Joint CDF of Continuous Random Variables¶

Definition¶

The joint CDF of a random pair $(X,Y)$ is

\begin{aligned} F_{X,Y}(x,y)=\mathbb{P}(X\leq x,Y\leq y). \end{aligned}

It exists for every random pair, including pairs without a joint PDF. This page develops its form for pairs with a joint density; the discrete form is on Joint CDF of Discrete Random Variables.

Interpretation¶

The joint CDF $F_{X,Y}(x,y)$ is the probability that the pair lands in the lower-left quadrant $\{(u,v)\colon u\le x,\ v\le y\}$ of the plane. This one function determines the whole joint distribution, and a joint density allows it to be calculated by a double integral.

The unit square with the lower-left rectangle up to the point (0.4, 0.5) shaded, and the diagonal y = x, contrasting independent uniforms, where F equals 0.2, with Y = X, where F equals 0.4.

For independent uniforms, probability is spread evenly over the square, so $F(0.4,0.5)$ is the area $0.4\cdot0.5=0.2$ $0.2$ . For $Y=X$ $X$ , all probability lies on the diagonal, and the quadrant contains the diagonal piece up to $(0.4,0.4)$ , so $F(0.4,0.5)=\mathbb{P}(X\le0.4)=0.4$ $0.4$ . Both pairs have $\operatorname{Unif}(0,1)$ marginals, so the marginals alone do not determine the joint CDF.

As a function of two variables, $F_{X,Y}$ is nondecreasing in each argument, tends to $0$ when either argument tends to $-\infty$ , and tends to the marginal CDF $F_X(x)$ when $y\to\infty$ $\infty$ . Where it is smooth enough, the joint density is recovered as the mixed partial derivative $\partial^2F_{X,Y}/\partial x\,\partial y$ ; for independent uniforms, $F(x,y)=xy$ on the unit square and this derivative is $1$ . Moreover, $X$ and $Y$ are independent if and only if $F_{X,Y}(x,y)=F_X(x)F_Y(y)$ for all $x,y$ .

The definition is the same as for discrete pairs; only the way of computing it changes, from a double sum of a joint PMF to a double integral of a joint density.

The upper-right quadrant is not the complement of the lower-left one: $\mathbb{P}(X>x,Y>y)=1-F_X(x)-F_Y(y)+F_{X,Y}(x,y)$ $F_{X, Y} (x, y)$ , not $1-F_{X,Y}(x,y)$ $F_{X, Y} (x, y)$ . For independent uniforms at $(0.4,0.5)$ this is $0.3$ , not $0.8$ .

Key Results¶

With a joint density, $F_{X,Y}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{X,Y}(u,v)\,\mathrm{d}v\,\mathrm{d}u$ $\int_{- \infty}^{x} \int_{- \infty}^{y} f_{X, Y} (u, v) d v d u$ .
Rectangle probabilities follow by inclusion-exclusion: $F(b,d)-F(a,d)-F(b,c)+F(a,c)$ for $(a,b]\times(c,d]$ $(c, d]$ .

Examples¶

For independent $\operatorname{Unif}(0,1)$ variables, $F_{X,Y}(0.4,0.5)=0.2$ $0.2$ .
If $Y=X\sim\operatorname{Unif}(0,1)$ $Unif (0, 1)$ , then $F_{X,Y}(0.4,0.5)=0.4$ $0.4$ ; this pair has no joint PDF.

Prerequisites¶

Cumulative Distribution Function

Sources¶

Nicky van Foreest · Probability Distributions · Week 2

Keywords: joint CDF, joint distribution function Symbols: infinity, integral, partial

Last updated: 20 October 2026 03:02