Suppose we have $X$ and $Y$ two random variables and $U$=$f_1(X,Y)$ and $V$=$f_2(X,Y)$.
Let's suppose we have the two or one of these densities $f_{(X,Y)}$ and $f_{(U,V)}$. What I don't get is when trying one in function of the other is :
- I sometimes have trouble understanding where to multiply the determinant but what I got is that when we are calculating the jacobian determinant's elements if I derive with respect to $x$ and $y$ then the equality is $f_{(X,Y)}$=$|Je|$$f_{(U,V)}$ and if I derive with respect to $u$ and $v$ then the equality $f_{(U,V)}$=$|Je|$$f_{(X,Y)}$ ???
- Suppose that the two functions $f_{(X,Y)}$ and $f_{(U,V)}$ don't map to the same space. When writing the double integrals when do I change the interval I'm integrating in?
I just want an intuitive way to understand these things so I could memorize them and never forget them.
Jacobian determinants admit the intuitive notation $$\frac{\partial u\partial v}{\partial x\partial y}:=\frac{dudv}{dxdy}=\left|\begin{array}{cc} \tfrac{\partial u}{\partial x} & \tfrac{\partial v}{\partial x}\\ \tfrac{\partial u}{\partial y} & \tfrac{\partial v}{\partial y} \end{array}\right|.$$Then $dudv=\frac{\partial u\partial v}{\partial x\partial y}dxdy$ is a multivariant extension of the univariate chain rule $du=\frac{\partial u}{\partial x}dx$.