What is a Random Variable?

A great many real questions boil down to attaching a number to the outcome of an experiment whose result we cannot predict with certainty. "How many heads show up when a coin is tossed 20 times?", "how many defective bulbs are there in a box of 50?", "how many customers walk into a shop in an hour?" — in every case, chance decides the outcome, but what we actually care about is a number derived from that outcome.

This idea is captured by the notion of a random variable. Formally, if $S$ is the sample space of a random experiment, a random variable is simply a function $X : S \to \mathbb{R}$ that assigns a real number to every outcome in $S$. It does not have to be the outcome itself — it can be any numerical feature we choose to track, such as a count, a sum, or a score.

Worked Example. Suppose two ordinary six-sided dice are rolled together. The sample space $S$ consists of all $36$ ordered pairs $(i,j)$ with $i,j \in \{1,2,3,4,5,6\}$. Define $X$ to be the sum of the two numbers shown, so $X(i,j) = i+j$. This $X$ is a random variable — a function from $S$ into the real numbers. Its possible values (its range) are $2,3,4,\dots,12$. For instance, $X(3,4) = 7$, and there are six outcomes in $S$ that give $X = 7$, namely $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$. …