# Poisson distribution

${\displaystyle {\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}}}$, or ${\displaystyle e^{-\lambda }\sum _{i=0}^{\lfloor k\rfloor }{\frac {\lambda ^{i}}{i!}}\ }$, or ${\displaystyle Q(\lfloor k+1\rfloor ,\lambda )}$

${\displaystyle \lambda +e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\log(k!)}{k!}}}$ (for large ${\displaystyle \lambda }$)

In probability theory and statistics, the Poisson distribution (French pronunciation: ​; in English often rendered /ˈpwɑːsɒn/), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event.[1] The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. If receiving any particular piece of mail does not affect the arrival times of future pieces of mail, i.e., if pieces of mail from a wide range of sources arrive independently of one another, then a reasonable assumption is that the number of pieces of mail received in a day obeys a Poisson distribution.[2] Other examples that may follow a Poisson include the number of phone calls received by a call center per hour and the number of decay events per second from a radioactive source.

The Poisson distribution is popular for modelling the number of times an event occurs in an interval of time or space.

The Poisson distribution may be useful to model events such as

The Poisson distribution is an appropriate model if the following assumptions are true.

If these conditions are true, then k is a Poisson random variable, and the distribution of k is a Poisson distribution.

## Related Topics

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