### Introduction

In this Section we introduce a probability model which can be used when the outcome of an experiment is a random variable taking on positive integer values and where the only information available is a measurement of its average value. This has widespread applications, for example in analysing traffic flow, in fault prediction on electric cables and in the prediction of randomly occurring accidents. We shall look at the Poisson distribution in two distinct ways. Firstly, as a distribution in its own right. This will enable us to apply statistical methods to a set of problems which cannot be solved using the binomial distribution. Secondly, as an approximation to the binomial distribution $X\sim B\left(n,p\right)$ in the case where $n$ is large and $p$ is small. You will find that this approximation can often save the need to do much tedious arithmetic.

#### Prerequisites

- understand the concepts of probability
- understand the concepts and notation for the binomial distribution

#### Learning Outcomes

- recognise and use the formula for probabilities calculated from the Poisson model
- use the recurrence relation to generate a succession of probabilities
- use the Poisson model to obtain approximate values for binomial probabilities

#### Contents

1 The Poisson approximation to the binomial distribution1.1 Practical considerations

2 The Poisson distribution

2.1 Definition of a Poisson process

3 Expectation and variance of the poisson distribution