Distribution, Poisson
Distribution, Poisson
The Poisson distribution (named after Siméon Denis Poisson, 1781–1840) is used to describe certain events in time or in space. It is derived from the binomial distribution with the extension that time (or space) is thought to be continuous instead of discrete. In the case of discrete time (binomial distribution), let the probability that a certain event occurs during one period of length 1 be a constant p ; then the number of events Z ∊ {0, 1, 2, …) happening during a certain number n of such periods is binomially distributed with parameters n and p. No event influences any other event—that is, the events are mutually independent. If the length of the period is decreased to Δt, and if we are still interested in the number of events happening during a time span of length n, then this time span consists of n/Δt periods within each of which the event probability is p Δt. For t → 0 and np = λ, the formula for the binomial distribution
is transformed into the formula of the Poisson distribution
The difference between the two distributions is shown in two graphs drawn from a sample of 25,000 simulated random events:
Random variables X _{t} form a Poisson process if the following conditions hold: The probability that exactly one event occurs during a time span of length Δt is proportional to the length of the time span; the probability that more than one event occurs during this time span is negligible; and the number of events occurring in disjoint time intervals are mutually independent. X _{s} is the number of events that occurred from t = 0 until t = s. The increments (X_{t} – X_{s} ) follow the Poisson distribution.
The Poisson distribution is also called a “distribution of rare events,” because for a fixed λ and a large n the probability of the individual event occurring per time unit is, of course, small. An important application of the
Poisson distribution and of the Poisson process (a stochastic process whose random increments are Poisson distributed) is queuing theory; in this case, the number of new customers arriving during a fixed period is the random increment. The parameter λ of the Poisson distribution is called “intensity” and yields the expected value of customers arriving per time unit in this example.
Another example is the number of accidents that occur in a certain area during a given period. The distribution of this number will also follow the Poisson distribution. If one compares the number of car accidents with passengers killed with and without a seat belt, one has to compare the parameters of the two Poisson distributions and would find out whether using seat belts had a significant influence on the number of deadly accidents.
Given the conditions under which events are Poisson distributed, one could argue that customers arriving at an airport often arrive in pairs or even larger groups—which would violate the applicability of the model to realworld scenarios. Another obvious violation is due to the fact that the arrival intensities at airports (or in emergency units) are not the same over the entire day or the week, such that the model can be applied only to selected times spans during which the arrival intensities are more or less constant over a considerable period of time. Physical processes such as the decay of radioactive material are less prone to violations of this kind than processes where humans are involved; but even here—at least with the help of computer simulation—intensities that vary over time can be successfully modelled.
SEE ALSO Central Tendencies, Measures of; Distribution, Normal; Distribution, Uniform; Variables, Random
BIBLIOGRAPHY
Hoel, Paul G. 1984. Introduction to Mathematical Statistics. 5th ed. Hoboken, NJ: Wiley.
Poisson, Siméon D. [1837] 2003. Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Paris: Editions Jacques Gabay.
Klaus G. Troitzsch
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Distribution, Poisson." International Encyclopedia of the Social Sciences. . Encyclopedia.com. 13 Jul. 2018 <http://www.encyclopedia.com>.
"Distribution, Poisson." International Encyclopedia of the Social Sciences. . Encyclopedia.com. (July 13, 2018). http://www.encyclopedia.com/socialsciences/appliedandsocialsciencesmagazines/distributionpoisson
"Distribution, Poisson." International Encyclopedia of the Social Sciences. . Retrieved July 13, 2018 from Encyclopedia.com: http://www.encyclopedia.com/socialsciences/appliedandsocialsciencesmagazines/distributionpoisson
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.
Poisson distribution
Poisson distribution The basic discrete probability distribution for data in the form of counts of random events. If each event occurs with the same probability and the mean frequency of events is μ, the probability that exactly r events will occur is e^{–μ}μ^{r}/r!
The Poisson distribution is discrete, taking the values r = 0, 1, 2,… and it can be obtained as a limiting case of the binomial distribution as n tends to infinity while np is held fixed. The mean and variance of the Poisson distribution are both equal to μ.
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Poisson distribution." A Dictionary of Computing. . Encyclopedia.com. 13 Jul. 2018 <http://www.encyclopedia.com>.
"Poisson distribution." A Dictionary of Computing. . Encyclopedia.com. (July 13, 2018). http://www.encyclopedia.com/computing/dictionariesthesaurusespicturesandpressreleases/poissondistribution
"Poisson distribution." A Dictionary of Computing. . Retrieved July 13, 2018 from Encyclopedia.com: http://www.encyclopedia.com/computing/dictionariesthesaurusespicturesandpressreleases/poissondistribution
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.
Poisson distribution
Poisson distribution The basis of a method whereby the distribution of a particular attribute in a population can be calculated from its mean occurrence in a random sample of the population, provided that the population is large and the probability that the attribute will occur is less than 0.1. For a given mean the distribution can be calculated, giving the probabilities that a sample will contain 0,1, 2, 3, … examples of the particular attribute. The distribution is named after the French mathematician S. D. Poisson (1781–1840).
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Poisson distribution." A Dictionary of Zoology. . Encyclopedia.com. 13 Jul. 2018 <http://www.encyclopedia.com>.
"Poisson distribution." A Dictionary of Zoology. . Encyclopedia.com. (July 13, 2018). http://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/poissondistribution2
"Poisson distribution." A Dictionary of Zoology. . Retrieved July 13, 2018 from Encyclopedia.com: http://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/poissondistribution2
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.
Poisson distribution
Poisson distribution The basis of a method whereby the distribution of a particular attribute in a population can be calculated from its mean occurrence in a random sample of the population, provided that the population is large and the probability that the attribute will occur is less than 0.1. For a given mean, the distribution can be calculated, giving the probability that a sample will contain 0, 1, 2, 3,… examples of the particular attribute. The distribution is named after the French mathematician S. D. Poisson (1781–1840).
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Poisson distribution." A Dictionary of Plant Sciences. . Encyclopedia.com. 13 Jul. 2018 <http://www.encyclopedia.com>.
"Poisson distribution." A Dictionary of Plant Sciences. . Encyclopedia.com. (July 13, 2018). http://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/poissondistribution1
"Poisson distribution." A Dictionary of Plant Sciences. . Retrieved July 13, 2018 from Encyclopedia.com: http://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/poissondistribution1
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.
Poisson distribution
Poisson distribution The basis of a method whereby the distribution of a particular attribute in a population can be calculated from its mean occurrence in a random sample of that population, provided the population is large and there is a less than 0.1 probability that the attribute will occur. For a given mean the distribution can be calculated, giving the probabilities that a sample will contain 0, 1, 2, 3,…examples of the particular attribute. The distribution is named after the French mathematician S. D. Poisson (1781–1840).
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Poisson distribution." A Dictionary of Ecology. . Encyclopedia.com. 13 Jul. 2018 <http://www.encyclopedia.com>.
"Poisson distribution." A Dictionary of Ecology. . Encyclopedia.com. (July 13, 2018). http://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/poissondistribution0
"Poisson distribution." A Dictionary of Ecology. . Retrieved July 13, 2018 from Encyclopedia.com: http://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/poissondistribution0
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.
Poisson distribution
Poisson distribution A probability distribution applied in studies of rare events, such as earthquakes, or when the probability of an event taking place is small. It approximates to the binomial distribution, in certain cases, but is otherwise highly skewed.
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Poisson distribution." A Dictionary of Sociology. . Encyclopedia.com. 13 Jul. 2018 <http://www.encyclopedia.com>.
"Poisson distribution." A Dictionary of Sociology. . Encyclopedia.com. (July 13, 2018). http://www.encyclopedia.com/socialsciences/dictionariesthesaurusespicturesandpressreleases/poissondistribution
"Poisson distribution." A Dictionary of Sociology. . Retrieved July 13, 2018 from Encyclopedia.com: http://www.encyclopedia.com/socialsciences/dictionariesthesaurusespicturesandpressreleases/poissondistribution
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.
Poisson distribution
Poisson distribution In statistics, a discrete probability distribution which is applied to the number of times an event occurs.
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Poisson distribution." A Dictionary of Earth Sciences. . Encyclopedia.com. 13 Jul. 2018 <http://www.encyclopedia.com>.
"Poisson distribution." A Dictionary of Earth Sciences. . Encyclopedia.com. (July 13, 2018). http://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/poissondistribution
"Poisson distribution." A Dictionary of Earth Sciences. . Retrieved July 13, 2018 from Encyclopedia.com: http://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/poissondistribution
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.