The Joint CDF of Two Discrete Random Variables

Defines the joint cumulative distribution function (CDF) of two discrete random variables, computes it from a joint PMF, recovers marginal CDFs by limiting in one variable, and uses the rectangle (inclusion-exclusion) formula to compute probabilities of rectangular events and individual PMF entries from the joint CDF.

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Introduction

The joint cumulative distribution function (joint CDF) of two discrete random variables $X$ and $Y$ is defined as

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

In terms of the joint PMF $p_{X,Y}$ , this is the sum of the PMF over all pairs $(x_i, y_j)$ in the bottom-left rectangle with corner at $(x, y)$ :

F_{X,Y}(x,y) = \sum\limits_{x_i \le x}\, \sum\limits_{y_j \le y} p_{X,Y}(x_i, y_j).

For example, suppose $X$ and $Y$ have the joint PMF

\begin{array}{c|cc} p_{X,Y}(x,y) & y=0 & y=1 \\ \hline x=0 & 0.2 & 0.3 \\ x=1 & 0.1 & 0.4 \end{array}

To find $F_{X,Y}(0,1)$ , we sum the joint PMF over all pairs $(x_i, y_j)$ with $x_i \le 0$ and $y_j \le 1$ :

F_{X,Y}(0,1) = p_{X,Y}(0,0) + p_{X,Y}(0,1) = 0.2 + 0.3 = 0.5.

Notice that $0 \le F_{X,Y}(x,y) \le 1$ for every $(x,y)$ . The joint CDF equals $1$ once $x$ and $y$ exceed the largest values $X$ and $Y$ can take, and equals $0$ when either is below the smallest possible value.