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Poisson distribution
In probability theory and statistics, the Poisson distribution (French pronunciation (:pwasɔ̃); in English usually ), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space 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.
For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. If receiving any particular piece of mail doesn't affect the arrival times of future pieces of mail, i.e., if pieces of mail from a wide range of sources arrive independently of one another, then a reasonable assumption is that the number of pieces of mail received per day obeys a Poisson distribution. Other examples that may follow a Poisson: the number of phone calls received by a call center per hour, the number of decay events per second from a radioactive source, or the number of pedicabs in queue in a particular street in a given hour of a day.
== History ==
The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published, together with his probability theory, in 1837 in his work ''Recherches sur la probabilité des jugements en matière criminelle et en matière civile'' ("Research on the Probability of Judgments in Criminal and Civil Matters").〔S.D. Poisson, ''Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés'' (Paris, France: Bachelier, 1837), (page 206 ).〕 The work theorized about the number of wrongful convictions in a given country by focusing on certain random variables ''N'' that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a time-interval of given length. The result had been given previously by Abraham de Moivre (1711) in ''De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus'' in Philosophical Transactions of the Royal Society, p. 219.〔 This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.〔(Stigler, Stephen M. "Poisson on the Poisson distribution." Statistics & Probability Letters 1.1 (1982): 33-35. )〕〔(Hald, Anders, Abraham de Moivre, and Bruce McClintock. "A. de Moivre:'De Mensura Sortis' or'On the Measurement of Chance'." International Statistical Review/Revue Internationale de Statistique (1984): 229-262 )〕
A practical application of this distribution was made by Ladislaus Bortkiewicz in 1898 when he was given the task of investigating the number of soldiers in the Prussian army killed accidentally by horse kicks; this experiment introduced the Poisson distribution to the field of reliability engineering.〔Ladislaus von Bortkiewicz, ''Das Gesetz der kleinen Zahlen'' (law of small numbers ) (Leipzig, Germany: B.G. Teubner, 1898). On (page 1 ), Bortkiewicz presents the Poisson distribution. On (pages 23–25 ), Bortkiewicz presents his famous analysis of "4. Beispiel: Die durch Schlag eines Pferdes im preussischen Heere Getöteten." (4. Example: Those killed in the Prussian army by a horse's kick.).〕
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