Joint Probability density function

[marina87]

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<∞

 

[haruspex]

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

∫∫f(x,y)=1, or it would not be a pdf.
You can use the Jacobian, provided you consider it separately in each quadrant (so that the |x| and |y| can be resolved).

 

[Ray Vickson]

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<∞

Why did you post this same question in two different threads and ignore the response in your first thread?

 

[marina87]

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