In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1 (e.g., number of events in a given area or volume). The Poisson distribution is named after French mathematician Siméon Denis Poisson. It plays an important role for discrete-stable distributions. Under a Poisson distribution with the expectation of λ events in a given interval, the probability of k events in the same interval is:
λ k e − λ k ! . {\displaystyle {\frac {\lambda ^{k}e^{-\lambda }}{k!}}.}
For instance, consider a call center which receives an average of λ = 3 calls per minute at all times of day. If the number of calls received in any two given disjoint time intervals is independent, then the number k of calls received during any minute has a Poisson probability distribution. Receiving k = 1 to 4 calls then has a probability of about 0.77, while receiving 0 or at least 5 calls has a probability of about 0.23. A classic example used to motivate the Poisson distribution is the number of radioactive decay events during a fixed observation period.
History The introduction of the Poisson distribution is credited to French mathematician and physicist Siméon Denis Poisson (1781–1840), who published it together with his probability theory in Recherches sur la probabilité des jugements en matière criminelle et en matière civile (1837). This work theorizes about the number of wrongful convictions in a given country by focusing on certain random variables N that count the number of events that take place during a time interval of given length. However, similar results had already been given in 1711 by Abraham de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus . 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. In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space. A further practical application was made by Ladislaus Bortkiewicz in 1898. Bortkiewicz showed that the frequency with which soldiers in the Prussian army were accidentally killed by horse kicks could be well modeled by a Poisson distribution..
Definitions
Probability mass function A discrete random variable X is said to have a Poisson distribution with parameter λ > 0 {\displaystyle \lambda >0} if it has a probability mass function given by:
f ( k ; λ ) = Pr ( X = k ) = λ k e − λ k ! , {\displaystyle f(k;\lambda )=\Pr(X{=}k)={\frac {\lambda ^{k}e^{-\lambda }}{k!}},}
where
k is the number of occurrences ( k = 0 , 1 , 2 , … {\displaystyle k=0,1,2,\ldots } ) e is Euler's number ( e = 2.71828 … {\displaystyle e=2.71828\ldots } ) k! = k(k–1) ··· (3)(2)(1) is the factorial. The positive real number λ is equal to the expected value of X and also to its variance.
λ = E ( X ) = Var ( X ) . {\displaystyle \lambda =\operatorname {E} (X)=\operatorname {Var} (X).}
The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. The number of such events that occur during a fixed time interval is, under the right circumstances, a random number with a Poisson distribution. The equation can be adapted if, instead of the average number of events λ , {\displaystyle \lambda ,} we are given the average rate r {\displaystyle r} at which events occur. Then λ = r t , {\displaystyle \lambda =rt,} and:
P ( k events in interval t ) = ( r t ) k e − r t k ! . {\displaystyle P(k{\text{ events in interval }}t)={\frac {(rt)^{k}e^{-rt}}{k!}}.}
Examples
The Poisson distribution may be useful to model events such as:
the number of meteorites greater than one-meter diameter that strike Earth in a year; the number of laser photons hitting a detector in a particular time interval; the number of students achieving a low and high mark in an exam; and locations of defects and dislocations in materials. Examples of the occurrence of random points in space are: the locations of asteroid impacts with earth (2-dimensional), the locations of imperfections in a material (3-dimensional), and the locations of trees in a forest (2-dimensional).
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