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Math Help - joint pdf and independence

  1. #1
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    joint pdf and independence

    X~Gamma(theta,alpha)
    y~Gamma(theta,beta)
    U=X/Y
    V=X+Y

    Find fX,Y(x,y)
    State whether U and V are independent
    Find marginal pdf



    i am having trouble finding fX,Y(x,y) using information given that X and Y are of Gamma distribution.

    And what is the theorem used to determine the independence of U and V?
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  2. #2
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    Quote Originally Posted by appleting View Post
    X~Gamma(theta,alpha)
    y~Gamma(theta,beta)
    U=X/Y
    V=X+Y

    Find fX,Y(x,y)
    State whether U and V are independent
    Find marginal pdf



    i am having trouble finding fX,Y(x,y) using information given that X and Y are of Gamma distribution.

    And what is the theorem used to determine the independence of U and V?
    Are X and Y independent?

    If U and V are independent then the joint pdf g(u, v) can be written as g(u, v) = g_U(u) \cdot g_V(v).

    Use the change-of-variables formula to find g(u, v). Obviously f(x, y) is needed first, whch means that the answer to my quesiton is needed.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Are X and Y independent?

    If U and V are independent then the joint pdf g(u, v) can be written as g(u, v) = g_U(u) \cdot g_V(v).

    Use the change-of-variables formula to find g(u, v). Obviously f(x, y) is needed first, whch means that the answer to my quesiton is needed.
    oops! yes, X and Y are independent!
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  4. #4
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    Quote Originally Posted by appleting View Post
    oops! yes, X and Y are independent!
    OK. So the joint pdf of X and X should be obvious. Then you can use the change-of-variables formula to get the joint pdf of U and V, g(u, v).

    Then see whether of not g(u, v) can be written as a product g(u, v) = g_U(u) \cdot g_V(v). If yes, then U and V are independent. If no, then they're not.
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    OK. So the joint pdf of X and X should be obvious. Then you can use the change-of-variables formula to get the joint pdf of U and V, g(u, v).

    Then see whether of not g(u, v) can be written as a product g(u, v) = g_U(u) \cdot g_V(v). If yes, then U and V are independent. If no, then they're not.
    hmm im having problem with the gamma distribution in this question

    fx,y(X,Y) = {[(theta)^(alpha+beta)][x^(alpha-1)][y^(beta-1)][e^(-theta*(x+y))]}/[T(alpha+beta)] ??

    or is it

    fx,y(X,Y) = {[(theta)^(alpha+beta)][x^(alpha-1)][y^(beta-1)][e^(-theta*(x+y))]}/[T(alpha)T(beta)] ??

    or am i completely wrong? help~

    (T is that sign thats like a an upside down mirrored L)
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  6. #6
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    Quote Originally Posted by appleting View Post
    hmm im having problem with the gamma distribution in this question

    fx,y(X,Y) = {[(theta)^(alpha+beta)][x^(alpha-1)][y^(beta-1)][e^(-theta*(x+y))]}/[T(alpha+beta)] ??

    or is it

    fx,y(X,Y) = {[(theta)^(alpha+beta)][x^(alpha-1)][y^(beta-1)][e^(-theta*(x+y))]}/[T(alpha)T(beta)] ??

    or am i completely wrong? help~

    (T is that sign thats like a an upside down mirrored L)
    What you need is here: Gamma distribution - Wikipedia, the free encyclopedia. Multiply the pdf's of X and Y together.
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