# Statistics - Poisson (Process|distribution)

The Poisson process is a stochastic process that counts the number of events in a given interval (mostly time).

The process is named after the Poisson distribution introduced by French mathematician Siméon Denis Poisson. The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.

Distribution of counts that occur at a certain “rate”

The value (mostly time) between each pair of consecutive events has an exponential distribution with parameter λ and each of these inter-arrival times is assumed to be independent of other inter-arrival times.

## 3 - Assumption

The process is random (ie the events are independent)

## 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 useful for modeling punctual phenomena where events occur independently from each other:

• telephone calls at a call center,
• 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

## 6 - Interval

The Poisson distribution can be used with time and/or space intervals.

• time,
• distance,
• area
• volume.