Why is the derivative of the joint CDF the joint PDF? Is there a "multivariable" form of FTC?

Question

The definition of the joint PDF with two random variables X and Y is the derivative of the joint CDF with respect to X and then Y (or vice versa). That is what Wikipedia says. However, I am confused, because in the single-variable case, this wasn't really a definition: the fact that the derivative of the CDF is the PDF is a theorem, not a definition, that involved the application of the Fundamental Theorem of Calculus. Thus, is there a "multivariable" form of FTC that explains the fact the derivative of the joint CDF is the joint PDF? If there isn't, what's the rationale behind the definition of the joint PDF?

Answer 1

Presenting something as a definition or as a theorem is a matter of taste. If you define the pdf in one variable as the limit of $$ \frac{P(X\in (x,x+h))}{h} $$ as $h\to 0$, then the cdf is its integral by FTC. But you can also define the pdf as the derivative of the cdf and easily obtain the above definition. In the multidimensional case, yo can define the joint pdf in a similar way, as a density or limit of $$ \frac{P((X,Y)\in (x,x+h)\times(y,y+h))}{h^2} $$ as $h\to 0$ and then prove that is is the mixed second derivative of the cdf, or vice versa.

The pdf is the density associated to a given cdf. In the absolutely continuous case, such density exists as a locally integrable function, but in general, say if your cdf has jumps, the pdf may contain $\delta$-distributions at some points.