I would like to proof the expression ($P[X>x, Y>y]$) for two continuous random variables $X$ and $Y$.
$P[X>x, Y>y]$ = $1- P[X \leq x, Y \leq y]$ (From the definition of probability).
(As I understand: $P[X \leq x, Y \leq y] = P[X \leq x] + P[Y \leq y] - P[X \leq x, Y \leq y]]$) hence,
= $1 - [P[X \leq x]+P[Y \leq y] - P[X \leq x, Y \leq y]]$
= $1 - P[X \leq x] -P[Y \leq y] + P[X \leq x, Y \leq y]]$
= $1 - F(x) - G(y) + H(x, y)$
Where, $F(x)$, $G(y)$ and $H(x,y)$ are the margins of $X$, $Y$ and the joint distribution, respectively.
I am not sure if my proof is correct or not. Any help, please?