Properties of the Joint CDF of Two Continuous Random Variables

The joint CDF of two continuous random variables $X$ and $Y$ is

$$F_{X,Y}(x,y) = P(X \le x,\; Y \le y).$$

The joint CDF satisfies four basic properties:

1. Limits at $-\infty$: $\lim\limits_{x\to -\infty} F_{X,Y}(x,y) = 0$ for every $y,$ and $\lim\limits_{y\to -\infty} F_{X,Y}(x,y) = 0$ for every $x.$
2. Limit at $+\infty$: $\lim\limits_{x,y\to \infty} F_{X,Y}(x,y) = 1.$
3. Monotonicity: $F_{X,Y}$ is non-decreasing in each argument.
4. Marginal CDFs: The CDFs of $X$ and $Y$ alone are recovered by sending the other variable to $+\infty{:}$

$$F_X(x) = \lim_{y\to\infty} F_{X,Y}(x,y), \qquad F_Y(y) = \lim_{x\to\infty} F_{X,Y}(x,y).$$

For instance, suppose

$$F_{X,Y}(x,y) = (1-e^{-x})(1-e^{-y}) \quad \text{for } x,y \ge 0.$$

Sending $y\to\infty,$ the factor $1-e^{-y} \to 1,$ so the marginal CDF of $X$ is

$$F_X(x) = (1-e^{-x})\cdot 1 = 1 - e^{-x}, \qquad x\ge 0.$$

Sending both $x,y\to\infty$ gives $F_{X,Y}(\infty,\infty) = 1\cdot 1 = 1,$ confirming property 2.