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:

enter image description here

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?

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    $\begingroup$ I'm guessing the notation is $\partial,$ (\partial) not $\delta$ (\delta). It is pronounced "the mixed second partial derivative of the CDF" or "the second partial derivative of the CDF with respect to $x$ and $y$". $\endgroup$ – spaceisdarkgreen Sep 22 '18 at 20:01
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    $\begingroup$ Also, that is only the plot of a CDF for one variable, not a joint CDF for two variables $\endgroup$ – spaceisdarkgreen Sep 22 '18 at 20:02

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.

  • $\begingroup$ what is wrong with my description/statement? $\endgroup$ – user366312 Sep 22 '18 at 22:19
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    $\begingroup$ @yahoo.com It's the rate of change in the direction of $y$ of the rate of change in the direction of $x$. $\endgroup$ – Jack M Sep 22 '18 at 22:31

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