Chapter 2

Random Variables

# 2.1 Introduction

We saw in the previous chapter that many random experiments have numerical outcomes. Even if the outcome itself is not numerical, such as the case is Example 1.4, where a coin is flipped twice, we often consider events that can be described in terms of numbers, for example, {the number of heads equals 2}. It would be convenient to have some mathematical notation to avoid the need to spell out all events in words. For example, instead of writing {the number of heads equals 1} and {the number of heads equals 2}, we could start by denoting the number of heads by X and consider the events {X = 1} and {X = 2}. The quantity X is then something whose value is not known before the experiment but becomes known after.

There is a more formal definition that defines a random variable as a real-valued function on the sample space. If X denotes the number of heads in two coin flips, we would thus, for example, have X(HH) = 2. In a more advanced treatment of probability theory, this formal definition is necessary, but for our purposes, Definition 2.1 is enough.

A random variable X is thus something that does not have a value until after the experiment. Before the experiment, we can only describe the set of possible values, that is, the range of X and the associated probabilities. Let us look at a simple example.

Example 2.1. Flip a coin ...