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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?

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The first equality

$$P[X>x,Y>y] =1- P[X \leq x, Y \leq y]$$

is not right.

The issue is that the complement of the set $S=\{X>x, Y>y\} =\{(X,Y)\in \mathbb R^2 \ ; \ X>x \text{ and } Y>y\}$ is not $S^\prime=\{(X,Y)\in \mathbb R^2 \ ; \ X\le x \text{ and } Y\le y\}$ but $S^c=\{(X,Y)\in \mathbb R^2 \ ; \ X\le x \text{ or } Y\le y\}$.

However, you did a second error in writing $$P[X \leq x, Y \leq y] = P[X \leq x] + P[Y \leq y] - P[X \leq x, Y \leq y]],$$ which luckily corrected the first one.

To correct the overall proof, you should use following sets equalities $$(X\cap Y)^c = X^c \cup Y^c \text{ and } Y =(Y\setminus X\cap Y) \cup (X\cap Y)$$

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