Share on Facebook Share on Twitter Email
Answers.com

Poisson distribution

 
Dictionary: Pois·son distribution   (pwä-sôN') pronunciation
n. Statistics

A probability distribution which arises when counting the number of occurrences of a rare event in a long series of trials.

[After Siméon Denis Poisson (1781-1840), French mathematician.]


Search unanswered questions...
Enter a question here...
Search: All sources Community Q&A Reference topics
Statistics Dictionary: Poisson distribution
Top

A random variable X, whose set of possible values consists of the non-negative integers, with probability function given by




where λ is a positive constant, is said to have a Poisson distribution, or to be a Poisson variable, with parameter λ. (By convention, λ0=1 and 0!=1.) The distribution was named after Poisson, though the first derivation was by de Moivre in 1711. If we note that P(X=0)=e, successive probabilities can be calculated by using the recurrence relation



The mean and variance of the distribution are both λ. If λ is not an integer the mode is the value of the integer r for which r−1<λ<r. If λ is an integer then P(X=λ−1)=P(X=λ) and both (λ−1) and λ are modes. If λ<1 the graph of the probability function decreases steadily, whereas if λ>1 the graph increases steadily to the value at the mode, then decreases steadily, tending to 0 as r → ∞. A short table of cumulative probabilities for small values of λ is given in Appendix V.

For large values of λ a normal approximation to the Poisson distribution may be used:



and Φ is the cumulative distribution function for a standard normal variable. The '½' is a continuity correction. The approximation may be described as 'For large values of λ a Poisson variable with mean λ is approximately N(λ, λ)'.

In a Poisson process the number of events in a given region, or a given time interval, has a Poisson distribution. See diagram overleaf.



Poisson distribution. The outlines of Poisson distribution with parameter λ increasingly resemble that of a normal distribution as λ increases. Both the mean and the variance of a Poisson distribution with parameter λ are equal to λ.



Computer Desktop Encyclopedia: Poisson distribution
Top

A statistical method developed by the 18th century French mathematician S. D. Poisson, which is used for predicting the probable distribution of a series of events. For example, when the average transaction volume in a communications system can be estimated, Poisson distribution is used to determine the probable minimum and maximum number of transactions that can occur within a given time period.

Download Computer Desktop Encyclopedia to your iPhone/iTouch

Business Dictionary: Poisson Distribution
Top

Type of probability distribution typically used in studies concerned with the count or number of occurrences of events.

Political Dictionary: Poisson distribution
Top

A probability distribution for the frequency of a particular event in a given period of time. It is named after Siméon-Denis Poisson (1781-1840) and famously describes the distributions of deaths due to horse kicks in the Prussian cavalry. In political science it is used for modelling variables such as the number of vetoes cast by a president in a year, or the number of strikes in a factory.

— Stephen Fisher

Veterinary Dictionary: Poisson distribution
Top

A statistical distribution which often describes the sampling frequency of individual, isolated counts in time and space.

Wikipedia: Poisson distribution
Top
Poisson
Probability mass function
Plot of the Poisson PMF
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
Plot of the Poisson CDF
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 \lambda \in (0,\infty)
Support k \in \{0,1,2,\ldots\}
Probability mass function (pmf) \frac{e^{-\lambda} \lambda^k}{k!}\!
Cumulative distribution function (cdf) \frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}\! for k\ge 0 or e^{-\lambda} \sum_{i=0}^{k} \frac{\lambda^i}{i!}\

(where \Gamma(x, y)\,\! is the Incomplete gamma function and \lfloor k\rfloor is the floor function)

Mean \lambda\,\!
Median \approx\lfloor\lambda+1/3-0.02/\lambda\rfloor
Mode \lfloor\lambda\rfloor and λ − 1 if λ is an integer
Variance \lambda\,\!
Skewness \lambda^{-1/2}\,
Excess kurtosis \lambda^{-1}\,
Entropy \lambda[1\!-\!\log(\lambda)]\!+\!e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k\log(k!)}{k!}

(for large λ) \frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -
                     \frac{19}{360 \lambda^3} + O(\frac{1}{\lambda^4})

Moment-generating function (mgf) \exp(\lambda (e^t-1))\,
Characteristic function \exp(\lambda (e^{it}-1))\,

In probability theory and statistics, the Poisson distribution (pronounced [pwasõ]) (or Poisson law of large numbers[1]) 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 first introduced 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, the 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

f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!},\,\!

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

Poisson noise and characterizing small occurrences

The parameter λ is not only the mean number of occurrences \scriptstyle\langle k \rangle, but also its variance \scriptstyle\sigma_k^2 \ =\   \langle k^{2} \rangle - \langle k \rangle^{2} (see Table). Thus, the number of observed occurrences fluctuates about its mean λ with a standard deviation \scriptstyle\sigma_{k}\, =\, \sqrt{\lambda}. 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 \scriptstyle\sigma_{I} = e\sqrt{N}/t\ (i.e. the standard deviation of the Poisson process), the charge e can be estimated from the ratio \scriptstyle\sigma_{I}^{2}/I. 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.

\Pr(N_t=k)=f(k;\lambda t)=\frac{e^{-\lambda t} (\lambda t)^k}{k!}\,\!

Related distributions

  • If X_1 \sim \mathrm{Pois}(\lambda_1)\, and X_2 \sim \mathrm{Pois}(\lambda_2)\, then the difference Y = X1X2 follows a Skellam distribution.
  • If X_1 \sim \mathrm{Pois}(\lambda_1)\, and X_2 \sim \mathrm{Pois}(\lambda_2)\, are independent, and Y = X1 + X2, then the distribution of X1 conditional on Y = y is a binomial. Specifically, X_1|(Y=y) \sim \mathrm{Binom}(y, \lambda_1/(\lambda_1+\lambda_2))\,. More generally, if X1, X2,..., Xn are independent Poisson random variables with parameters λ1, λ2,..., λn then
X_i \left|\sum_{j=1}^n X_j\right. \sim \mathrm{Binom}\left(\sum_{j=1}^nX_j,\frac{\lambda_i}{\sum_{j=1}^n\lambda_j}\right)
  • 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. (This can be done by taking λ = pn and using formula for binomial distribution with n approaching \infty.) 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.[2]
  • For sufficiently large values of λ, (say λ>1000), the normal distribution with mean λ and variance λ (standard deviation \sqrt{\lambda}), 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).
F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,
  • Variance-stabilizing transformation: When a variable is Poisson distributed, its square root is approximately normally distributed with expected value of about \sqrt \lambda and variance of about 1/4.[3] Under this transformation, the convergence to normality is far faster than the untransformed variable. Other, slightly more complicated, variance stabilizing transformations are available,[4] 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 inter-arrival times follow the Exponential distribution, with mean 1 / λ.

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:

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 more-cumbersome 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

\lim_{n\to\infty}\left(1-{\lambda \over n}\right)^n=e^{-\lambda},

and the definition of the Binomial distribution

P(X=k)={n \choose k} p^k (1-p)^{n-k}.

If the binomial probability can be defined such that p = λ / n, we can evaluate the limit of P as n goes large:


\begin{align}

\lim_{n\to\infty} P(X=k)&=\lim_{n\to\infty}{n \choose k} p^k (1-p)^{n-k} \\
 &=\lim_{n\to\infty}{n! \over (n-k)!k!} \left({\lambda \over n}\right)^k \left(1-{\lambda\over n}\right)^{n-k}\\
&=\lim_{n\to\infty}
\underbrace{\left[\frac{n!}{n^k\left(n-k\right)!}\right]}_F
\left(\frac{\lambda^k}{k!}\right)
\underbrace{\left(1-\frac{\lambda}{n}\right)^n}_{\to\exp\left(-\lambda\right)}
\underbrace{\left(1-\frac{\lambda}{n}\right)^{-k}}_{\to 1} \\
&= F\left(\frac{\lambda^k}{k!}\right)\exp\left(-\lambda\right)
\end{align}

To evaluate the F term, we first take its logarithm


\lim_{n\to\infty}\log\left(F\right) =
\log\left(n!\right) - k\log\left(n\right) - \log\left[\left(n-k\right)!\right].

Using Stirling's approximation


\lim_{n\to\infty}\log\left(n!\right) \to n\log\left(n\right) - n,

the expression for \log\left(F\right) can be further simplified to


\lim_{n\to\infty}\log\left(F\right) =
\left[n\log\left(n\right) -n\right] -
\left[k\log\left(n\right)\right] -
\left[\left(n-k\right)\log\left(n-k\right)-\left(n-k\right)\right]

=\left(n-k\right)\log\left(\frac{n}{n-k}\right) - k

= \underbrace{-\left(1-\frac{k}{n}\right)}_{\to -1}
\underbrace{\log\left(1-\frac{k}{n}\right)^{n}}_{\to-k}
-k
= k - k = 0.

Therefore \lim_{n\to\infty}F =
\exp\left(0\right) = 1.


Consequently, the limit of the distribution becomes

{\lambda^k \exp\left(-\lambda\right) \over k!},

which now assumes the Poisson distribution.

More generally, whenever a sequence of independent binomial random variables with parameters n and pn is such that

\lim_{n\rightarrow\infty} np_n = \lambda,

the sequence converges in distribution to a Poisson random variable with mean λ (see, e.g. law of rare events).

2-dimensional Poisson process

 P(N(D)=k)=\frac{(\lambda|D|)^k e^{-\lambda|D|}}{k!}

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
  • D is the 2-dimensional region
  • N(D) is the number of points in the process in region D

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 \scriptstyle\lfloor \lambda \rfloor, 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 X_i \sim \mathrm{Pois}(\lambda_i)\, follow a Poisson distribution with parameter \lambda_i\, and Xi are independent, then
Y = \sum_{i=1}^N X_i \sim \mathrm{Pois}\left(\sum_{i=1}^N \lambda_i\right)\,
also follows a Poisson distribution whose parameter is the sum of the component parameters.
  • The sum of normalised square deviations is approximately distributed as chi-square if the mean is of a moderate size (λ > 5 is suggested).[5] If X_1,\dots,X_N are observations from independent Poisson distributions with means \lambda_1,\dots,\lambda_N then \sum_{i=1}^N \frac{(X_i-\lambda_i)^2}{\lambda_i}\sim \chi^2
  • The moment-generating function of the Poisson distribution with expected value λ is
\mathrm{E}\left(e^{tX}\right)=\sum_{k=0}^\infty e^{tk} f(k;\lambda)=\sum_{k=0}^\infty e^{tk} {\lambda^k e^{-\lambda} \over k!} =e^{\lambda(e^t-1)}.
D_{\mathrm{KL}}(\lambda\|\lambda_0) = \lambda_0 - \lambda + \lambda \log \frac{\lambda}{\lambda_0}.

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 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.

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


\begin{align}
L(\lambda) & = \log \prod_{i=1}^n f(k_i \mid \lambda) \\
& = \sum_{i=1}^n \log\!\left(\frac{e^{-\lambda}\lambda^{k_i}}{k_i!}\right) \\
& = -n\lambda + \left(\sum_{i=1}^n k_i\right) \log(\lambda) - \sum_{i=1}^n \log(k_i!). \end{align}

Take the derivative of L with respect to λ and equate it to zero:

\frac{\mathrm{d}}{\mathrm{d}\lambda} L(\lambda) = 0
\iff -n + \left(\sum_{i=1}^n k_i\right) \frac{1}{\lambda} = 0. \!

Solving for λ yields the a stationary point, which if the second derivative is negative is the maximum-likelihood estimate of λ:

\widehat{\lambda}_\mathrm{MLE}=\frac{1}{n}\sum_{i=1}^n k_i. \!

Checking the second derivative, it is found that it is negative for all λ and ki greater than zero, therefore this stationary point is indeed a maximum of the initial likelihood function:

\frac{\partial^2 L}{\partial \lambda^2} =  \sum_{i=1}^n -\lambda^{-2} k_i

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

\lambda \sim \mathrm{Gamma}(\alpha, \beta) \!

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:

 g(\lambda \mid \alpha,\beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \; \lambda^{\alpha-1} \; e^{-\beta\,\lambda} \qquad \text{ for } \lambda>0 \,\!.

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

\lambda \sim \mathrm{Gamma}(\alpha + \sum_{i=1}^n k_i, \beta + n). \!

The posterior mean E[λ] approaches the maximum likelihood estimate \widehat{\lambda}_\mathrm{MLE} in the limit as \alpha\to 0,\ \beta\to 0.

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.[6]

See also

Online visualization tools

Notes

  1. ^ p963-965, Jan Gullberg, Mathematics from the birth of numbers, W. W. Norton & Company; ISBN 039304002X ISBN 978-0393040029
  2. ^ NIST/SEMATECH, '6.3.3.1. Counts Control Charts', e-Handbook of Statistical Methods, accessed 25 October 2006
  3. ^ McCullagh, Peter; Nelder, John (1989). Generalized Linear Models. London: Chapman and Hall. ISBN 0-412-31760-5.  page 196 gives the approximation and the subsequent terms.
  4. ^ Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN 0-471-54897-9, p163
  5. ^ Box, Hunter and Hunter. Statistics for experimenters. Wiley. p. 57. 
  6. ^ Good, I.J., Some statistical applications of Poisson's work, Statist. Sci. 1 (2) (1986), 157–180. JSTOR link

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): 223–246. doi:10.1007/BF02293108. 
  • Joachim H. Ahrens, Ulrich Dieter (1982). "Computer Generation of Poisson Deviates". ACM Transactions on Mathematical Software 8 (2): 163–179. doi:10.1145/355993.355997. 
  • Ronald J. Evans, J. Boersma, N. M. Blachman, A. A. Jagers (1988). "The Entropy of a Poisson Distribution: Problem 87-6". SIAM Review 30 (2): 314–317. doi:10.1137/1030059. 

Best of the Web: Poisson distribution
Top

Some good "Poisson distribution" pages on the web:


Math
mathworld.wolfram.com
 
 
 

 

Copyrights:

Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.  Read more
Statistics Dictionary. A Dictionary of Statistics. Second edition revised. Copyright © Oxford University Press, 2008. All rights reserved.  Read more
Computer Desktop Encyclopedia. THIS COPYRIGHTED DEFINITION IS FOR PERSONAL USE ONLY.
All other reproduction is strictly prohibited without permission from the publisher.
© 1981-2009 Computer Language Company Inc.  All rights reserved.  Read more
Business Dictionary. Dictionary of Business Terms. Copyright © 2000 by Barron's Educational Series, Inc. All rights reserved.  Read more
Political Dictionary. The Concise Oxford Dictionary of Politics. Copyright © 1996, 2003 by Oxford University Press. All rights reserved.  Read more
Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved.  Read more
Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Poisson distribution" Read more