Introduction

Attaching a Number to Chance

A great many real questions boil down to attaching a number to the outcome of an experiment whose result cannot be predicted with certainty: how many heads turn up in 20 coin tosses, how many defective items are in a batch of 50, how many customers arrive at a shop in an hour. In every case, chance decides the underlying outcome, but what we actually care about is a number derived from it.

This idea is formalised through the notion of a random variable — a function that assigns a real number to every outcome of a random experiment. A closely related idea is the Bernoulli trial: any experiment with exactly two possible outcomes, conventionally labelled success (S) and failure (F). If $p$ is the probability of success and $q$ the probability of failure, then necessarily $p+q=1$, since together they exhaust every possibility.

What This Chapter Covers

Building on the probability theory from earlier chapters, this chapter defines random variables precisely, distinguishes discrete random variables (whose values can be listed or counted) from continuous ones, and introduces the probability distribution of a discrete random variable — the table or rule that says how the total probability of $1$ is spread across its possible values. It then develops the two most important discrete distributions built on repeated Bernoulli trials: the binomial distribution and the Poisson distribution, along with their means and variances.