If X \sim P_o(\lambda) \quad then
P(X=x) = \displaystyle \frac{e^{-\lambda} \lambda^x}{x!}

The classic Poisson example is the data set of von Bortkiewicz (1898), for the chance of a Prussian cavalryman being killed by the kick of a horse. Ten army corps were observed over 20 years, giving a total of 200 observations of one corps for a one year period. The period or module of observation is thus one year. The total deaths from horse kicks were 122, and the average number of deaths per year per corps was thus 122/200 = 0.61. This is a rate of less than 1. It is also obvious that it is meaningless to ask how many times per year a cavalryman was not killed by the kick of a horse. In any given year, we expect to observe, well, not exactly 0.61 deaths in one corps (that is not possible; deaths occur in modules of 1), but sometimes none, sometimes one, occasionally two, perhaps once in a while three, and (we might intuitively expect) very rarely any more. Here, then, is the classic Poisson situation: a rare event, whose average rate is small, with observations made over many small intervals of time.

The classic Poisson example is the data set of von Bortkiewicz (1898), for the chance of a Prussian cavalryman being killed by the kick of a horse. Ten army corps were observed over 20 years, giving a total of 200 observations of one corps for a one year period. The period or module of observation is thus one year. The total deaths from horse kicks were 122, and the average number of deaths per year per corps was thus 122/200 = 0.61. This is a rate of less than 1. It is also obvious that it is meaningless to ask how many times per year a cavalryman was not killed by the kick of a horse. In any given year, we expect to observe, well, not exactly 0.61 deaths in one corps (that is not possible; deaths occur in modules of 1), but sometimes none, sometimes one, occasionally two, perhaps once in a while three, and (we might intuitively expect) very rarely any more. Here, then, is the classic Poisson situation: a rare event, whose average rate is small, with observations made over many small intervals of time.

## Summary/Background

The Poisson distribution was discovered by Siméon-Denis Poisson (21 June 1781 – 25 April 1840) and published, together with his probability theory, in 1838 in his work Recherches sur la probabilite des jugements en matieres criminelles et matiere civile ("Research on the Probability of Judgments in Criminal and Civil Matters"). In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event.

He published between 300 and 400 mathematical works in all. Despite this exceptionally large output, he worked on one topic at a time.

He published between 300 and 400 mathematical works in all. Despite this exceptionally large output, he worked on one topic at a time.

## Software/Applets used on this page

## Glossary

### event

any set of possible outcomes of a statistical experiment

### period

the horizontal length of one complete cycle

### poisson distribution

A distribution used to estimate probabilities of random events which have a small probability of occuring, for example volcanic eruptions, accident rates.

### union

The union of two sets A and B is the set containing all the elements of A and B.

### work

Equal to F x s, where F is the force in Newtons and s is the distance travelled and is measured in Joules.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|

AQA A-Level (UK - Pre-2017) | S2 | The Poisson Distribution | Probabilities in context | - |

AQA AS Further Maths 2017 | Statistics | Poisson Distribution | Poisson Probabilities in Context | - |

AQA AS/A2 Further Maths 2017 | Statistics | Poisson Distribution | Poisson Probabilities in Context | - |

CCEA A-Level (NI) | S1 | The Poisson Distribution | Probabilities in context | - |

CIE A-Level (UK) | S2 | The Poisson Distribution | Probabilities in context | - |

Edexcel A-Level (UK - Pre-2017) | S2 | The Poisson Distribution | Probabilities in context | - |

Edexcel AS Further Maths 2017 | Further Statistics 1 | Poisson Distributions | Poisson Probabilities in Context | - |

Edexcel AS/A2 Further Maths 2017 | Further Statistics 1 | Poisson Distributions | Poisson Probabilities in Context | - |

I.B. Higher Level | 7 | The Poisson Distribution | Probabilities in context | - |

Methods (UK) | M15 | The Poisson Distribution | Probabilities in context | - |

OCR A-Level (UK - Pre-2017) | S2 | The Poisson Distribution | Probabilities in context | - |

OCR AS Further Maths 2017 | Statistics | Poisson Distribution | Poisson Probabilities in Context | - |

OCR MEI AS Further Maths 2017 | Statistics A | Poisson Distribution | Poisson Probabilities in Context | - |

OCR-MEI A-Level (UK - Pre-2017) | S2 | The Poisson Distribution | Probabilities in context | - |

Universal (all site questions) | P | The Poisson Distribution | Probabilities in context | - |

WJEC A-Level (Wales) | S1 | The Poisson Distribution | Probabilities in context | - |