The Joint CDF of Two Continuous Random Variables

The joint cumulative distribution function of two continuous random variables, defined as $F_{X,Y}(x,y) = P(X \le x, Y \le y)$ . Compute the joint CDF by integrating the joint PDF, recover the joint PDF via the mixed partial derivative $\partial^2 F/\partial x\,\partial y$ , extract marginal CDFs by sending one argument to infinity, and check independence using the factorization $F_{X,Y}(x,y) = F_X(x)\,F_Y(y)$ .

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Introduction

The joint cumulative distribution function (CDF) of two continuous random variables $X$ and $Y$ is defined exactly as in the discrete case:

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

If $X$ and $Y$ have joint PDF $f_{X,Y},$ then the joint CDF is recovered by integrating the PDF over the region $\{(s,t) : s \le x,\, t \le y\}{:}$

F_{X,Y}(x,y) = \int_{-\infty}^{x} \int_{-\infty}^{y} f_{X,Y}(s,t)\,dt\,ds.

Geometrically, $F_{X,Y}(x,y)$ is the volume under the joint density surface lying over the region to the lower-left of the point $(x,y).$

For example, suppose $f_{X,Y}(s,t) = 1$ on the unit square $[0,1]^2$ (and zero elsewhere). Then for $(x,y) \in [0,1]^2,$

F_{X,Y}(x,y) = \int_0^x \int_0^y 1\,dt\,ds = \int_0^x y\,ds = xy.