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Thread: Help finding equation to differentiate

  1. #1
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    Help finding equation to differentiate

    I am having trouble figuring out my next step to this problem. I believe I need to differentiate an equation but I am unsure of how to get that equation. So far I have gotten X=z/Y but I do not know what X is. I do however know that the limits are (0,1) for z. The problem is:
    Let X and Y be continuous random variables for which f_x,y(x,y)=x+y. Find f_z(z), where Z=XY.
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  2. #2
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    Re: Help finding equation to differentiate

    ok... so you've got a joint PDF

    $f_{XY}(x,y) = x+y,~x,y \in [0,1]$

    $Z=XY$

    $F_Z(z) = P[Z < z] = P[X Y < z] = \displaystyle{\int_0^1 \int_0^{\min(1,z/x)}}~x+y~dy~dx,~z \in [0,1]$

    $F_Z(z) = \displaystyle{\int_0^z \int_0^1}~(x+y)~dy~dx + \displaystyle{\int_z^1 \int_0^{z/x}}~(x+y)~dy~dx= 2z - z^2,~z \in [0,1]$

    so

    $f_Z(z) = \dfrac {d}{dz} F_Z(z) = 2(1-z),~z\in [0,1]$
    Thanks from ballerninja29
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