1. ## Transformations

Hello,
A question from my homework asks if u,v,w are iid exponentially and y=u/w and z=v/w find the pdf of (y,z). I know generally how to do transformations but in this case I cant find a Jacobian since the transformation is from 3 to 2 variables. How would I go about solving this problem.
Thanks

2. Hello,

You're correct, a transformation has to be from n to n variables. (it has to be a $\mathcal{C}^1$-diffeomorphism)
Thus just make the following transformation : (u,v,w) -> (y,z,w), then integrate with respect to w to get the "marginal" pdf of (y,z)

3. Thank you,
I made the necessary substitutions and found that the inverse Jacobian is W^2. I also found that (u,v,w)= (x*w,y*w,w). Where do I go from there to find f(x,y,w).
Thank you

4. Change of Variable -- the Jacobian

what's preventing you from continuing ?