Poisson distribution

In probability theory and statistics, the Poisson distribution (/ˈpwɑːsɒn/; French pronunciation: ​[pwasɔ̃]), 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 mean 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.

Poisson Distribution
Probability mass function
Poisson pmf.svg
The horizontal axis is the index k, the number of occurrences. λ is the expected rate of occurrences. The vertical axis is the probability of k occurrences given λ. The function is defined only at integer values of k; the connecting lines are only guides for the eye.
Cumulative distribution function
Poisson cdf.svg
The horizontal axis is the index k, the number of occurrences. The CDF is discontinuous at the integers of k and flat everywhere else because a variable that is Poisson distributed takes on only integer values.
Parameters (rate)
Support (Natural numbers starting from 0)

, or , or

(for , where is the upper incomplete gamma function, is the floor function, and Q is the regularized gamma function)
Ex. kurtosis

(for large )

Fisher information

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 distribution 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.


Probability mass functionEdit

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

A discrete random variable X is said to have a Poisson distribution with parameter λ > 0, if, for k = 0, 1, 2, ..., the probability mass function of X is given by:[3]:60



The positive real number λ is equal to the expected value of X and also to its variance[4]


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 a time rate for the number of events   to happen. Then   (showing   number of events per unit of time), and



The Poisson distribution may be useful to model events such as

  • The number of meteorites greater than 1 meter diameter that strike Earth in a year
  • The number of patients arriving in an emergency room between 10 and 11 pm
  • The number of laser photons hitting a detector in a particular time interval

Assumptions and validityEdit

The Poisson distribution is an appropriate model if the following assumptions are true:[5]

  • k is the number of times an event occurs in an interval and k can take values 0, 1, 2, ....
  • The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
  • The average rate at which events occur is independent of any occurrences. For simplicity, this is usually assumed to be constant, but may in practice vary with time.
  • Two events cannot occur at exactly the same instant; instead, at each very small sub-interval exactly one event either occurs or does not occur.

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

The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial equals λ divided by the number of trials, as the number of trials approaches infinity (see Related distributions).

Examples of probability for Poisson distributionsEdit

Once in an interval events: The special case of λ = 1 and k = 0Edit

Suppose that astronomers estimate that large meteorites (above a certain size) hit the earth on average once every 100 years (λ = 1 event per 100 years), and that the number of meteorite hits follows a Poisson distribution. What is the probability of k = 0 meteorite hits in the next 100 years?


Under these assumptions, the probability that no large meteorites hit the earth in the next 100 years is roughly 0.37. The remaining 1 − 0.37 = 0.63 is the probability of 1, 2, 3, or more large meteorite hits in the next 100 years. In an example above, an overflow flood occurred once every 100 years (λ = 1). The probability of no overflow floods in 100 years was roughly 0.37, by the same calculation.

In general, if an event occurs on average once per interval (λ = 1), and the events follow a Poisson distribution, then P(0 events in next interval) = 0.37. In addition, P(exactly one event in next interval) = 0.37, as shown in the table for overflow floods.

Examples that violate the Poisson assumptionsEdit

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 (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups).

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 Poission 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.


Descriptive statisticsEdit

  • The mode of a Poisson-distributed random variable with non-integer λ is equal to  , which is the largest integer less than or equal to λ. This is also written as floor(λ). When λ is a positive integer, the modes are λ and λ − 1.
  • All of the cumulants of the Poisson distribution are equal to the expected value λ. The nth factorial moment of the Poisson distribution is λn.
  • The expected value of a Poisson process is sometimes decomposed into the product of intensity and exposure (or more generally expressed as the integral of an "intensity function" over time or space, sometimes described as “exposure”).[8]


Bounds for the median ( ) of the distribution are known and are sharp:[9]


Higher momentsEdit

where the {braces} denote Stirling numbers of the second kind.[10][1]:6 The coefficients of the polynomials have a combinatorial meaning. In fact, when the expected value of the Poisson distribution is 1, then Dobinski's formula says that the nth moment equals the number of partitions of a set of size n.

For the non-centered moments we define  , then[11]


where   is some absolute constant greater than 0.

Sums of Poisson-distributed random variablesEdit

If   for   are independent, then  .[12]:65 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.[13][14]

Other propertiesEdit

  • The Poisson distributions are infinitely divisible probability distributions.[15]:233[7]:164
  • The directed Kullback–Leibler divergence of   from   is given by
  • Bounds for the tail probabilities of a Poisson random variable   can be derived using a Chernoff bound argument.[16]:97-98
  • The upper tail probability can be tightened (by a factor of at least two) as follows:[17]
where   is the directed Kullback–Leibler divergence, as described above.
  • Inequalities that relate the distribution function of a Poisson random variable   to the Standard normal distribution function   are as follows:[17]
where   is again the directed Kullback–Leibler divergence.

Poisson racesEdit

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 (See the entry on bounds on tails of binomial distributions for details). 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..[18]

Related distributionsEdit


  • If   and   are independent, then the difference   follows a Skellam distribution.
  • If   and   are independent, then the distribution of   conditional on   is a binomial distribution.
Specifically, if  , then  .
More generally, if X1, X2,..., Xn are independent Poisson random variables with parameters λ1, λ2,..., λn then
given    . In fact,  .
  • If   and the distribution of  , conditional on X = k, is a binomial distribution,  , then the distribution of Y follows a Poisson distribution  . In fact, if  , conditional on X = k, follows a multinomial distribution,  , then each   follows an independent Poisson distribution  .
  • 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 n is sufficiently large and p is sufficiently small. There is a rule of thumb stating that the Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and np ≤ 10.[19]
  • The Poisson distribution is a special case of the discrete compound Poisson distribution (or stuttering Poisson distribution) with only a parameter.[20][21] The discrete compound Poisson distribution can be deduced from the limiting distribution of univariate multinomial distribution. It is also a special case of a compound Poisson distribution.
  • For sufficiently large values of λ, (say λ>1000), the normal distribution with mean λ and variance λ (standard deviation  ) is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate continuity correction is performed, i.e., if P(X ≤ x), where x is a non-negative integer, is replaced by P(X ≤ x + 0.5).
Under this transformation, the convergence to normality (as   increases) is far faster than the untransformed variable.[citation needed] Other, slightly more complicated, variance stabilizing transformations are available,[7]:168 one of which is Anscombe transform.[23] See Data transformation (statistics) for more general uses of transformations.
  • If for every t > 0 the number of arrivals in the time interval [0, t] follows the Poisson distribution with mean λt, then the sequence of inter-arrival times are independent and identically distributed exponential random variables having mean 1/λ.[24]:317–319
  • The cumulative distribution functions of the Poisson and chi-squared distributions are related in the following ways:[7]:167

Poisson ApproximationEdit

Assume   where  , then[25]   is multinomially distributed   conditioned on  .

This means[16]:101-102, among other things, that for any nonnegative function  , if   is multinomially distributed, then


where  .

The factor of   can be removed if   is further assumed to be monotonically increasing or decreasing.

Bivariate Poisson distributionEdit

This distribution has been extended to the bivariate case.[26] The generating function for this distribution is




The marginal distributions are Poisson(θ1) and Poisson(θ2) and the correlation coefficient is limited to the range


A simple way to generate a bivariate Poisson distribution   is to take three independent Poisson distributions   with means   and then set  . The probability function of the bivariate Poisson distribution is


Free Poisson distributionEdit

The free Poisson distribution[27] with jump size   and rate   arises in free probability theory as the limit of repeated free convolution


as N → ∞.

In other words, let   be random variables so that   has value   with probability   and value 0 with the remaining probability. Assume also that the family   are freely independent. Then the limit as   of the law of   is given by the Free Poisson law with parameters  .

This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process.

The measure associated to the free Poisson law is given by[28]




and has support  .

This law also arises in random matrix theory as the Marchenko–Pastur law. Its free cumulants are equal to  .

Some transforms of this lawEdit

We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher[29]

The R-transform of the free Poisson law is given by


The Cauchy transform (which is the negative of the Stieltjes transformation) is given by


The S-transform is given by


in the case that  .

Statistical InferenceEdit

Parameter estimationEdit

Given a sample of n measured values  , for i = 1, ..., n, we wish to estimate the value of the parameter λ of the Poisson population from which the sample was drawn. The maximum likelihood estimate is [30]


Since each observation has expectation λ so does the sample mean. Therefore, the maximum likelihood estimate is an unbiased estimator of λ. It is also an efficient estimator since its variance achieves the Cramér–Rao lower bound (CRLB).[citation needed] Hence it is minimum-variance unbiased. Also it can be proven that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for λ.

To prove sufficiency we may use the factorization theorem. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one that depends solely on the sample   (called  ) and one that depends on the parameter   and the sample   only through the function  . Then   is a sufficient statistic for  .


The first term,  , depends only on  . The second term,  , depends on the sample only through  . Thus,   is sufficient.

To find the parameter λ that maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function:


We take the derivative of   with respect to λ and compare it to zero:


Solving for λ gives a stationary point.


So λ is the average of the ki values. Obtaining the sign of the second derivative of L at the stationary point will determine what kind of extreme value λ is.


Evaluating the second derivative at the stationary point gives:


which is the negative of n times the reciprocal of the average of the ki. This expression is negative when the average is positive. If this is satisfied, then the stationary point maximizes the probability function.

For completeness, a family of distributions is said to be complete if and only if   implies that   for all  . If the individual   are iid  , then  . Knowing the distribution we want to investigate, it is easy to see that the statistic is complete.


For this equality to hold,   must be 0. This follows from the fact that none of the other terms will be 0 for all   in the sum and for all possible values of  . Hence,   for all   implies that  , and the statistic has been shown to be complete.

Confidence intervalEdit

The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution, and this leads to an alternative expression. Given an observation k from a Poisson distribution with mean μ, a confidence interval for μ with confidence level 1 – α is


or equivalently,


where   is the quantile function (corresponding to a lower tail area p) of the chi-squared distribution with n degrees of freedom and   is the quantile function of a gamma distribution with shape parameter n and scale parameter 1.[7]:176-178[31] This interval is 'exact' in the sense that its coverage probability is never less than the nominal 1 – α.

When quantiles of the gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the Wilson–Hilferty transformation):[32]


where   denotes the standard normal deviate with upper tail area α / 2.

For application of these formulae in the same context as above (given a sample of n measured values ki each drawn from a Poisson distribution with mean λ), one would set


calculate an interval for μ = , and then derive the interval for λ.

Bayesian inferenceEdit

In Bayesian inference, the conjugate prior for the rate parameter λ of the Poisson distribution is the gamma distribution.[33] Let


denote that λ is distributed according to the gamma density g parameterized in terms of a shape parameter α and an inverse scale parameter β:


Then, given the same sample of n measured values ki as before, and a prior of Gamma(α, β), the posterior distribution is


The posterior mean E[λ] approaches the maximum likelihood estimate   in the limit as  , which follows immediately from the general expression of the mean of the gamma distribution.

The posterior predictive distribution for a single additional observation is a negative binomial distribution,[34]:53 sometimes called a gamma–Poisson distribution.

Simultaneous estimation of multiple Poisson meansEdit

Suppose   is a set of independent random variables from a set of   Poisson distributions, each with a parameter  ,  , and we would like to estimate these parameters. Then, Clevenson and Zidek show that under the normalized squared error loss  , when  , then, similar as in Stein's example for the Normal means, the MLE estimator   is inadmissible. [35]

In this case, a family of minimax estimators is given for any   and   as[36]


Occurrence and applicationsEdit

Applications of the Poisson distribution can be found in many fields including:[37]

The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete properties (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or space. Examples of events that may be modelled as a Poisson distribution include:

  • The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry. This example was used in a book by Ladislaus Bortkiewicz (1868–1931).[40]:23-25
  • The number of yeast cells used when brewing Guinness beer. This example was used by William Sealy Gosset (1876–1937).[41][42]
  • The number of phone calls arriving at a call centre within a minute. This example was described by A.K. Erlang (1878–1929).[43]
  • Internet traffic.
  • The number of goals in sports involving two competing teams.[44]
  • The number of deaths per year in a given age group.
  • The number of jumps in a stock price in a given time interval.
  • Under an assumption of homogeneity, the number of times a web server is accessed per minute.
  • The number of mutations in a given stretch of DNA after a certain amount of radiation.
  • The proportion of cells that will be infected at a given multiplicity of infection.
  • The number of bacteria in a certain amount of liquid.[45]
  • The arrival of photons on a pixel circuit at a given illumination and over a given time period.
  • The targeting of V-1 flying bombs on London during World War II investigated by R. D. Clarke in 1946.[46]

Gallagher showed in 1976 that the counts of prime numbers in short intervals obey a Poisson distribution[47] provided a certain version of the unproved prime r-tuple conjecture of Hardy-Littlewood[48] is true.

Law of rare eventsEdit

Comparison of the Poisson distribution (black lines) and the binomial distribution with n = 10 (red circles), n = 20 (blue circles), n = 1000 (green circles). All distributions have a mean of 5. The horizontal axis shows the number of events k. As n gets larger, the Poisson distribution becomes an increasingly better approximation for the binomial distribution with the same mean.

The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval. Let this total number be  . Divide the whole interval into   subintervals   of equal size, such that   >   (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in an interval   for each   is equal to  . Now we assume that the occurrence of an event in the whole interval can be seen as a Bernoulli trial, where the   trial corresponds to looking whether an event happens at the subinterval   with probability  . The expected number of total events in   such trials would be  , the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form  . As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as   goes to infinity. In this case the binomial distribution converges to what is known as the Poisson distribution by the Poisson limit theorem.

In several of the above examples—such as, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the binomial distribution, that is


In such cases n is very large and p is very small (and so the expectation np is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution[citation needed]


This approximation is sometimes known as the law of rare events,[49]:5since each of the n individual Bernoulli events rarely occurs. The name may be misleading because the total count of success events in a Poisson process need not be rare if the parameter np is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour.

The word law is sometimes used as a synonym of probability distribution, and convergence in law means convergence in distribution. Accordingly, the Poisson distribution is sometimes called the "law of small numbers" because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. The Law of Small Numbers is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898.[40][50]

Poisson point processEdit

The Poisson distribution arises as the number of points of a Poisson point process located in some finite region. More specifically, if D is some region space, for example Euclidean space Rd, for which |D|, the area, volume or, more generally, the Lebesgue measure of the region is finite, and if N(D) denotes the number of points in D, then


Poisson regression and negative binomial regressionEdit

Poisson regression and negative binomial regression are useful for analyses where the dependent (response) variable is the count (0, 1, 2, ...) of the number of events or occurrences in an interval.

Other applications in scienceEdit

In a Poisson process, the number of observed occurrences fluctuates about its mean λ with a standard deviation  . These fluctuations are denoted as Poisson noise or (particularly in electronics) as shot noise.

The correlation of the mean and standard deviation in counting independent discrete occurrences is useful scientifically. By monitoring how the fluctuations vary with the mean signal, one can estimate the contribution of a single occurrence, even if that contribution is too small to be detected directly. For example, the charge e on an electron can be estimated by correlating the magnitude of an electric current with its shot noise. If N electrons pass a point in a given time t on the average, the mean current is  ; since the current fluctuations should be of the order   (i.e., the standard deviation of the Poisson process), the charge   can be estimated from the ratio  .[citation needed]

An everyday example is the graininess that appears as photographs are enlarged; the graininess is due to Poisson fluctuations in the number of reduced silver grains, not to the individual grains themselves. By correlating the graininess with the degree of enlargement, one can estimate the contribution of an individual grain (which is otherwise too small to be seen unaided).[citation needed] Many other molecular applications of Poisson noise have been developed, e.g., estimating the number density of receptor molecules in a cell membrane.


In Causal Set theory the discrete elements of spacetime follow a Poisson distribution in the volume.

Computational methodsEdit

The Poisson distribution poses two different tasks for dedicated software libraries: Evaluating the distribution  , and drawing random numbers according to that distribution.

Evaluating the Poisson distributionEdit

Computing   for given   and   is a trivial task that can be accomplished by using the standard definition of   in terms of exponential, power, and factorial functions. However, the conventional definition of the Poisson distribution contains two terms that can easily overflow on computers: λk and k!. The fraction of λk to k! can also produce a rounding error that is very large compared to e−λ, and therefore give an erroneous result. For numerical stability the Poisson probability mass function should therefore be evaluated as


which is mathematically equivalent but numerically stable. The natural logarithm of the Gamma function can be obtained using the lgamma function in the C standard library (C99 version) or R, the gammaln function in MATLAB or SciPy, or the log_gamma function in Fortran 2008 and later.

Some computing languages provide built-in functions to evaluate the Poisson distribution, namely

  • R: function dpois(x, lambda);
  • Excel: function POISSON( x, mean, cumulative), with a flag to specify the cumulative distribution;
  • Mathematica: univariate Poisson distribution as PoissonDistribution[ ],[51] bivariate Poisson distribution as MultivariatePoissonDistribution[ ,{  ,  }],.[52]

Random drawing from the Poisson distributionEdit

The less trivial task is to draw random integers from the Poisson distribution with given  .

Solutions are provided by:

Generating Poisson-distributed random variablesEdit

A simple algorithm to generate random Poisson-distributed numbers (pseudo-random number sampling) has been given by Knuth:[53]:137-138

algorithm poisson random number (Knuth):
        Let L ← e−λ, k ← 0 and p ← 1.
        k ← k + 1.
        Generate uniform random number u in [0,1] and let p ← p × u.
    while p > L.
    return k − 1.

The complexity is linear in the returned value k, which is λ on average. There are many other algorithms to improve this. Some are given in Ahrens & Dieter, see § References below.

For large values of λ, the value of L = e−λ may be so small that it is hard to represent. This can be solved by a change to the algorithm which uses an additional parameter STEP such that e−STEP does not underflow:[citation needed]

algorithm poisson random number (Junhao, based on Knuth):
        Let λLeft ← λ, k ← 0 and p ← 1.
        k ← k + 1.
        Generate uniform random number u in (0,1) and let p ← p × u.
        while p < 1 and λLeft > 0:
            if λLeft > STEP:
                p ← p × eSTEP
                λLeft ← λLeft − STEP
                p ← p × eλLeft
                λLeft ← 0
    while p > 1.
    return k − 1.

The choice of STEP depends on the threshold of overflow. For double precision floating point format, the threshold is near e700, so 500 shall be a safe STEP.

Other solutions for large values of λ include rejection sampling and using Gaussian approximation.

Inverse transform sampling is simple and efficient for small values of λ, and requires only one uniform random number u per sample. Cumulative probabilities are examined in turn until one exceeds u.

algorithm Poisson generator based upon the inversion by sequential search:[54]:505
        Let x ← 0, p ← e−λ, s ← p.
        Generate uniform random number u in [0,1].
    while u > s do:
        x ← x + 1.
        p ← p × λ / x.
        s ← s + p.
    return x.


The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile(1837).[55]:205-207 The work theorized about the number of wrongful convictions in a given country by focusing on certain random variables N that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a time-interval of given length. The result had already been given in 1711 by Abraham de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus .[56]:219[57]:14-15[58]:193[7]:157 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.[59][60]

In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space.[61] A further practical application of this distribution was made by Ladislaus Bortkiewicz in 1898 when he was given the task of investigating the number of soldiers in the Prussian army killed accidentally by horse kicks;[40]:23-25 this experiment introduced the Poisson distribution to the field of reliability engineering.

See alsoEdit



  1. ^ a b Haight, Frank A. (1967), Handbook of the Poisson Distribution, New York, NY, USA: John Wiley & Sons, ISBN 978-0-471-33932-8
  2. ^ Brooks, E. Bruce (2007-08-24), Statistics | The Poisson Distribution, Warring States Project, Umass.edu, retrieved 2014-04-18
  3. ^ Yates, Roy D.; Goodman, David J. (2014), Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers (2nd ed.), Hoboken, USA: Wiley, ISBN 978-0-471-45259-1
  4. ^ For the proof, see : Proof wiki: expectation and Proof wiki: variance
  5. ^ Koehrsen, William (2019-01-20), The Poisson Distribution and Poisson Process Explained, Towards Data Science, retrieved 2019-09-19
  6. ^ Ugarte, Maria Dolores; Militino, Ana F.; Arnholt, Alan T. (2016), Probability and Statistics with R (Second ed.), Boca Raton, FL, USA: CRC Press, ISBN 978-1-4665-0439-4
  7. ^ a b c d e f g h i Johnson, Norman L.; Kemp, Adrienne W.; Kotz, Samuel (2005), "Poisson Distribution", Univariate Discrete Distributions (3rd ed.), New York, NY, USA: John Wiley & Sons, Inc., pp. 156–207, doi:10.1002/0471715816, ISBN 978-0-471-27246-5
  8. ^ Helske, Jouni (2017). "KFAS: Exponential family state space models in R". arXiv:1612.01907 [stat.CO].
  9. ^ Choi, Kwok P. (1994), "On the medians of gamma distributions and an equation of Ramanujan", Proceedings of the American Mathematical Society, 121 (1): 245–251, doi:10.2307/2160389, JSTOR 2160389
  10. ^ Riordan, John (1937), "Moment Recurrence Relations for Binomial, Poisson and Hypergeometric Frequency Distributions" (PDF), Annals of Mathematical Statistics, 8 (2): 103–111, doi:10.1214/aoms/1177732430, JSTOR 2957598
  11. ^ Jagadeesan, Meena (2017). "Simple analysis of sparse, sign-consistent JL". arXiv:1708.02966 [cs.DS].
  12. ^ Lehmann, Erich Leo (1986), Testing Statistical Hypotheses (second ed.), New York, NJ, USA: Springer Verlag, ISBN 978-0-387-94919-2
  13. ^ Raikov, Dmitry (1937), "On the decomposition of Poisson laws", Comptes Rendus de l'Académie des Sciences de l'URSS, 14: 9–11
  14. ^ von Mises, Richard (1964), Mathematical Theory of Probability and Statistics, New York, NJ, USA: Academic Press, doi:10.1016/C2013-0-12460-9, ISBN 978-1-4832-3213-3
  15. ^ Laha, Radha G.; Rohatgi, Vijay K. (1979), Probability Theory, New York, NJ, USA: John Wiley & Sons, ISBN 978-0-471-03262-5
  16. ^ a b Mitzenmacher, Michael; Upfal, Eli (2005), Probability and Computing: Randomized Algorithms and Probabilistic Analysis, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-83540-4
  17. ^ a b Short, Michael (2013), "Improved Inequalities for the Poisson and Binomial Distribution and Upper Tail Quantile Functions", ISRN Probability and Statistics, 2013: 412958, doi:10.1155/2013/412958
  18. ^ Kamath, Govinda M.; Şaşoğlu, Eren; Tse, David (2015), "Optimal Haplotype Assembly from High-Throughput Mate-Pair Reads", 2015 IEEE International Symposium on Information Theory (ISIT), 14–19 June, Hong Kong, China, pp. 914–918, arXiv:1502.01975, doi:10.1109/ISIT.2015.7282588, S2CID 128634
  19. ^ Prins, Jack (2012), " Counts Control Charts", e-Handbook of Statistical Methods, NIST/SEMATECH, retrieved 2019-09-20
  20. ^ Zhang, Huiming; Liu, Yunxiao; Li, Bo (2014), "Notes on discrete compound Poisson model with applications to risk theory", Insurance: Mathematics and Economics, 59: 325–336, doi:10.1016/j.insmatheco.2014.09.012
  21. ^ Zhang, Huiming; Li, Bo (2016), "Characterizations of discrete compound Poisson distributions", Communications in Statistics - Theory and Methods, 45 (22): 6789–6802, doi:10.1080/03610926.2014.901375, S2CID 125475756
  22. ^ McCullagh, Peter; Nelder, John (1989), Generalized Linear Models, Monographs on Statistics and Applied Probability, 37, London, UK: Chapman and Hall, ISBN 978-0-412-31760-6
  23. ^ Anscombe, Francis J. (1948), "The transformation of Poisson, binomial and negative binomial data", Biometrika, 35 (3–4): 246–254, doi:10.1093/biomet/35.3-4.246, JSTOR 2332343
  24. ^ Ross, Sheldon M. (2010), Introduction to Probability Models (tenth ed.), Boston, MA, USA: Academic Press, ISBN 978-0-12-375686-2
  25. ^ "1.7.7 – Relationship between the Multinomial and Poisson | STAT 504".
  26. ^ Loukas, Sotirios; Kemp, C. David (1986), "The Index of Dispersion Test for the Bivariate Poisson Distribution", Biometrics, 42 (4): 941–948, doi:10.2307/2530708, JSTOR 2530708
  27. ^ Free Random Variables by D. Voiculescu, K. Dykema, A. Nica, CRM Monograph Series, American Mathematical Society, Providence RI, 1992
  28. ^ James A. Mingo, Roland Speicher: Free Probability and Random Matrices. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.
  29. ^ Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher, pp. 203–204, Cambridge Univ. Press 2006
  30. ^ Paszek, Ewa. "Maximum Likelihood Estimation – Examples".
  31. ^ Garwood, Frank (1936), "Fiducial Limits for the Poisson Distribution", Biometrika, 28 (3/4): 437–442, doi:10.1093/biomet/28.3-4.437, JSTOR 2333958
  32. ^ Breslow, Norman E.; Day, Nick E. (1987), Statistical Methods in Cancer Research: Volume 2—The Design and Analysis of Cohort Studies, Lyon, France: International Agency for Research on Cancer, ISBN 978-92-832-0182-3, archived from the original on 2018-08-08, retrieved 2012-03-11
  33. ^ Fink, Daniel (1997), A Compendium of Conjugate Priors
  34. ^ Gelman; Carlin, John B.; Stern, Hal S.; Rubin, Donald B. (2003), Bayesian Data Analysis (2nd ed.), Boca Raton, FL, USA: Chapman & Hall/CRC, ISBN 1-58488-388-X
  35. ^ Clevenson, M. Lawrence; Zidek, James V. (1975), "Simultaneous Estimation of the Means of Independent Poisson Laws", Journal of the American Statistical Association, 70 (351): 698–705, doi:10.1080/01621459.1975.10482497, JSTOR 2285958
  36. ^ Berger, James O. (1985), Statistical Decision Theory and Bayesian Analysis, Springer Series in Statistics (2nd ed.), New York, NJ, USA: Springer-Verlag, doi:10.1007/978-1-4757-4286-2, ISBN 978-0-387-96098-2
  37. ^ Rasch, Georg (1963), "The Poisson Process as a Model for a Diversity of Behavioural Phenomena" (PDF), 17th International Congress of Psychology, 2, Washington, DC, USA, August 20th – 26th, 1963: American Psychological Association, doi:10.1037/e685262012-108CS1 maint: location (link)
  38. ^ Flory, Paul J. (1940), "Molecular Size Distribution in Ethylene Oxide Polymers", Journal of the American Chemical Society, 62 (6): 1561–1565, doi:10.1021/ja01863a066
  39. ^ Lomnitz, Cinna (1994), Fundamentals of Earthquake Prediction, New York: John Wiley & Sons, ISBN 0-471-57419-8, OCLC 647404423
  40. ^ a b c von Bortkiewitsch, Ladislaus (1898), Das Gesetz der kleinen Zahlen [The law of small numbers] (in German), Leipzig, Germany: B. G. Teubner, p. On page 1, Bortkiewicz presents the Poisson distribution. On pages 23–25, Bortkiewitsch presents his analysis of "4. Beispiel: Die durch Schlag eines Pferdes im preußischen Heere Getöteten." (4. Example: Those killed in the Prussian army by a horse's kick.)
  41. ^ Student (1907), "On the Error of Counting with a Haemacytometer", Biometrika, 5 (3): 351–360, doi:10.2307/2331633, JSTOR 2331633
  42. ^ Boland, Philip J. (1984), "A Biographical Glimpse of William Sealy Gosset", The American Statistician, 38 (3): 179–183, doi:10.1080/00031305.1984.10483195, JSTOR 2683648
  43. ^ Erlang, Agner K. (1909), "Sandsynlighedsregning og Telefonsamtaler" [Probability Calculation and Telephone Conversations], Nyt Tidsskrift for Matematik (in Danish), 20 (B): 33–39, JSTOR 24528622
  44. ^ Hornby, Dave (2014), Football Prediction Model: Poisson Distribution, Sports Betting Online, retrieved 2014-09-19
  45. ^ Koyama, Kento; Hokunan, Hidekazu; Hasegawa, Mayumi; Kawamura, Shuso; Koseki, Shigenobu (2016), "Do bacterial cell numbers follow a theoretical Poisson distribution? Comparison of experimentally obtained numbers of single cells with random number generation via computer simulation", Food Microbiology, 60: 49–53, doi:10.1016/j.fm.2016.05.019, PMID 27554145
  46. ^ Clarke, R. D. (1946), "An application of the Poisson distribution" (PDF), Journal of the Institute of Actuaries, 72 (3): 481, doi:10.1017/S0020268100035435
  47. ^ Gallagher, Patrick X. (1976), "On the distribution of primes in short intervals", Mathematika, 23 (1): 4–9, doi:10.1112/s0025579300016442
  48. ^ Hardy, Godfrey H.; Littlewood, John E. (1923), "On some problems of "partitio numerorum" III: On the expression of a number as a sum of primes", Acta Mathematica, 44: 1–70, doi:10.1007/BF02403921
  49. ^ Cameron, A. Colin; Trivedi, Pravin K. (1998), Regression Analysis of Count Data, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-63567-7
  50. ^ Edgeworth, Francis Y. (1913), "On the use of the theory of probabilities in statistics relating to society", Journal of the Royal Statistical Society, 76 (2): 165–193, doi:10.2307/2340091, JSTOR 2340091
  51. ^ "Wolfram Language: PoissonDistribution reference page". wolfram.com. Retrieved 2016-04-08.
  52. ^ "Wolfram Language: MultivariatePoissonDistribution reference page". wolfram.com. Retrieved 2016-04-08.
  53. ^ Knuth, Donald Ervin (1997), Seminumerical Algorithms, The Art of Computer Programming, 2 (3rd ed.), Addison Wesley, ISBN 978-0-201-89684-8
  54. ^ Devroye, Luc (1986), "Discrete Univariate Distributions" (PDF), Non-Uniform Random Variate Generation, New York, NJ, USA: Springer-Verlag, pp. 485–553, doi:10.1007/978-1-4613-8643-8_10, ISBN 978-1-4613-8645-2
  55. ^ Poisson, Siméon D. (1837), Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés [Research on the Probability of Judgments in Criminal and Civil Matters] (in French), Paris, France: Bachelier
  56. ^ de Moivre, Abraham (1711), "De mensura sortis, seu, de probabilitate eventuum in ludis a casu fortuito pendentibus" [On the Measurement of Chance, or, on the Probability of Events in Games Depending Upon Fortuitous Chance], Philosophical Transactions of the Royal Society (in Latin), 27 (329): 213–264, doi:10.1098/rstl.1710.0018
  57. ^ de Moivre, Abraham (1718), The Doctrine of Chances: Or, A Method of Calculating the Probability of Events in Play, London, Great Britain: W. Pearson
  58. ^ de Moivre, Abraham (1721), "Of the Laws of Chance", in Motte, Benjamin (ed.), The Philosophical Transactions from the Year MDCC (where Mr. Lowthorp Ends) to the Year MDCCXX. Abridg'd, and Dispos'd Under General Heads (in Latin), Vol. I, London, Great Britain: R. Wilkin, R. Robinson, S. Ballard, W. and J. Innys, and J. Osborn, pp. 190–219
  59. ^ Stigler, Stephen M. (1982), "Poisson on the Poisson Distribution", Statistics & Probability Letters, 1 (1): 33–35, doi:10.1016/0167-7152(82)90010-4
  60. ^ Hald, Anders; de Moivre, Abraham; McClintock, Bruce (1984), "A. de Moivre: 'De Mensura Sortis' or 'On the Measurement of Chance'", International Statistical Review / Revue Internationale de Statistique, 52 (3): 229–262, doi:10.2307/1403045, JSTOR 1403045
  61. ^ Newcomb, Simon (1860), "Notes on the theory of probabilities", The Mathematical Monthly, 2 (4): 134–140