What is the intuitive explanation of the Joint PDF formula?

Question

I know that $$f_{X,Y}(x,y)=\frac{\partial}{\partial x\partial y}F_{X,Y}(x,y)$$ I am not exactly sure of the intuitive description of doing two partial derivatives. In the one-dimensional case, the idea to me is clear. The slope of the CDF essentially gives the "density" under that spot. But does taking two partials give the same "density" as in the single dimension case? It seems hard for me to visualize. Taking $\frac{\partial}{\partial y}$ in my head gives me a new function that is full of how the CDF changes with respect to the $y$ direction. However, the next $\frac{\partial}{\partial x}$ does not make sense to me. Why would me measure the change in the changes of $y$ in the $x$ direction?

Answer 1

You want to find the "local" probability of the tuple $(X, Y)$ lies in the square with vertices $(x,y),(x+dx,y),(x,y+dy),(x+dx,y+dy)$. This probability is (sum and sustract the regions carefully!) $$ F(x+dx,y+dy)-(F(x+dx,y)-F(x,y))-(F(x,y+dy)-F(x,y))-F(x,y)\\ =F(x+dx,y+dy)-F(x+dx,y)-F(x,y+dy)+F(x,y)\\ =\frac{\partial F}{\partial y}(x+dx,y) dy- \frac{\partial F}{\partial y}(x,y)dy\\ =\frac{\partial^2 F}{\partial x\partial y}(x,y) dxdy. $$

So, $\frac{\partial^2 F}{\partial x\partial y}(x,y)$ is the PDF you want.

The reason why we have $\frac{\partial}{\partial x}$ is that we are simply applying the definition of partial derivative in two directions. We first evaluate the change in $y$ direction, giving two terms involving the partial derivative $\partial/\partial y$; then we group them together by taking $\frac{\partial}{\partial x}$. So we are counting the probability in two directions separately.