3.1: Introduction to Random Variables

Now that we have formally defined probability and the underlying structure, we add another layer: random variables. Random variables allow characterization of outcomes, so that we do not need to focus on each outcome specifically. We begin with the formal definition.

Definition \(\PageIndex{1}\)

A random variable is a function from a sample space \(S\) to the real numbers \(\mathbb{R}\). We denote random variables with capital letters, e.g., $$X: S \rightarrow \mathbb{R}.\notag$$

Informally, a random variable assigns numbers to outcomes in the sample space. So, instead of focusing on the outcomes themselves, we highlight a specific characteristic of the outcomes.

Example \(\PageIndex{1}\)

Consider again the context of Example 1.1.1, where we recorded the sequence of heads and tails in two tosses of a fair coin. The sample space for this random experiment is given by
$$S = \{hh, ht, th, tt\}.\notag$$
Suppose we are only interested in tosses that result in heads. We can define a random variable \(X\) that tracks the number of heads obtained in an outcome. So, if outcome \(hh\) is obtained, then \(X\) will equal 2. Formally, we denote this as follows:

\begin{align*}
X: S & \rightarrow \mathbb{R} \\
s & \mapsto\ \text{number of}\ h\text{'s in}\ s
\end{align*}

Since there are only four outcomes in \(S\), we can list the value of \(X\) for each outcome individually:

\begin{align*}
\text{inputs:}\ S\ &\stackrel{\text{function:}\ X}{\longrightarrow}\ \text{outputs:}\ \mathbb{R} \\
hh &\quad\stackrel{X}{\mapsto}\quad 2 \\
th &\quad\stackrel{X}{\mapsto}\quad 1 \\
ht &\quad\stackrel{X}{\mapsto}\quad 1 \\
tt &\quad\stackrel{X}{\mapsto}\quad 0
\end{align*}

We can also write the above as follows:

$$X(hh) = 2,\quad X(ht) = X(th) = 1,\quad X(tt) = 0.\notag$$

The advantage to defining the random variable \(X\) in this context is that the two outcomes \(ht\) and \(th\) are both assigned a value of \(1\), meaning we are not focused on the actual sequence of heads and tails that resulted in obtaining one heads.

---

In Example 3.1.1, note that the random variable we defined only equals one of three possible values: \({0, 1, 2}\). This is an example of what we call a discrete random variable. We will also encounter another type of random variable: continuous. The next definitions make precise what we mean by these two types.

Definition \(\PageIndex{2}\)

A discrete random variable is a random variable that has only a finite or countably infinite (think integers or whole numbers) number of possible values.

Definition \(\PageIndex{3}\)

A continuous random variable is a random variable with infinitely many possible values (think an interval of real numbers, e.g., \([0,1]\)).

In this chapter, we take a closer look at discrete random variables, then in Chapter 4 we consider continuous random variables.