How can we show conditional density function with change of variable technique? I want to find a conditional density fucntion p(x|y) using a change of variable technique.
First, i tried to estimate p(x|y) based on the formular of bivariate change of variable technique:
Suppose that the transformation $$(z = f_1(x,y), y = f_2(x,y))$$ is defferentiable and has inverse transformation $$(x = f_1^{-1}(z,y), y = f_2^{-1}(z,y)) $$.
Then as i think, we can estimate p(x|y) with change of variable techinque:
$$
p(x,y) = p(z,y)\begin{vmatrix}{}
  dz/dx& dz/dy\\
  dy/dx& dy/dy
\end{vmatrix}
$$
Finally, 
$$p(x,y)/p(y)=p(x|y) = p(z,y)/p(y)= p(z|y)/p(y)\begin{vmatrix}{}
  dz/dx& dz/dy\\
  dy/dx& dy/dy
\end{vmatrix}
$$
However, the value determinant of Jacovian is zero like followings: $$\begin{vmatrix}{}
  dz/dx& dz/dy\\
  dy/dx& dy/dy
\end{vmatrix}=dz/dx * dy/dy - dz/dy*dy/dx=dz/dx  - dz/dy*dy/dx=0
$$


*

*In this process, is there something wrong ?
And, if i am wrong about estimating conditional density function, how can i derive condtional density function with change of varible techniques?
 A: In general, this is a correct way to compute the conditional density, provided the transformation is actually non-singular as you indicate. But in this case, you should not wind up with a Jacobian that vanishes identically. 
One problem that jumps out is that you have given the same name $y$ to one of the transformed coordinates and one of the original coordinates. You should give it a new name, perhaps writing $w=f_2(x,y)$ rather than $y=f_2(x,y)$. Once you do that you won't have the apparent nonsense of your Jacobian vanishing identically cause of coincidences.
I'm not sure why you wrote $y=f_2(x,y)$ in the first place. Perhaps in the concrete situation you had, we had $f_2(x,y)=y$. Still here, in the abstract, you need to write $w=f_2(x,y)$ (or be very careful when thinking about the inversion of the functions). 
On a side note, it's almost never a good idea to be abstract and "pre-process" your question. This makes the problem you're having harder to diagnose and can lead to issues similar to the XY problem.
