Statistical Inference: A Short Course

5.1 The Discrete Probability Distribution

A probability distribution is termed discrete if the underlying random variable X is discrete. (Remember that a random variable X is a real-valued function defined on the elements of a sample space S—its role is to define the possible outcomes of a random experiment.) In this regard, the number of values that X can assume forms a countable set: either there exists a finite number of elements or the elements are countably infinite (can be put in one-to-one correspondence with the positive integers or counting numbers).

For instance, let X represent the outcome (the face showing) on the roll of a single six-sided die. Then X = {1, 2, 3, 4, 5, 6}—the X values form a countable set. However, what if we roll the die repeatedly until a five appears for the first time? Then if X represents the number of rolls it takes to get a five for the first time, clearly, X = {1, 2, 3, . . .}, that is, the X values form a countably infinite set.

This said, a discrete probability distribution is a distribution exhibiting the values of a discrete random variable X and their associated probabilities. In order to construct a discrete probability distribution, we need three pieces of information:

1. The random experiment: this enables us to obtain the sample space S = {E_i, i = 1, . . . , n}, a countable collection of simple events.

2. A discrete random variable X defined on S, which also results in a countable set of values (real numbers) {X₁, X₂, . . . , X