Poisson point process: Difference between revisions

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===Discovery===

There are a number of claims for early uses or discoveries of the Poisson point process.<ref name="Stirzaker2000"/><ref name="GuttorpThorarinsdottir2012"/> For example, [[John Michell]] in 1767, a decade before Poisson was born, was interested in the probability a star being within a certain region of another star under the erroneous assumption that the stars were "scattered by mere chance", and studied an example consisting of the six brightest [[star]]s in the [[Pleiades]], without deriving the Poisson distribution. This work inspired [[Simon Newcomb]] to study the problem and to calculate the Poisson distribution as an

approximation for the binomial distribution in 1860.<ref name="GuttorpThorarinsdottir2012"/>

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In Sweden 1903, [[Filip Lundberg]] published a thesis containing work, now considered fundamental and pioneering, where he proposed to model insurance claims with a homogeneous Poisson process.<ref name="EmbrechtsFrey2001page367">{{cite book|last1=Embrechts|first1=Paul|title=Stochastic Processes: Theory and Methods|last2=Frey|first2=Rüdiger|last3=Furrer|first3=Hansjörg|chapter=Stochastic processes in insurance and finance|volume=19|year=2001|page=367|issn=0169-7161|doi=10.1016/S0169-7161(01)19014-0|series=Handbook of Statistics|isbn=9780444500144}}</ref><ref name="Cramér1969">{{cite journal|last1=Cramér|first1=Harald|title=Historical review of Filip Lundberg's works on risk theory|journal=Scandinavian Actuarial Journal|volume=1969|issue=sup3|year=1969|pages=6–12|issn=0346-1238|doi=10.1080/03461238.1969.10404602}}</ref>

In [[Denmark]] in 1909 another discovery occurred when [[A.K. Erlang]] derived the Poisson distribution in 1909 when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time awareunaware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent toof each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.<ref name="Stirzaker2000"/>

In 1910 [[Ernest Rutherford]] and [[Hans Geiger]] published experimental results on counting alpha particles. Their experimental work had mathematical contributions from [[Harry Bateman]], who derived Poisson probabilities as a solution to a family of differential equations, though the solution had been derived earlier, resulting in the independent discovery of the Poisson process.<ref name="Stirzaker2000"/> After this time, there were many studies and applications of the Poisson process, but its early history is complicated, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and various physical scientists.<ref name="Stirzaker2000"/>

===Early applications===