Joint distribution function of z and |z|

[snazmn]

X and Y are 2 independent gaussian random variables with parameter a.
Z = XY / (X-Y)
W = XY / |X-Y|
I am to find the joint distribution function of z and w.
I know how to find pdf of Z but how could I use it to find the joint distribution function of z and w?

 

[bpet]

Here Z and W have no joint pdf (because Z=+/-W) so you'll have to calculate the joint cdf directly. I'd suggest breaking it up into special cases, for example P[Z<=z,W<=w]=(1/2)P[Z<=z] if -z<=w<z.