Poisson Distribution

Domain	Definitions
Aerospace	A one-parameter discrete frequency distribution giving the probability the n points (or events) will be (or occur) in an interval (or time) x , provided that these points are individually independent and that the number occurring in a subinterval does not influence the number occurring in any other nonoverlapping subinterval. It has the form f(n,x) = e-sx(sx)n/n!The mean and variance are both sx , and s is the average density (or rate) with which the events occur. When sx is large, the Poisson distribution approaches the normal distribution. The binomial distribution approaches the Poisson when the number of events n becomes large and the probability of success P becomes small in such a way that nP aproaches sx. (references)
Geological	A probability distribution that characterizes discrete events occurring independently of one another in time. (references)
Health	A distribution function used to describe the occurrence of rare events or to describe the sampling distribution of isolated counts in a continuum of time or space. (references)
Statistics	The distribution of number of events in a given time, arising from a Poisson process. This differs from the binomial distribution in that there is no upper limit, corresponding to the parameter'n'of a binomial process, to the number of events which may occur. Source: European Union. (references)
Source: compiled by the editor from various references; see credits.

Specialty Definition: Poisson distribution

In statistics and probability theory, the Poisson distribution is a discrete probability distribution (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) belonging to 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. The probability that there are exactly k occurrences (k being a natural number including 0, k = 0, 1, 2, ...) is:

Where:

e is the base of the natural logarithm (e = 2.71828...),
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 every 2 minutes, and you are interested in the number of events occcurring in a 10 minute interval, you would use as model a Poisson distribution with λ = 5.

Sometimes λ is taken to be the rate, i.e., the average number of occurrences per unit time. In that case, if N_t is the number of occurrences before time t then we have

and the waiting time T until the first occurrence is a continuous random variable with an exponential distribution; this probability distribution may be deduced from the fact that

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 include:

The number of unstable nuclei that decayed within a given period of time in a piece of radioactive substance.
The number of cars that pass through a certain point on a road during a given period of time.
The number of spelling mistakes a secretary makes while typing a single page.
The number of phone calls you get per day.
The number of times your web server is accessed per minute.
- For instance, the number of edits per hour recorded on Wikipedia's Recent Changes page follows an approximately Poisson distribution.
The number of roadkill you find per mile of road.
The number of mutations in a given stretch of DNA after a certain amount of radiation.
The number of pine trees per square mile 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 (an example made famous by a book of Ladislaus Josephovich Bortkiewicz (1868-1931)).
The number of bombs falling on each square mile of London during a German air raid in the early part of the Second World War.

How does this distribution arise? -- The limit theorem

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.

Properties

The expected value of a Poisson distributed random variable is equal to λ and so is its variance.

The most likely value ("mode") of a Poisson distributed random variable is equal to the largest integer ≤ λ, which is also written as floor(λ).

If λ is big enough (λ > 10 say), then the normal distribution with mean λ and standard deviation √ λ is an excellent approximation to the Poisson distribution.

If N and M are two independent random variables, both following a Poisson distribution with parameters λ and μ, respectively, then N + M follows a Poisson distribution with parameter λ + μ.

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 the title of 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.

Commercial Usage: Poisson Distribution

Expressions: Poisson Distribution

Frequency of Internet Keywords: Poisson Distribution

Modern Translation: Poisson Distribution

Language		Translations for "Poisson distribution"; alternative meanings/domain in parentheses.
Danish		Poisson-fordeling (Poisson's exponential limit). (various references)

Dutch		poisson-verdeling (Poisson's exponential limit), poissonverdeling (Poisson's exponential limit). (various references)

Finnish		ryhmitelty Poissonin jakauma (grouped Poisson distribution), kaksinkertainen Poissonin jakauma (double Poisson distribution, Thomas distribution), kaksimuuttujainen Poisson-jakauma (bivariate Poisson distribution). (various references)

French		loi de Poisson (Poisson's exponential limit), distribution de Poisson (Poisson's exponential limit). (various references)

German		Poisson-Verteilung (Poisson's exponential limit). (various references)

Greek		κατανομή POISSON (Poisson's exponential limit). (various references)

Italian		legge di Poisson (Poisson's exponential limit), distribuzione di Poisson (Poisson's exponential limit). (various references)

Pig Latin		oissonpay istributionday

Portuguese		distribuição de Poisson (Poisson's exponential limit). (various references)

Spanish		distribución de Poisson (Poisson's exponential limit). (various references)

Swedish		grupperadPoissonfördelning (grouped Poisson distribution), grupperad Poissonfördelning (grouped Poisson distribution), dubbel Poissonfördelning (double Poisson distribution, Thomas distribution), bivariatPoissonfördelning (bivariate Poisson distribution), bivariat Poisson-fördelning (bivariate Poisson distribution). (various references)
Source: compiled by the editor from various translation references.

Anagrams: Poisson Distribution

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "b-d-i-i-i-i-n-n-o-o-o-p-r-s-s-s-t-t-u"
	-5 letters: indispositions.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

Alternative Orthography: Poisson Distribution

Hexadecimal (or equivalents, 770AD-1900s) (references)

50 6F 69 73 73 6F 6E 44 69 73 74 72 69 62 75 74 69 6F 6E

Leonardo da Vinci (1452-1519; backwards) (references)

Binary Code (1918-1938, probably earlier) (references)

01010000 01101111 01101001 01110011 01110011 01101111 01101110 00100000 01000100 01101001 01110011 01110100 01110010 01101001 01100010 01110101 01110100 01101001 01101111 01101110

HTML Code (1990) (references)

&#80 &#111 &#105 &#115 &#115 &#111 &#110 &#32 &#68 &#105 &#115 &#116 &#114 &#105 &#98 &#117 &#116 &#105 &#111 &#110

ISO 10646 (1991-1993) (references)

0050 006F 0069 0073 0073 006F 006E 0044 0069 0073 0074 0072 0069 0062 0075 0074 0069 006F 006E

Encryption (beginner's substitution cypher): (references)

Domain	Title
Books	Quality Sampling and Reliability: New Uses for the Poisson Distribution (reference) Tests of the adequacy of fit for the bivariate poisson distribution (reference) (more book examples)
Source: compiled by the editor from various references; see credits.

Expression	Frequency per Day
poisson distribution	77
Source: compiled by the editor from various references; see credits.

Language	Coverage	Language Translations
Danish	ordbog, deskriptordefinition, oversættelse	tanskalainen, danois, dänisch, δανικόσ, δανόσ, danese, dinamarquês, danés, dansk
Dutch	woordenboek, definitie, translatie	hollandsk, hollantilainen, néerlandais, holländisch, ολλανδικόσ, ολλανδόσ, olandese, holandês, holandés, holländsk
Finnish	määritelmä, translaatio, taajuusmuutos	suomi, suomalainen, finnois, Finlandaise, finlandais, finnisch, φινλανδικόσ, finlandese, finlandês, finês, finlandés, finés, finsk
French	dictionnaire, définition, traduction	ranskalainen, français, französisch, γαλλικόσ, γαλλική γλώσσα, γαλλίδα, γάλλοσ, francese, francês, francés, fransk, franska
German	Übersetzung, Wörterbuch, Definition	tysker, Duitse, saksalainen, allemand, �"ερμανός, tedesco, alemão, alemán, tysk
Greek	λεξικό, ορισμός, μετάφραση	græker, kreikkalainen, grec, grieche, ελληνικόσ, 'Ελληνας, greco, grego, griego, grek
Italian	dizionario, definizione, traduzione	italiener, italialainen, italien, italienisch, Ιταλός, italiano, italiensk, italienska, italienare
Portuguese	dicionário, definição, tradução	portugiser, portugalilainen, portugais, portugiesisch, πορτογάλοσ, � ορτογάλος, portoghese, português, portugués, portugis
Spanish	diccionario, definición, traducción	Spaans, espanjalainen, espagnol, spanisch, ισπανικά, ισπανικόσ, ισπανοί, spagnolo, espanhol, español, spanska språk, spansk
Swedish	ordbok, lexikon, översättning	Zweeds, ruotsalainen, suédois, schwedisch, σουηδικόσ, σουηδικά, svedese, sueco, svensk
English	Dictionary, Definition, Translation	englantia, englantilainen, anglais, englisch, εγγλέζοσ, αγγλικόσ, inglese, inglês, inglés, engelsk

Expressions using "Poisson distribution": bivariate Poisson distribution ♦ double Poisson distribution. Additional references.
Source: compiled by the editor from various references; see credits.

Poisson Distribution

Definition: Poisson Distribution