Joint Distributions for Discrete Random Variables

A pair of discrete random variables $X$ and $Y$ has a joint probability mass function (joint pmf), defined by

$$p_{X,Y}(x,y) = P(X = x \text{ and } Y = y).$$

For each ordered pair $(x,y)$, $p_{X,Y}(x,y)$ gives the probability that $X$ takes the value $x$ and $Y$ takes the value $y$ at the same time.

A joint pmf is often displayed as a table. Suppose $X \in \{0,1\}$ and $Y \in \{0,1,2\}$ with the following joint pmf:

$$\begin{array}{c|ccc} & Y=0 & Y=1 & Y=2 \\ \hline X=0 & 0.10 & 0.15 & 0.25 \\ X=1 & 0.20 & 0.20 & 0.10 \end{array}$$

To find $P(X=0 \text{ and } Y=2)$, we read the entry in row $X=0$ and column $Y=2$:

$$P(X=0,\, Y=2) = p_{X,Y}(0,2) = 0.25.$$