# Process - Poisson Process

### Table of Contents

## 1 - About

The `Poisson process`

is a stochastic process in which events occur:

- continuously
- independently (of the time since the last event) - (ie random)
- at a constant / known average rate
- in a fixed interval of time and/or space

The process is named after the Poisson distribution introduced by French mathematician Siméon Denis Poisson.

## 2 - Articles Related

## 3 - Modeling

- The frequency rate on the interval follows a Poisson distribution
- The value (mostly time) between each pair of consecutive events follows an exponential distribution

## 4 - Example

For instance, suppose someone typically gets 4 pieces of mail per day on average.

There will be a certain spread:

- sometimes a little more,
- sometimes a little less,
- once in a while nothing at all.

Given only the average rate, for a certain period of observation (pieces of mail per day, phone calls per hour, etc.), and assuming that the process, or mix of processes, that produces the event flow is essentially random, the Poisson distribution specifies how likely it is that the count will be 3, or 5, or 11, or any other number, during one period of observation. That is, it predicts the degree of spread around a known average rate of occurrence

## 5 - Application

It is used to model random events (ie events that occur independently from each other) such as:

- the arrival of customers at a store,
- telephone calls at a call center,
- occurrence of earthquakes
- radioactive decay,
- requests on a web server,
- Observed frequency of a given term in a corpus
- Number of visits to web site in a fixed time interval
- Number of web site clicks in an hour

All this process can be seen as queue of event where the element have to wait (ie queuing theory)