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

## 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 |
---|---|---|---|

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

AQA AS Further Maths 2017 | Statistics | Poisson Distribution | Poisson Probabilities |

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

CBSE XII (India) | Probability | Probability | Repeated independent (Bernoulli) trials and binomial distribution |

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

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

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

Edexcel AS Further Maths 2017 | Further Statistics 1 | Poisson Distributions | Poisson Probabilities |

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

I.B. Higher Level | 7 | The Poisson Distribution | Probabilities |

Methods (UK) | M15 | The Poisson Distribution | Probabilities |

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

OCR AS Further Maths 2017 | Statistics | Poisson Distribution | Poisson Probabilities |

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

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

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

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