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Thread: Functionn of two random variables.

  1. #1
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    Functionn of two random variables.

    Could anyone help me with this question please?

    Given X and Y are continuous random variables with the following joint probability density function:

    $\displaystyle

    f(x,y) =
    \begin{cases}
    {\theta}^2 e^{-{\theta} y} & 0 < x < y \\
    0 & \text{otherwise}
    \end{cases}

    $

    i) Obtain the joint probability density function of X and Y − X.
    ii) Show that X and Y − X are independent random variables and write down their (marginal)
    probability density functions. Identify these distributions

    For part (i)

    I let U = X and Y = V - X,

    My Jacobian value was 1,
    and then I got $\displaystyle f(u,v) = {\theta}^2 e^{-{\theta} , (v-u)} & \text ,u > 0, (v-u) > u $ Is this the correct joint p.d.f for X and Y-X?

    Also for the marginal of X, I got $\displaystyle f X(x) = e^{-{\theta x} $ but I'm not quite sure.
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  2. #2
    MHF Contributor matheagle's Avatar
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    don't you want v=y-x?
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  3. #3
    MHF Contributor matheagle's Avatar
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    $\displaystyle f_X(x) = e^{-{\theta x} $ is not a valid density
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  4. #4
    MHF Contributor matheagle's Avatar
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    Let U = X and V = Y - X,

    then the Jacobian is 1

    and $\displaystyle f(u,v) = {\theta}^2 e^{-{\theta}(v+u)} I(u>0)I(v>0) $

    So $\displaystyle f_U(u) = \theta e^{-\theta u} I(u>0) $

    and

    $\displaystyle f_V(v) = \theta e^{-\theta v}I(v>0) $
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  5. #5
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    Quote Originally Posted by matheagle View Post
    Let U = X and V = Y - X,

    then the Jacobian is 1

    and $\displaystyle f(u,v) = {\theta}^2 e^{-{\theta}(v+u)} I(u>0)I(v>0) $

    So $\displaystyle f_U(u) = \theta e^{-\theta u} I(u>0) $

    and

    $\displaystyle f_V(v) = \theta e^{-\theta v}I(v>0) $
    Thanks, I thought I was close to it.
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