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Math Help - Function of random variables

  1. #1
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    Function of random variables

    can some one help in this question:

    If X is Normal( \mu, \sigma^2) and Y is Exponentioal ( \lambda)
    What is the distribution for:

    Z=X+Y
    Z=X-Y
    Z=X*y
    Z=x/Y

    ineed these distribution quickness

    please
    Last edited by doaa; May 3rd 2009 at 10:06 AM.
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  2. #2
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    Quote Originally Posted by doaa View Post
    can some one help in this question:

    If X is Normal( \mu, \sigma^2) and Y is Exponentioal ( \lambda)
    What is the distribution for:

    Z=X+Y
    Z=X-Y
    Z=X*y
    Z=x/Y

    ineed these distribution quickness

    please
    Are X and Y independent?
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  3. #3
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    Yes they independent
    please if you can help???
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  4. #4
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    Quote Originally Posted by doaa View Post
    Yes they independent
    please if you can help???
    Then can't you write down the joint pdf of X and Y and then calculate the cdf of Z in each case.

    (You could also use moment generating functions for Z = X + Y but then you'd need to identify the resulting mgf ....)
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  5. #5
    MHF Contributor matheagle's Avatar
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    This is a nasty problem. There is no nice answer, i.e., nice form.
    IF BOTH X and Y are normal or BOTH are exponential then you can use the MGF. BUT the MGF here won't be recognizable otherwise and that only works for the sum and difference too. So that's worthless. I would just do a 2-2 tranform and integrate out the dummy variable. BUT I wonder if the instructor wanted both of the same form.
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  6. #6
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    First thank you

    But my doctor give me as homework and i tried to solve it but the formulas is complicated
    and i though that my work is wrong
    and i dont know what to do
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  7. #7
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    Instead of 2-2 transform, it's probably easier to find F_Z(z).

    i) X + Y
    F_Z(z) = \int_{-\infty}^{\infty} \int_{-\infty}^{z-y}f_X(x)f_Y(y) dx dy

    ii) X-Y
    F_Z(z) = \int_{-\infty}^{\infty} \int_{-\infty}^{z+y}f_X(x)f_Y(y) dx dy

    iii) X*Y
    F_Z(z) = \int_{-\infty}^{\infty} \int_{z/y}^{0}f_X(x)f_Y(y) dx dy + \int_{-\infty}^{\infty} \int_{0}^{z/y}f_X(x)f_Y(y) dx dy

    iv) X/Y
    F_Z(z) = \int_{-\infty}^{\infty} \int_{-\infty}^{yz}f_X(x)f_Y(y) dx dy

    The upper and lower limit might not be correct but I hope you get the idea.
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