{{For|Poisson processes in higher dimensions|Poisson point process}}

In [[probability theory]], a '''Poisson process''' is a model for situations where certain events occur [[independence (probability)|idependently]] at random instants during an interval of time, at uniform rate; or that occur at random points within a region of some [[space (mathematics)|space]], with uniform [[density]].   Namely, the probability of an event occuring within any [[infinitesimal]] part of that time interval or region is some constant ''λ'' times the [[measure (mathematics)|measure]] (duration, length, [[area (mathematics)|area]], [[volume (mathematics)|volume]], etc.) of that part, independently of when or where the other events occurred.

The process is named after the French mathematician [[Siméon Denis Poisson]] (1781-1840).  It is one of the simplest [[stochastic process]]es (specifically, a [[point process]]), and is a good model of [[radioactive decay]],<ref>{{cite doi|10.1016/0020-708X(78)90101-1}}</ref> telephone calls,<ref>{{cite doi|10.1109/MCOM.2009.4804392}}</ref> requests for a particular document on a web server,<ref name="ArlittMartin">{{cite doi|10.1109/90.649565}}</ref> defects on electrical wires,<ref name="Gitlow.148">Howard S. Gitlow <nowiki>[Giltow]</nowiki>, Rosa Oppenheim, Alan Oppenheim, David Levine (1989), ''Tools and Methods for the Improvement of Quality''. CRC Press, page 148. ISBN 9780256056808</ref> raindrops hitting a flat surface,<ref name="Kingman.57">J. F. C. Kingman (1992), ''Poisson Processes', page 57. Oxford University Press. ISBN 9780191591242.</ref> and many more.

In a Poisson process on a one-[[dimension (mathematics)|dimensional]] domain, such as an interval of time or a length of wire, the separation between each pair of consecutive events has an [[exponential distribution]], whose parameter ''λ'' is the density of the events; and the separation of each pair is independent of the separations of the other pairs.  It is a [[continuous-time process]], the simplest example of a [[birth-death process]], with no death. In general, a Poisson process can be seen as a continuous ([[limit (mathematics)|limiting]]) version of a discrete [[Bernoulli process]].

==Definition==
Let ''R'' be a [[measurable space]] (like the real numbers, a rectangle in the Cartesian plane, the surface of a sphere, etc.) and let's say that a subset ''S'' of ''R'' is ''bounded'' if it has finite measure.  Consider a random variable ''X'' whose value is a subset of the points of ''R''.  The variable ''X'' is said to be a ''Poisson process with domain'' ''R'' if the following conditions are satisfied:
* for any bounded subset ''S'' of ''R'', the probability of ''S'' containing some point of ''X'' is strictly less than 1, and depends only on the measure of ''S'';
* for any two disjoint subsets ''S'' and ''T'' of ''R'', the probability of ''S'' containing some point of ''X'' is independent of whether ''T'' contains any point of ''X''. 
From these properties, it follows that, for any bounded subset ''S'' of ''R'', the number |''X''∩''S''| of points of ''X'' that lie in ''S'' is a random variable, whose [[probability distribution]] depends only on the measure ''s'' of ''S''.  Specifically, 
:<math> \mathop{\mathrm{Prob}}(|X\cap S| = n) = P(n,s) = \frac{e^{-\lambda s} (\lambda s)^n}{n!}</math>
Here ''λ'' is a positive [[real number]], called the ''intensity'' or ''density'' of the Poisson process, which is the limit of <math> \mathop{\mathrm{Prob}}(''X''∩''S'' ≠ ∅)/s</math> (or of <math> P(1,s)/s</math>) as the measure ''s'' of ''S'' tends to zero.  Moreover, for any two disjoint bounded subsets ''S'' and ''T'' of ''R'', the counts |''X''∩''S''| and |''X''∩''T''| are independent random variables.

==Properties==