Joint CDF of Discrete 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}

The definition applies to any random pair; this page develops its discrete form, and Joint CDF of Continuous Random Variables develops the form with a joint density.

Interpretation¶

The joint CDF measures the probability that two variables simultaneously lie below specified thresholds: $F_{X,Y}(x,y)$ is the probability that $X$ is at most $x$ and $Y$ is at most $y$ . For a discrete pair it is the sum of the joint masses in the lower-left corner below $(x,y)$ .

A 6 by 6 grid of two-dice outcomes with the lower-left block of x at most 2 and y at most 3 highlighted; its six outcomes give the joint CDF value 1/6.

For two fair dice the highlighted corner $\{X\le2,\,Y\le3\}$ contains $2\cdot3=6$ of the $36$ equally likely outcomes, so $F_{X,Y}(2,3)=6/36=\tfrac{1}{6}$ $\frac{1}{6}$ . Because the dice are independent, this also equals $F_X(2)F_Y(3)=\tfrac{2}{6}\cdot\tfrac{3}{6}$ $\frac{3}{6}$ .

The joint CDF is defined at every point $(x,y)$ of the plane, not only at possible values. Between possible values it is constant, so for the dice $F_{X,Y}(2.5,3.7)=F_{X,Y}(2,3)$ $F_{X, Y} (2, 3)$ . It is nondecreasing in each argument, and letting $y\to\infty$ removes the condition on $Y$ and leaves the CDF of $X$ . In general the joint CDF is not the product of the two marginal CDFs; it is for independent variables. For $X=Y\sim\operatorname{Bern}(0.4)$ $Bern (0.4)$ , $F_{X,Y}(0,0)=0.6$ $0.6$ , not $0.6^2$ .

Probabilities of rectangles follow from four corner values by inclusion–exclusion: for $x_1<x_2$ and $y_1<y_2$ $y_{2}$ ,

\mathbb{P}(x_1<X\le x_2,\,y_1<Y\le y_2) =F_{X,Y}(x_2,y_2)-F_{X,Y}(x_1,y_2) -F_{X,Y}(x_2,y_1)+F_{X,Y}(x_1,y_1).

\begin{aligned} &\mathbb{P}(x_1<X\le x_2,\,y_1<Y\le y_2)\\ &=F_{X,Y}(x_2,y_2)-F_{X,Y}(x_1,y_2)\\ &\quad-F_{X,Y}(x_2,y_1)+F_{X,Y}(x_1,y_1). \end{aligned}

For the dice, $F_{X,Y}(2,3)-F_{X,Y}(1,3)-F_{X,Y}(2,2)+F_{X,Y}(1,2)=(6-3-4+2)/36=\tfrac{1}{36}=\mathbb{P}(X=2,Y=3)$ $3)$ . The joint CDF of continuous variables has the same definition, with the sum replaced by an integral of a joint density.

Key Results¶

$\displaystyle F_{X,Y}(x,y)=\sum_{u\le x}\sum_{v\le y}p_{X,Y}(u,v)$ $u \leq x \sum v \leq y \sum p_{X, Y} (u, v)$ .
$F_X(x)=\lim_{y\to\infty}F_{X,Y}(x,y)$ and similarly for $Y$ .

Examples¶

For independent fair dice, $F_{X,Y}(2,3)=(2/6)(3/6)=1/6$ $1/6$ .
If $X=Y$ is Bernoulli with parameter $0.4$ , then $F_{X,Y}(0,0)=0.6$ $0.6$ , rather than $0.6^2$ .

Prerequisites¶

Cumulative Distribution Function

Sources¶

Nicky van Foreest · Probability Distributions · Week 1

Keywords: joint CDF, joint distribution function Symbols: infinity, sum

Last updated: 20 October 2026 03:02