Joint Probability Mass Function (JPMF)

Definition

Let $X, Y$ be two discrete rv's defined on the sample space $S$ of an experiment. The joint probability mass function $p (x, y)$ is defined for each pair of numbers $(x, y)$ by:

p (x, y) = P (X = x \land Y = y)

Here:

P ((X, Y) \in A) = \sum_{(x, y) \in A} p (x, y)

The support of $p (x, y)$ is defined by $(x, y) | p (x, y) > 0$ , namely the set of points with non-zero probability.

Definition

The marginal probability mass function of $X, Y$ denoted respectively $p_{X} (x)$ and $p_{Y} (y)$ are given by:

p_{X} (x) = \sum_{y} p (x, y); p_{Y} (y) = \sum_{x} p (x, y)

For example, to get $p_{X} (250)$ , you just do $\sum_{y} p (250, y)$ .

More than 2 Variables Case

The joint pmf of $n$ discrete rv's $x_{1}, \dots, x_{n}$ gives the probability of any $n$ -tuple of values $x_{1}, \dots, x_{n}$ :

p (x_{1}, x_{2}, \dots, x_{n}) = P (X_{1} = x_{1} \cap \dots \cap X_{n} = x_{n})