How does the following formula read in the context of Random Variable and probability?
$$f_{XY}(x, y) = \frac{\partial^2 \cdot F_{XY}(x, y)}{\partial x \, \partial y}$$
Since, the plot of a continuous CDF ($F_{XY}(x,y)$) looks like the following:
Source: Joseph K. Blitzstein, Jessica Hwang, Introduction to Probability. Page-287.
I am trying to translate the above formula in layman's terms. As far as I can tell, it means:
The Joint Probability Density Function(PDF) of the random variables $X,Y$ would be the rate of change of slope of the function $F_{XY}(x, y)$ in the both of the directions of $X$ and $Y.$
Am I correct?
A probability density function, as its name implies, can be interpreted as the density of probability with respect to length - the local "probability per meter" (or whatever the units might be). In the same way, your joint probability density function is the "probability per area" density.
Specifically, take a small rectangle $R$ with bottom-left corner $(x, y)$ of width $w$ and height $h$, and consider the probability that $(X, Y)$ is in $R$:
$$\begin{align} P((X,Y)\in R)&=P(X\in[x,x+w],Y\in[y,y+h])\\ &=F(x+w,y+h)-F(x+w,y)-F(x,y+h)+F(x,y) \end{align}$$
In the second equation I'm writing $F$ for $F_{X,Y}$, and the formula may not be very obvious, but it's easy to see by drawing a diagram. Now divide by the area to get the "probability density":
$$\frac{P((X,Y)\in R)}{wh}=\frac1w\left(\frac{F(x+w,y+h)-F(x+w,y)}h-\frac{F(x,y+h)-F(x,y)}h\right)$$
The right hand side, viewed as a formula in $(w, h)$, has a limit at zero equal to $\frac{\partial^2 \cdot F(x, y)}{\partial x \partial y}=f_{(X,Y)}(x, y)$, at least if $F$ is $C^2$. In other words, in the limit of an arbitrarily small rectangle, we get the PDF.