Is it a distribution function?

Question

Consider $F(x,y) = \mathbb{I}[x+y \geq 1] $ .

$\mathbb{I} $ denotes the indicator function . So , $F(x,y)$ is 1 iff $x+y \geq 1$ and $0$ otherwise .

Can we consider to be a bivariate distribution ? What are the necessary and sufficient conditions that are to be checked , and does this example satisfy all these requirements .

I am trying to understand bivariate distributions , so please try to bear with my (seemingly trivial ) questions .

Answer 1

A cumulative distribution function $F$ of two random variables $X$ and $Y$ satisfies the following properties:

1. $$\Delta_{a}^{b} F:= F(b_1,b_2)+F(a_1,a_2) - F(a_1,b_2) - F(b_1,a_2) \ge 0$$

where $a = (a_1,a_2)$, $b=(b_1,b_2)$ $\in \mathbb{R}^2$ with $a_1 \le b_1$ and $a_2 \le b_2$.

2. (Continuity from the right)

$$ \lim_{x\to x_0^{+}} F(x,y) = F(x_0,y)$$

$$\lim_{y\to y_0^{+}} F(x,y) = F(x,y_0)$$

3. $$\lim_{x\to-\infty}F(x,y) = \lim_{y\to-\infty}F(x,y) = 0$$

4.

$$\lim_{(x,y) \to (\infty,\infty)}F(x,y) = 1$$