Introduction

Many experiments in probability have outcomes that aren't numbers by themselves — the colour of a sweater drawn at random, the face shown on a tossed coin, and so on. A random variable bridges this gap: it is a real-valued function whose domain is the sample space of a random experiment, assigning a numerical value to every possible outcome.

For instance, if two sweaters are drawn one after another (with replacement) from a mix of black and white sweaters, the sample space consists of ordered pairs of colours. Defining $X$ as the number of white sweaters drawn turns every outcome into a number — $X$ could be $0$, $1$, or $2$ depending on the draw. Likewise, tossing a coin twice gives outcomes like $HH$, $HT$, $TH$, $TT$, and a variable such as the number of heads, or even the number of tails minus the number of heads, assigns a real number to each outcome. Note that more than one random variable can be defined on the same sample space — the choice depends on what you want …