distribution probability that represents the happening of unusual events in a large number of trials
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 are independent of the time since the last event. The Poisson distribution can also be used for other specified intervals such as: distance, area or volume. A classic example is the probability of a certain number of bombs striking a randomly selected area from a group of equally sized areas. This example was applied to German V-1 buzz bombs (a flying bomb, the precurser to the guided missile) striking South London during WW II. On paper, South London was divided geographically into 576 areas each having 0.25km2 areas. Assuming the 535 bombs launched toward South London were done so with random targeting. Therefore, the probability of any number of bombs (0 to 535) striking any area of the 576, at random, can be calculated. For use in the Poisson distribution, the mean, λ, is the quotient of number of bombs divided by number of equally sized areas. The distribution was discovered by
Siméon-Denis 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, a number of discrete occurrences (sometimes called "arrivals") that take place during a
time-interval 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 non-negative
integer, k = 0, 1, 2, ...) is equal to wheree 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 functionk! is the
factorial of kλ 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 every 4
minutes, and you are interested in the number of events occurring in a 10 minute interval, you would use as model a Poisson distribution with λ = 10/4 = 2.5. 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 is sometimes called a Poissonian, analogous to the term Gaussian for a Gauss or
normal distribution.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
variance 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.Related distributionsIf and , then the difference follows a
Skellam distribution.If and are independent, and , then the distribution of conditional on is a
binomial. Specifically, . More generally, if X1, X2,..., Xn are Poisson random variables with parameters λ1, λ2,..., λn 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. 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 (lower-case) x is a non-negative integer, is replaced by P(X ≤ x + 0.5). 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 cars that pass through a certain point on a road (sufficiently distant from traffic lights) during a given period of time.The number of spelling mistakes one makes while typing a single page.The number of phone calls at a
call center per minute.The number of times a
web server is accessed per minute.The number of
roadkill (animals killed) found per unit length of road.The number of
mutations in a given stretch of
DNA after a certain amount of radiation.The number of unstable
nuclei that decayed within a given period of time in a piece of
radioactive substance. The radioactivity of the substance will weaken with time, so the total time interval used in the model should be significantly less than the
mean lifetime of the substance.The number of pine trees per unit area of mixed forest.The number of
stars in a given volume of space.The number of soldiers killed by horse-kicks each year in each corps in the
Prussian cavalry. This example was made famous by a book of
Ladislaus Josephovich Bortkiewicz (
1868–
1931).The distribution of visual receptor cells in the
retina of the
human eye.The number of light bulbs that burn out in a certain amount of time.The number of viruses that can infect a cell in cell culture.The number of hematopoietic stem cells in a sample of unfractionated bone marrow cells.The
inventivity of an inventor over their career.The number of particles that "scatter" off of a target in a nuclear or high energy physics experiment. 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 limit is sometimes known as the law of rare events, although this name may be misleading because the events in a Poisson process need not be rare (the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution, but these events would not be considered rare). It provides a means by which to approximate random variables using the Poisson distribution rather than the more-cumbersome binomial distribution. Here are the details. First, recall from
calculus that Let p = λ/n. Then we have As n approaches ∞, the expression over the first underbrace approaches 1; the second remains constant since "n" does not appear in it at all; the third approaches e−λ; and the fourth expression approaches 1. Consequently the limit is More generally, whenever a sequence of binomial random variables with parameters n and pn is such that the sequence
converges in distribution to a Poisson random variable with mean λ (see, e.g.,
law of rare events). Properties The
expected value of a Poisson-distributed 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 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.Sums of Poisson-distributed random variables:If follow a Poisson distribution with parameter and are
independent, then also follows a Poisson distribution whose parameter is the sum of the component parameters.The
moment-generating function of the Poisson distribution with expected value λ isAll 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 Poisson distributions are
infinitely divisible probability distributions.The directed
Kullback-Leibler divergence between Poi(λ0) and Poi(λ) is given by Generating Poisson-distributed random variables A simple way to generate random Poisson-distributed 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 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. Parameter estimation Maximum likelihood Given a sample of n measured values ki 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 log-likelihood function Take the derivative of L with respect to λ and equate it to zero: Solving for λ yields the maximum-likelihood 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
Cramér-Rao lower bound (CRLB). 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 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 . The posterior predictive distribution of additional data is a
Gamma-Poisson (i.e.
negative binomial) distribution.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.See also
Compound Poisson distributionPoisson processPoisson regressionQueueing theoryErlang 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 formulaRobbins lemma, a lemma relevant to
empirical Bayes methods relying on the Poisson distributionReferences <references/>External links
Queueing Theory BasicsM/M/1 Queueing SystemEngineering Statistics Handbook: Poisson Distributionxkcd webcomic involving a Poisson distributionPoisson Distribution at QWikiPoisson Derivation 1: Continuous limit of a Binomial distributionPoisson Derivation 2: Generating function approachPoisson Derivation 3: Summation of the waiting-time distribution
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A mathematical statement of the probability that exactly k discrete events will take place during an interval of length t , expressed by where k is a non-negative integer, e is the base of the natural logarithms (e2.71828), is the constant rate that the events occur, and t is the expected number of events occurring during an interval of length t .
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