Poisson distribution

In probability theory and statistics, 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 if these events occur with a known constant mean rate and independently of the time since the last event. It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1.
The Poisson distribution is named after French mathematician Siméon Denis Poisson. It plays an important role for discrete-stable distributions.
Under a Poisson distribution with the expectation of events in a given interval, the probability of events in the same interval is:
For instance, consider a call center which receives an average of calls per minute at all times of day. If the number of calls received in any two given disjoint time intervals is independent, then the number of calls received during any minute has a Poisson probability distribution. Receiving calls then has a probability of about 0.77, while receiving 0 or at least 5 calls has a probability of about 0.23.
A classic example used to motivate the Poisson distribution is the number of radioactive decay events during a fixed observation period.

History

The introduction of the Poisson distribution is credited to French mathematician and physicist Siméon Denis Poisson, who published it together with his probability theory in Recherches sur la probabilité des jugements en matière criminelle et en matière civile . This work theorizes about the number of wrongful convictions in a given country by focusing on certain random variables that count the number of events that take place during a time interval of given length. However, similar results had already been given in 1711 by Abraham de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus. This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.
In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space.
A further practical application was made by Ladislaus Bortkiewicz in 1898. Bortkiewicz showed that the frequency with which soldiers in the Prussian army were accidentally killed by horse kicks could be well modeled by a Poisson distribution..

Definitions

Probability mass function

A discrete random variable is said to have a Poisson distribution with parameter if it has a probability mass function given by:
where

is the number of occurrences
is Euler's number
is the factorial.

The positive real number is equal to the expected value of and also to its variance.
The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. The number of such events that occur during a fixed time interval is, under the right circumstances, a random number with a Poisson distribution.
The equation can be adapted if, instead of the average number of events we are given the average rate at which events occur. Then and:

Examples

The Poisson distribution may be useful to model events such as:

the number of meteorites greater than one-meter diameter that strike Earth in a year;
the number of laser photons hitting a detector in a particular time interval;
the number of students achieving a low and high mark in an exam; and
locations of defects and dislocations in materials.

Examples of the occurrence of random points in space are: the locations of asteroid impacts with earth, the locations of imperfections in a material, and the locations of trees in a forest.

Assumptions and validity

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

, a nonnegative integer, is the number of times an event occurs in an interval.
The occurrence of one event does not affect the probability of a second event.
The average rate at which events occur is independent of any occurrences.
Two events cannot occur at exactly the same instant.

If these conditions are true, then is a Poisson random variable; the distribution of is a Poisson distribution.
The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial is, where is the expectation and is the number of trials, in the limit that with kept constant
:
The Poisson distribution may also be derived from the differential equations
with initial conditions and evaluated at

Examples of probability for Poisson distributions

On a particular river, overflow floods occur once every 100 years on average. Calculate the probability of = 0, 1, 2, 3, 4, 5, or 6 overflow floods in a 100-year interval, assuming the Poisson model is appropriate.
Because the average event rate is one overflow flood per 100 years,


0	0.368
1	0.368
2	0.184
3	0.061
4	0.015
5	0.003
6	0.0005

The probability for 0 to 6 overflow floods in a 100-year period.
In this example, it is reported that the average number of goals in a World Cup soccer match is approximately 2.5 and the Poisson model is appropriate.
Because the average event rate is 2.5 goals per match,


0	0.082
1	0.205
2	0.257
3	0.213
4	0.133
5	0.067
6	0.028
7	0.010

The probability for 0 to 7 goals in a match.

Examples that violate the Poisson assumptions

The number of students who arrive at the student union per minute will likely not follow a Poisson distribution, because the rate is not constant and the arrivals of individual students are not independent. The non-constant arrival rate may be modeled as a mixed Poisson distribution, and the arrival of groups rather than individual students as a compound Poisson process.
The number of magnitude 5 earthquakes per year in a country may not follow a Poisson distribution if one large earthquake increases the probability of aftershocks of similar magnitude.
Examples in which at least one event is guaranteed are not Poisson distributed; but may be modeled using a zero-truncated Poisson distribution.
Count distributions in which the number of intervals with zero events is higher than predicted by a Poisson model may be modeled using a zero-inflated model.

Properties

Descriptive statistics

The expected value of a Poisson random variable is.
The variance of a Poisson random variable is also.
The coefficient of variation is while the index of dispersion is 1.
The mean absolute deviation about the mean is
The mode of a Poisson-distributed random variable with non-integer is equal to which is the largest integer less than or equal to . This is also written as. When is a positive integer, the modes are and.
All of the cumulants of the Poisson distribution are equal to the expected value . The -th factorial moment of the Poisson distribution is.
The expected value of a Poisson process is sometimes decomposed into the product of intensity and exposure.
Median

Bounds for the median of the distribution are known and are sharp:

Higher moments

The higher non-centered moments of the Poisson distribution are Touchard polynomials in :
where the braces denote Stirling numbers of the second kind. In other words,
When the expected value is set to, Dobinski's formula implies that the ‑th moment is equal to the number of partitions of a set of size.
A simple upper bound is:

Sums of Poisson-distributed random variables

If for are independent, then A converse is Raikov's theorem, which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables.

Maximum entropy

It is a maximum-entropy distribution among the set of generalized binomial distributions with mean and, where a generalized binomial distribution is defined as a distribution of the sum of N independent but not identically distributed Bernoulli variables.

Other properties

The Poisson distributions are infinitely divisible probability distributions.
The directed Kullback–Leibler divergence of from is given by
If is an integer, then satisfies and
Bounds for the tail probabilities of a Poisson random variable can be derived using a Chernoff bound argument.
The upper tail probability can be tightened as follows: where is the Kullback–Leibler divergence of from.
Inequalities that relate the distribution function of a Poisson random variable to the Standard normal distribution function are as follows:

where is the Kullback–Leibler divergence of from and is the Kullback–Leibler divergence of from.

Poisson races

Let and be independent random variables, with then we have that
The upper bound is proved using a standard Chernoff bound.
The lower bound can be proved by noting that is the probability that where which is bounded below by where is relative entropy. Further noting that and computing a lower bound on the unconditional probability gives the result. More details can be found in the appendix of Kamath et al.

Related distributions

As a Binomial distribution with infinitesimal time-steps

The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed — see [|law of rare events] below. Therefore, it can be used as an approximation of the binomial distribution if is sufficiently large and is sufficiently small. The Poisson distribution is a good approximation of the binomial distribution if is at least 20 and is smaller than or equal to 0.05, and an excellent approximation if and. Letting and be the respective cumulative density functions of the binomial and Poisson distributions, one has:
One derivation of this uses probability-generating functions. Consider a Bernoulli trial whose probability of one success is within a given interval. Split the interval into parts, and perform a trial in each subinterval with probability. The probability of successes out of trials over the entire interval is then given by the binomial distribution
whose generating function is:
Taking the limit as increases to infinity and applying the product limit definition of the exponential function, this reduces to the generating function of the Poisson distribution: