# Chapter 6 Common Distributions of Discrete Random Variables

Discrete random variables take at most countably many possible values (e.g., \(0, 1, 2, \ldots\)). They are often counting variables (e.g., the number of Heads in 10 coin flips).

The probability mass function (pmf) of a discrete random variable \(X\), defined on a probability space with probability measure \(\textrm{P}\), is a function \(p_X:\mathbb{R}\mapsto[0,1]\) which specifies each possible value of the RV and the probability that the RV takes that particular value: \(p_X(x)=\textrm{P}(X=x)\) for each possible value of \(x\).

The axioms of probability imply that a valid pmf must satisfy \[\begin{align*} p_X(x) & \ge 0 \quad \text{for all $x$}\\ p_X(x) & >0 \quad \text{for at most countably many $x$ (the possible values, i.e., support)}\\ \sum_x p_X(x) & = 1 \end{align*}\]

The countable set of possible values of a discrete random variable \(X\), \(\{x: p_X(x)>0\}\), is called its **support**.

In this section we study some commonly used discrete distributions and their properties. When developing a probability model for a random process, certain assumptions are made about the process or the distribution of a corresponding RV. Some situations are so common that the corresponding distributions have special names.