How to remember the Jacobian

Question

the following is the problem that I was working on.

Let $f(x,y)=8xy$ for $0<x<y<1$. What is the joint density function of $W={X \over Y}$ and $Z=Y$?

Since I am self studying this material, I am not quite sure what is customary and here are some questions that I have.

1), Am I being asked what $f_{W,Z}(w,z)$ is or $f_{Z,W}(z,w)$ ? Is it irrelevant?

2), I am comfortable converting z and w in terms of x and y, but I have very hard time memorizing the Jacobian. I want to say that the rows are x and y and the columns are w and z, but due to the same question I have with 1), I get confused which derivative with respect to what goes where. Does anyone have a good solution to this?

Answer 1

1) is irrelevant.

2) I assume you're studying for exam P. From my experience, there isn't a good way to memorize the Jacobian form.

Answer 2

Answer:

$$f(x,y) = 8xy, 0<x<y<1$$

$$W = \frac{X}{Y}$$ $$Z = Y$$

Inverse Transformation is then

$$X = WZ$$ $$Y = Z$$

The ranges get transformed from $0<x<y<1 => 0<wz<z<1 => 0<w<1; 0<z<1$

Now take find the Jacobian:

$$D = \begin{bmatrix}\dfrac{\partial x}{\partial w} & \dfrac{\partial x}{\partial z}\\\dfrac{\partial y}{\partial w} & \dfrac{\partial y}{\partial z}\end{bmatrix}$$

$$D = \begin{bmatrix}z & w\\0 & 1\end{bmatrix}$$

Now take the determinant of the Jacobian $$|D| = det(D) = z$$

$$f_{W,Z}(w,z) = f_{X,Y}(wz,z)\cdot z$$ $$ f_{W,Z}(w,z) = 8(wz)\cdot z\cdot z$$ $$ f_{W,Z}(w,z) = 8wz^3; 0<z<1, 0<w<1$$

This is the joint density function of w,z

This is what is being asked.

Thanks

Satish