Consider a random variable $X$ and its density function $f_{X}(x)$, consider a bijective, differentiable function $H$, and let the random variable $Y=H(X)$. I am trying to compute the joint density function $f_{X,Y}(x,y)$.
According to the post Joint density of a random variable and its function, $f_{X,Y}(x,y)=0$ when $y\neq H(x)$ (which is obvious) and $f_{X,Y}(x,y)=f_{X}(x)$ when $y=H(x)$.
I don't know why the second conclusion is true. Suppose $f_{X,Y}(x,H(x))=f_{X}(x)$, then let $y=H(x)$, we should also have $f_{X,Y}(H^{-1}(y),y)=f_{Y}(y)$, which means $f_{Y}(y)=f_{X}(H^{-1}(y))$. However, according to the change of variable, $f_{Y}(y)=f_{X}(H^{-1}(y))|\det(\frac{\partial H^{-1}(z)}{\partial z}|_{z=y})|$.
Why these two things are contradict?
Thanks!