Poisson distribution
From Wikipedia, the free encyclopedia
Probability mass function The horizontal axis is the index k. The function is only defined at integer values of k (empty lozenges). The connecting lines are only guides for the eye. 

Cumulative distribution function The horizontal axis is the index k. The CDF is discontinuous at the integers of k and flat everywhere else because a variable that is Poisson distributed only takes on integer values. 

Parameters  

Support  
Probability mass function (pmf)  
Cumulative distribution function (cdf)  for or
(where is the Incomplete gamma function and is the floor function) 
Mean  
Median  
Mode  and λ − 1 if λ is an integer 
Variance  
Skewness  
Excess kurtosis  
Entropy 
(for large λ) 
Momentgenerating function (mgf)  
Characteristic function 
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
The distribution was discovered by SiméonDenis Poisson (1781–1840) and published, together with his probability theory, in 1838 in his work Recherches sur la probabilité des jugements en matières criminelles et matière civile ("Research on the Probability of Judgments in Criminal and Civil Matters"). The work focused on certain random variables N that count, among other things, the number of discrete occurrences (sometimes called "arrivals") that take place during a timeinterval of given length. If the expected number of occurrences in this interval is λ, then the probability that there are exactly k occurrences (k being a nonnegative integer, k = 0, 1, 2, ...) is equal to
where
 e is the base of the natural logarithm (e = 2.71828...)
 k is the number of occurrences of an event  the probability of which is given by the function
 k! is the factorial of k
 λ is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average 4 times per minute, and you are interested in the number of events occurring in a 10 minute interval, you would use as your model a Poisson distribution with λ = 10×4 = 40.
As a function of k, this is the probability mass function. The Poisson distribution can be derived as a limiting case of the binomial distribution.
The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. A classic example is the nuclear decay of atoms.
The Poisson distribution is sometimes called a Poissonian, analogous to the term Gaussian for a Gauss or normal distribution.
Contents 
[edit] Poisson noise and characterizing small occurrences
The parameter λ is not only the mean number of occurrences , but also its variance (see Table). Thus, 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 I = eN / t; since the current fluctuations should be of the order (i.e. the standard deviation of the Poisson process), the charge e can be estimated from the ratio . 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). Many other molecular applications of Poisson noise have been developed, e.g., estimating the number density of receptor molecules in a cell membrane.
[edit] Related distributions
 If and then the difference Y = X_{1} − X_{2} follows a Skellam distribution.
 If and are independent, and Y = X_{1} + X_{2}, then the distribution of X_{1} conditional on Y = y is a binomial. Specifically, . More generally, if X_{1}, X_{2},..., X_{n} are independent Poisson random variables with parameters λ_{1}, λ_{2},..., λ_{n} then
 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. 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. According to this rule the approximation is excellent if n ≥ 100 and np ≤ 10.^{[1]}
 For sufficiently large values of λ, (say λ>1000), the normal distribution with mean λ, and variance λ, 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., P(X ≤ x), where (lowercase) x is a nonnegative integer, is replaced by P(X ≤ x + 0.5).
 Variance stabilizing transformation: When a variable is Poisson distributed, its square root is approximately normally distributed with expected value of about and variance of about 1/4.^{[2]} Under this transformation, the convergence to normality is far faster than the untransformed variable. Other, slightly more complicated, variance stabilizing transformations are available,^{[3]} one of which is Anscombe transform. See Data transformation (statistics) for more general uses of transformations.
 If the number of arrivals in a given time interval [0,t] follows the Poisson distribution, with mean = λt, then the lengths of the interarrival times follow the Exponential distribution, with mean 1 / λ.
[edit] Occurrence
The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete nature (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 horsekicks each year in each corps in the Prussian cavalry. This example was made famous by a book of Ladislaus Josephovich Bortkiewicz (1868–1931).
 The number of phone calls at a call centre per minute.
 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.
[edit] How does this distribution arise? — The law of rare events
In several of the above examples—for example, 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. However, the binomial distribution with parameters n and λ/n, i.e., the probability distribution of the number of successes in n trials, with probability λ/n of success on each trial, approaches the Poisson distribution with expected value λ as n approaches infinity. This provides a means by which to approximate random variables using the Poisson distribution rather than the morecumbersome binomial distribution.
This limit is sometimes known as the law of rare events, since each of the individual Bernoulli events rarely triggers. The name may be misleading because the total count of success events in a Poisson process need not be rare if the parameter λ 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 the average member of the population who is very unlikely to make a call to that switchboard in that hour.
The proof may proceed as follows. First, recall from calculus
and the definition of the Binomial distribution
If the binomial probability can be defined such that p = λ / n, we can evaluate the limit of P as n goes large:
The F term can be written as
and then note that, since k is fixed, this is a rational function of n with limit 1.
Consequently, the limit of the distribution becomes
which now assumes the Poisson distribution.
More generally, whenever a sequence of independent binomial random variables with parameters n and p_{n} is such that
the sequence converges in distribution to a Poisson random variable with mean λ (see, e.g. law of rare events).
[edit] Properties
 The expected value of a Poissondistributed random variable is equal to λ and so is its variance. The higher moments of the Poisson distribution are Touchard polynomials in λ, whose coefficients 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.
 The mode of a Poissondistributed random variable with noninteger 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.
 Sums of Poissondistributed random variables:
 If follow a Poisson distribution with parameter and X_{i} are independent, then
 also follows a Poisson distribution whose parameter is the sum of the component parameters.
 The momentgenerating function of the Poisson distribution with expected value λ is
 All of the cumulants of the Poisson distribution are equal to the expected value λ. The nth factorial moment of the Poisson distribution is λ^{n}.
 The Poisson distributions are infinitely divisible probability distributions.
 The directed KullbackLeibler divergence between Pois(λ) and Pois(λ_{0}) is given by
[edit] Generating Poissondistributed random variables
A simple way to generate random Poissondistributed numbers is given by Knuth, see References below.
algorithm poisson random number (Knuth): init: Let L ← e^{−λ}, k ← 0 and p ← 1. do: k ← k + 1. Generate uniform random number u in [0,1] and let p ← p × u. while p ≥ L. return k − 1.
While simple, the complexity is linear in λ. There are many other algorithms to overcome this. Some are given in Ahrens & Dieter, see References below.
[edit] Parameter estimation
[edit] Maximum likelihood
Given a sample of n measured values k_{i} we wish to estimate the value of the parameter λ of the Poisson population from which the sample was drawn. To calculate the maximum likelihood value, we form the loglikelihood function
Take the derivative of L with respect to λ and equate it to zero:
Solving for λ yields the maximumlikelihood estimate of λ:
Since each observation has expectation λ so does this sample mean. Therefore it is an unbiased estimator of λ. It is also an efficient estimator, i.e. its estimation variance achieves the CramérRao lower bound (CRLB).
[edit] Bayesian inference
In Bayesian inference, the conjugate prior for the rate parameter λ of the Poisson distribution is the Gamma distribution. 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 k_{i} 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 .
The posterior predictive distribution of additional data is a GammaPoisson (i.e. negative binomial) distribution.
[edit] The "law of small numbers"
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. Some historians of mathematics have argued that the Poisson distribution should have been called the Bortkiewicz distribution.^{[4]}
[edit] See also
 Compound Poisson distribution
 Tweedie distributions
 Poisson process
 Poisson regression
 Poisson sampling
 Queueing theory
 Erlang distribution which describes the waiting time until n events have occurred. For temporally distributed events, the Poisson distribution is the probability distribution of the number of events that would occur within a preset time, the Erlang distribution is the probability distribution of the amount of time until the nth event.
 Skellam distribution, the distribution of the difference of two Poisson variates, not necessarily from the same parent distribution.
 Incomplete gamma function used to calculate the CDF.
 Dobinski's formula (on combinatorial interpretation of the moments of the Poisson distribution)
 Schwarz formula
 Robbins lemma, a lemma relevant to empirical Bayes methods relying on the Poisson distribution
 Coefficient of dispersion, a simple measure to assess whether observed events are close to Poisson
[edit] Online visualization tools
[edit] Notes
 ^ NIST/SEMATECH, '6.3.3.1. Counts Control Charts', eHandbook of Statistical Methods, accessed 25 October 2006
 ^ McCullagh, Peter; Nelder, John (1989). Generalized Linear Models. London: Chapman and Hall. ISBN 0412317605. page 196 gives the approximation and the subsequent terms.
 ^ Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN 0471548979, p163
 ^ p.e. I J Good, Some statistical applications of Poisson's work, Statist. Sci. 1 (2) (1986), 157180. JSTOR link
[edit] References
 Donald E. Knuth (1969). Seminumerical Algorithms. The Art of Computer Programming, Volume 2. Addison Wesley.
 Joachim H. Ahrens, Ulrich Dieter (1974). "Computer Methods for Sampling from Gamma, Beta, Poisson and Binomial Distributions". Computing 12 (3): 223246. doi: .
 Joachim H. Ahrens, Ulrich Dieter (1982). "Computer Generation of Poisson Deviates". ACM Transactions on Mathematical Software 8 (2): 163179. doi: .
 Ronald J. Evans, J. Boersma, N. M. Blachman, A. A. Jagers (1988). "The Entropy of a Poisson Distribution: Problem 876". SIAM Review 30 (2): 314317. doi: .
[edit] External links
