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Homework Help: Joint density

  1. Nov 4, 2012 #1
    1. The problem statement, all variables and given/known data

    f_xy(x,y)= 24xy for o<x<1 and 0<y<1-x

    let z=[1-x]/y and w=y

    determine joint density of wz
    and E(z)
    2. Relevant equations



    3. The attempt at a solution
    E[z] = Integral [0,1] integral [0,1-x] 24xy*(1-x)/y dydx = 2

    The joint distribution doing a transformation to x=1-zy and y =w so x = 1-wz

    Jacobian = -w

    so f_wz (wz) = 24(1-zw)w *|-w| = 24 (1-zw)w^2 the new bounds are 0<1-zw<1 => 1>zw>0 and 0<w<1-(1-zw) => 0<w<wz

    the bounds are 0<w<1 and o<z<1 but the density is not equaling 1 so I am doing something wrong.
     
  2. jcsd
  3. Nov 4, 2012 #2

    haruspex

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    I don't understand how you got from:
    which is right, to
     
  4. Nov 4, 2012 #3
    So we have 1>zw>0 and zw>w>0

    So 1>zw>w>0 => 1/w > z > 1 and it seems 1>w>0 but the integration does not work.
     
  5. Nov 4, 2012 #4

    haruspex

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    Your expression for joint pdf integrates to 1 for me. Please show your working,
     
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