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LaPlacian joint distribution

  1. May 7, 2013 #1
    A joint pdf is given as pxy(x,y)=(1/4)^2 exp[-1/2 (|x| + |y|)] for x and y between minus and plus infinity.

    Find the joint pdf W=XY and Z=Y/X.

    f(w,z)=∫∫f(x,y)=∫∫(1/4)^2*e^(-(|x|+|y|)/2)dxdy -∞<x,y<∞
    Someone told me I can not use Jacobian because of the absolute value. Is that true?
    So far this is what I have but I feel like I am not going anywhere.

    f(w,z)=(1/4)^2∫∫e^(-(|x|+|y|)/2)dxdy
    =(1/4)^2∫∫[e^-|x|/2]*e^-|y|/2]
    =(1/4)^2∫[e^-|x|/2]∫e^-|y|/2]

    =(1/4)^2[∫[e^(-x/2)+∫e^(x/2))] * [∫[e^(-y/2)+∫e^(y/2))] the limits from -∞<x,y<0 and 0<x,y<∞

    =[(1/4)^2 ]*4*[ ∫ [e^(x/2)dx] + ∫ [e^(y/2)dy] ] 0<x,y<∞
     
  2. jcsd
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