# 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.