How to derive probability mass function from cumulative distribution function

For $X, Y$ discrete, could you please show me how to derive $P(X = x, Y = y)$ from the cumulative distribution function $F_{XY}(x,y) = P(X <= x, Y <= y) $ ?

Moreover, why is this derivation more difficult than finding density function from the cumulative distribution function of 2 continous random variables ? I think it is due to the fact that we just need to take partial derivative, right ?

Thank you very much for your help!

Solution 1:

You can rewrite the event $\{X=x,Y=y\}$ in terms of inequalities.

Start from the event $\{X\leq x,Y\leq y\}$. This is obviously a larger event than you want; what is the extra?

One bad case, for instance, is $\{X\leq x, Y<y\}$; assuming we're thinking these variables are integer-valued, you can rewrite $Y<y$ as $Y\leq y-1$. Thus $\{X\leq x, Y<y\}=\{X\leq x, Y\leq y-1\}$.

Continuing on in this way, you can identify the event $\{X=x, Y=y\}$ as $\{X\leq x, Y\leq y\}$ less a collection of disjoint "bad" subsets that can be written in terms of $F$.