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Math Help - Joint Random Variables

  1. #1
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    Joint Random Variables

    Say X and Y are independent, exponentially distributed random variables - with possibly different parameters

    Determine the density func. of Z = X / Y

    Thanks for any help
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  2. #2
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    Re: Joint Random Variables

    Quote Originally Posted by RiemannManifold View Post
    Say X and Y are independent, exponentially distributed random variables - with possibly different parameters
    Determine the density func. of Z = X / Y
    Let U=X and V=\tfrac{X}{Y}.

    Then we write:
    u=x=h_u(x,y)
    v=\tfrac{x}{y}=h_v(x,y)
    x=u=h^{-1}_x(u,v)
    y=\tfrac{u}{v}=h^{-1}_y(u,v)

    J=\det \begin{bmatrix}\dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v}  \\ & \\  \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{bmatrix} = \det \begin{bmatrix} 1&0 \\ & \\ \dfrac{1}{v} & \dfrac{-u}{v^2} \end{bmatrix} = \dfrac{-u}{v^2} \neq 0 WHY?

    f_{U,V}(u,v) = f_{X,Y}\large{(}h^{-1}_x(u,v), h^{-1}_y(u,v)\large{)}|J|

    Remember that independence allows us to write the jpdf as the product of marginal pdfs.

    Can you take it from here?
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