Univariate Discrete Distributions: An Overview

Introduction

A random variable (rv) is said to be discrete if it can take a finite or a countably infinite number of values, i.e., has a discrete state space. These values need not be equally spaced but almost all discrete random variables of use in statistics take equally spaced values and so are said to have lattice distributions. Examples are numbers of aircraft accidents, numbers of bank failures, cosmic ray counts (x = 0, 1, 2, …), counts of occupants per car (x = 1, …), and numbers of albino children in families of 6 children (x = 0, 1, …, 6). Lattice variables are not restricted to count events; they can also be obtained by discretization of continuous measurements, e.g., flood heights (x = 0, 0. 5, 1, 1. 5, … meters).

The set of all possible outcomes from an experiment or sampling scheme is called the sample space, Ω. In univariate situations a single real value is associated with every outcome. The function Xthat determines these numerical values is the random variable and...