mate,

    the most favourable property of your pdf is that it doesnt involve terms in x,y and thus forth the transformed function V(u,v) will be have exactly the same values, however different domains,

    i.e.
    V(u,v) = 2 (new domain 1) and V(u,v) = 0 (elsewhere)

    So the only issue you have to deal with is the transformations of the boundarys,
    you have given
    u = Y-X, v = X
    thus as v = X we have two new variables,
    X = v and Y = u + X = u + v

    Therefore modify the domian with the transformed variables and your done!

    Hope this helps,

    David
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  3. #3
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    Quote Originally Posted by appleting View Post
    i've been looking at examples and have a thorough idea of the process of transformation of variables using the jacobian method, but still cannot solve this question. after solving the determinant i resulted with a zero, which i am sure is incorrect?

    this is the question:

    "X and Y have joint pdf fX,Y(x,y)=2 for 0<x<y<1 and fX,Y(x,y)=0 elsewhere. Find the joint pdf for U=Y-X and V=X, (and hence find the marginal pdf for U.)"

    my main problem is finding fU,V(u,v). If anyone can show me how to solve this question, id highly appreciate it!! thanks!!! =P
    U = Y - X and V = X give Y = U + V and X = V.

    Therefore \frac{\partial X}{\partial U} = 0, \frac{\partial X}{\partial V} = 1, \frac{\partial Y}{\partial U} = 1 and \frac{\partial Y}{\partial V} = 1.

    Therefore the transformation Jacobian is 1.

    So f_{U,V}(u, v) = 2 over the region of the UV-plane defined by 0 < V < U + V < 1, and zero elsewhere. You should think about what this region looks like (so that you can calculate the marginal pdf for U).
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  4. #4
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    Quote Originally Posted by mr fantastic View Post
    U = Y - X and V = X give Y = U + V and X = V.

    Therefore \frac{\partial X}{\partial U} = 0, \frac{\partial X}{\partial V} = 1, \frac{\partial Y}{\partial U} = 1 and \frac{\partial Y}{\partial V} = 1.

    Therefore the transformation Jacobian is 1.

    So f_{U,V}(u, v) = 2 over the region of the UV-plane defined by 0 < V < U + V < 1, and zero elsewhere. You should think about what this region looks like (so that you can calculate the marginal pdf for U).
    silly me... i see where i've gone wrong now... thanks!
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