When is the Poisson distribution applicable?

Poisson distribution

 
Probability function of the Poisson distribution for = 1, 5 and 9

The Poisson distribution (named after the mathematician Siméon Denis Poisson) is a probability distribution that can be used to model the number of events that occur independently of one another in a fixed time interval or spatial area at a constant mean rate. It is a univariate discrete probability distribution that represents a frequently occurring limit value of the binomial distribution for an infinite number of attempts. However, it can also be derived axiomatically from fundamental process properties.

The gains of a Poisson process are Poisson-distributed random variables. Extensions to the Poisson distribution such as the generalized Poisson distribution and the mixed Poisson distribution are mainly used in the field of actuarial mathematics.

definition

The Poisson distribution is a discrete probability distribution. It is given by a real parameter which describes the expected value and at the same time the variance of the distribution. It assigns the natural numbers the probabilities

to, where Euler's number and the faculty of designated. The parameter Describes the expected frequency of events during an observation. The Poisson distribution then gives the probability of a certain number of events on a case-by-case basis if the mean event rate is known.

properties

Distribution function

The distribution function is the Poisson distribution

and gives the probability for that, at most Find events where to go expected on average. Here designated the regularized gamma function of the lower limit.

Expectation, variance, moment

Is the random variable Poisson distributed, so , so is expectation and variance at the same time, because it applies

as

According to the shift formula it now follows:

The same applies to the third centered moment .

Median

It seems reasonable to assume that the median close to lies. An exact formula does not exist, however, which is the most accurate estimate possible

Coefficient of variation

The coefficient of variation is obtained immediately from the expected value and the variance

.

Obliqueness and curvature

The crookedness arises too

.

The curvature can also be shown closed as

.

Higher moments

The kth moment can be expressed as a polynomial of degree k in and is the kth complete Bell polynomial evaluated at the k positions :

.

Accumulators

The cumulant producing function of the Poisson distribution is

.

This means that all accumulators are the same

Characteristic function

The characteristic function has the form

.

Probability generating function

For the probability generating function one obtains

.

Moment generating function

The moment generating function of the Poisson distribution is

> Reproductivity

The Poisson distribution is reproductive, that is, the sum stochastically independent Poisson distributed random variable with the parameters is again Poisson distributed with the parameter . The following applies to the convolution

The Poisson distributions thus form a convolution half-group. This result follows directly from the characteristic function of the Poisson distribution and the fact that the characteristic function of a sum of independent random variables is the product of the characteristic functions.

The Poisson distribution is therefore also infinitely divisible. According to a sentence by the Soviet mathematician D. A. Raikow, the reverse also applies: is a Poisson-distributed random variable the sum of two independent random variables and , then are the summands and also Poisson distributed. A Poisson-distributed random variable can only be broken down into Poisson-distributed independent summands. This theorem is an analogue of Cramer's theorem for normal distribution.

Thinning

Often stochastic experiments occur in which the events are actually Poisson distributed, but the counting only takes place if an additional condition is met. For example, the number of eggs an insect lays might be Poisson distributed, but there is only a certain probability that a larva will hatch from each egg. An observer of this Poisson distributed random variable with parameter So each event only counts with one probability (independently of each other).

Alternatively, an error in the count could mean that the event is not registered. So if originally Events are present according to the binomial distribution just Events counted. In this case it is the real value unknown and varies between the measured value (seen all existing events) and infinite (there were more events than were seen). The probability of a reading is then found by means of the product of the probability of a successful measurement and the original Poisson distribution , summed over all possible values :

.

The values ​​found if there is a probability of proof are again Poisson distributed. The probability of detection reduces the parameter to the original Poisson distribution . This is also known as the thinning of the Poisson distribution.

calculation

The calculation of can be done recursively as follows. First you determine , then arise one after the other . With growing the probabilities become greater as long as is. Becomes , they shrink. The mode, i.e. the value with the greatest probability, is , if is not an integer, otherwise there are two neighboring ones (see diagram above right).

If the calculation of because of too large values ​​of and Causes problems, then the following approximation obtained with the Stirling formula can help:

Poisson-distributed random numbers are usually generated using the inversion method.

Parameter estimation

Maximum likelihood estimator

From a sample of Observations For should the parameter the Poisson population can be estimated. The maximum likelihood estimator is given by the arithmetic mean

.

The maximum likelihood estimator is an undistorted, efficient and sufficient estimator for the parameter .

Confidence interval

The confidence interval for is obtained from the relationship between Poisson and Chi-square distributions. Is a sample value before then is a confidence interval for at the confidence level given by

,

in which the quantile function of the chi-square distribution with Degrees of freedom.

Forecast interval

The task of the forecast interval is to predict an area in which the realization of an estimation function can be found with a high degree of probability before a sample is drawn. The number Poisson distributed events with a given probability is not exceeded can be calculated from the inversion of the distribution function:

It can be done again through the regularized gamma function express. An elementary form of inversion of the distribution function or the gamma function is unknown. In this case, a two-columned one does a good job Table of values ​​that can be easily calculated using the sum given above in the Distribution function section and shows the probabilities of certain values ​​of assigned.

Relationship to other distributions

Relationship to the binomial distribution

Like the binomial distribution, the Poisson distribution predicts the expected result of a series of Bernoulli experiments. The latter are random experiments that only know two possible results (for example “success” and “failure”), that is, they have a dichotomous event space. If the temporal or spatial observation interval is subdivided more and more, the number of attempts increases. The progressive subdivision causes a decrease in the probability of success such that the product against a finite limit converges. Accordingly, the binomial probability distribution approximates the mathematically somewhat simpler Poisson distribution.

The Poisson distribution can be derived from the binomial distribution. It is the limit distribution of the binomial distribution with very small proportions of the interesting features and a very large sample size: and under the secondary condition that the product assumes a value that is neither zero nor infinite. is then the expectation value for all binomial distributions considered in the limit value formation as well as for the resulting Poisson distribution.

Both the Poisson distribution and the binomial distribution are special cases of the Panjer distribution.

Relationship to the generalized binomial distribution

The generalized binomial distribution can also be approximated for large samples and small success probabilities using the Poisson approximation.

Relationship to normal distribution

The Poisson probabilities for λ = 30 are approximated by a normal distribution density

The Poisson distribution has for small values ​​of a strongly asymmetrical shape. For things that get bigger becomes more symmetrical and resembles from about a Gaussian normal distribution with and :

Relationship to the Erlang distribution

.

Relationship to the chi-square distribution

The distribution functions of the Poisson distribution and the chi-square distribution with Degrees of freedom are related in the following ways:

The probability, or to find more events in an interval within which one averages Expected events is equal to the probability that the value of is. So it applies

.

This follows from With and as regularized gamma functions.

Relationship to the Skellam distribution

Against this is the difference two stochastically independent Poisson-distributed random variables and with the parameters and not again Poisson distributed, but Skellam distributed. The following applies:

,

in which denotes the modified Bessel function.

Further Poisson distributions

Some other distributions are sometimes called "Poisson" and are generalizations of the Poisson distribution described here:

Free Poisson distribution

In free probability theory there is a free analogue to the Poisson distribution, the free Poisson distribution. In analogy to a corresponding limit value theorem for the Poisson distribution, it is used as the limit value of the iterated free convolution For Are defined.

Bivariate Poisson distribution

The bivariate Poisson distribution is defined by

The marginal distributions are Poisson distributed with the parameters and and it applies . The difference is Skellam-distributed with the parameters and .

This means that it is relatively easy to introduce dependencies between Poisson-distributed random variables if one knows or can estimate the mean values ​​of the marginal distributions and the covariance. One can then use the bivariate Poisson distribution simply generate it by taking three independent Poisson random variables defined with parameters and then puts.

The multivariate Poisson distribution can be defined analogously.

Application examples

"Rare" events

The classic example comes from Ladislaus von Bortkewitsch, who, when examining the number of deaths from hoofbashes in the individual cavalry units of the Prussian army per year, was able to prove that these numbers can be well described by a Poisson distribution.

In general, the following conditions must apply to the individual counting events (in the example, the individual deaths from hoofbeats) so that the number is Poisson distributed:

  1. Single events: The probability that two events will occur in a short period of time is negligible.
  2. Proportionality: The probability of observing an event in a short period of time is proportional to the length of the period.
  3. Homogeneity: The probability of observing an event in a short period of time is independent of the location of the period.
  4. Independence: The probability of observing an event in a short period of time is independent of the probability of an event in other non-overlapping periods of time.

Alternatively, these conditions can also be explained by the fact that the waiting time between two events is exponentially distributed. Since this is memoryless, the events occur almost randomly and independently of one another.

It must be checked in each individual case whether the conditions are met, but typical examples are:

  • Number of printing errors on a book page
  • Number of incoming calls per hour in a switchboard
  • Number of radioactive decays of a substance in a given time interval (provided that the decay rate does not decrease noticeably, i.e. the measurement time is short compared to the half-life)
  • Number of lightning strikes per hectare and year
  • Number of vaccination damages per year
  • the bombing of London

According to Palm-Chinchin's theorem, even general renewal processes converge against a Poisson process under relatively mild conditions, i.e. here, too, the Poisson distribution results for the number of events. This means that the conditions given above can still be weakened considerably.

Customer arrivals

In queuing systems, customers or orders arrive in the system to be served. In queuing theory, the different models are described in Kendall notation. In particular, the number of customers who arrive in a certain time interval is often modeled with a Poisson distribution (abbreviated as M. for exponentially distributed inter-arrival times). This modeling is very attractive, since this assumption often results in simple analytical solutions.

Often this assumption can also be justified approximately. Here, an example should be used to illustrate what this assumption means: For example, a department store is entered by a customer every 10 seconds on average on a Saturday. If the new people are counted every minute, then an average of 6 people would be expected to enter the store per minute. The choice of the length of the interval is up to the observer. If one were to choose an hour as the observation interval, this would result , with an interval of 1 second . The relative fluctuation in the number of customers () increases with increasing interval and consequently increasing from. The longer interval allows, in principle, a more precise observation over the longer averaging, but is associated with more effort and cannot record changes in conditions that occur within the interval (e.g. arrival of a bus with tourists willing to shop).

A Poisson distribution could exist under the following boundary conditions:

  1. Customers have to arrive individually. In reality, however, groups of people often arrive together.
  2. The probability that a customer will arrive could be proportional to the length of the observation period.
  3. There are certainly rush hours with increased customer traffic and lulls throughout the day.
  4. The customer arrivals in different time periods are not necessarily independent. E.g. if the department store is overcrowded, customers could be put off.

In this example, the assumption of the Poisson distribution is difficult to justify, so there are queue models, e.g. with group arrivals, finite queues or other arrival distributions, in order to model this arrival process more realistically. Fortunately, some important metrics, such as the average number of customers in the system according to Little's Law, do not depend on the specific distribution, i.e. even if assumptions are violated, the same result applies.

Ball fan model

In the area of ​​counting combinatorics, a standard task is to distribute balls or spheres on compartments and to count how many possibilities there are. If you order the Balls the Fan randomly, you get a binomial distribution for the number of balls in a fixed compartment . One application is, for example, the distribution of raisins on a cake, with the aim that each piece contains a minimum number of raisins.

Grains of rice randomly scattered on the ground.

The picture on the right shows a section of a floor with square tiles on which grains of rice were randomly scattered. The Fields contain each Grains of rice and total are located Grains of rice in the viewed section. The probabilities can now be determined directly via the binomial distribution, but the requirements of the Poisson approximation are also met.

The comparison between experiment and calculated Poisson distribution , in which Grains of rice / squares intuitively shows a good match. Statistically one could check the quality with an adaptation test.

Distribution of the example, counted (blue) and according to Poisson (red)
counted

0

15

12,7

1

15

17,2

2

11

11,6

3

5

5,2

4

1

1,7

5

2

0,5

The probability that a given field will be left blank is around 26%:

Sports results

In many sports, a competition is about getting more counting events than the opponent within a certain period of time. The physicist Metin Tolan extensively examined the applicability of the Poisson distribution in sport in his book on the soccer game.

The (temporal) constancy of the event probability - a sufficient prerequisite for the application of Poisson statistics (see above under Poisson assumptions) - is generally at most approximate for sports results. But if you are only interested in the pure count, e.g. the number of goals scored by a team, a Poisson distribution results even with a time-dependent goal rate. It is more difficult to justify the assumption that is often made that the scores or scores of two teams are independent. If this assumption cannot be adequately substantiated statistically, e.g. by hypothesis or adaptation tests for agreement of the data with the Poisson distribution, one can, for example, switch to the bivariate Poisson distribution and introduce a dependency by estimating the covariance.

Metin Tolan explains that the number of goals a team has scored in a soccer game can be assumed to be a Poisson distribution as a good approximation. In his approach, however, he only takes into account the average number of goals per game and team, i.e. he does not consider the strength of the opposing team, for example. He has also proven that over 70% of the variance in the distribution of points in the Bundesliga can be explained by chance. This also proves from a stochastic point of view why football is exciting.

For the 2015 cup final, for example, based on the previous Bundesliga season, Tolan would have estimated 2.12 goals for VfL Wolfsburg and 1.38 goals for Borussia Dortmund. Andreas Heuer goes one step further and defines the strength of a team as the average goal difference of a team when playing against an average opponent on a neutral pitch. Using the data from the previous Bundesliga season, one would have estimated an average goal difference of 1 for VfL Wolfsburg and 0.15 for Borussia Dortmund. In order to come to a game prognosis, one has to consider the average number of goals per game according to Heuer. For these two teams that would be 2.92 and Heuer would estimate 1.885 goals for VfL Wolfsburg and 1.035 goals for Borussia Dortmund. For seasonal forecasts, Heuer also takes into account other parameters in its complete model, such as home strength, market value or the performance of the teams in the previous seasons. In practice, the final ended with 3 goals for Wolfsburg and one goal for Dortmund.

2/3 law in roulette

The Poisson distribution gives a good estimate of how many different numbers will be hit in 37 roulette games.

Based on an article in Wikipedia.de
 
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Date of the last change: Jena, the: 10.05. 2021