LaPlacian 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<∞
I have problems with transforming the limits for the new functions.

 

[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<∞



I have problems with transforming the limits for the new functions.

Of course you can use the Jacobian, but you need to be careful to ensure that the variables therein do not cross zero; that is, |x| is perfectly well differentiable as long as x > 0 or x < 0. So, for positive w and z, what is the WY-region of {w < W < w+dw, z < Z < z+dz}? You ought to be able to find it as the union of two disjoint subregions, and in each subregion you can certainly use a Jacobian.

Now you need to look at the other cases where w> 0 and z > 0 do not both hold.

 

[marina87]

That is the part where I got stuck.
What is the best way to solve this problem? should I use Jacobian or should I use the distribution properties?

But my biggest question and where I need help is with the boundaries.