I have two jointly distributed random variables, $X$ and $Y$, for which I'm supposed to prove the following statement:
$$P(X > x, Y > y) = 1 - F_X(x) - F_Y(y) + F_{X,Y}(x,y)$$
If I understand correctly, the capital $F$ refers to the common distribution function (CDF), and $F_X$ and $F_Y$ refer to the marginal CDFs for $X$ and $Y$, respectively. I don't really understand what this equation overall is trying to "say" though, i.e., what its significance is. And I also have no idea where to start with proving it.
$\mathbb P(X> x , Y>y) = 1 - \mathbb P( (X \le x, Y\le y) \cup (X \le x, Y > y) \cup (X > x, Y\le y))$
$= 1 - (\mathbb P(X \le x, Y\le y) + \mathbb P(X \le x, Y > y) + \mathbb P(X > x, Y\le y)) $
$= 1 - [(\mathbb P(X \le x, Y\le y) + \mathbb P(X \le x, Y > y)) + (\mathbb P(X \le x, Y\le y) + \mathbb P(X > x, Y \le y)) - \mathbb P(X \le x, Y \le y)] $
$ = 1 - ( \mathbb P(X \le x, Y \in \mathbb R) + \mathbb P( X \in \mathbb R, Y \le y) - \mathbb P ( X \le x, Y \le y) ) $
$= 1 - \mathbb P (X \le x) - \mathbb P ( Y \le y) + \mathbb P ( X \le x, Y \le y) $
$=1 - F_X(x) - F_Y(y) + F_{(X,Y)}(x,y)$
If you want some explanation: First line is just use of $\mathbb P(A) = 1- \mathbb P(A^c)$, then I used the fact, that I have $3$ disjoint events. In the longest line, I've just added (for further purpose) $\mathbb P(X\le x, Y\le y) - \mathbb P(X\le x, Y\le y)$ . Then I grouped things together, because : $\mathbb P(X \le x) = \mathbb P(X \le x, Y \in \mathbb R) = \mathbb P (X \le x, Y > y) + \mathbb P (X \le x, Y \le y)$ (and similar with $\mathbb P(Y \le y)$. At the end, just came the definition of CDF