Poisson point process facts for kids

A Poisson process is a stochastic process. It counts the number of occurrences of an event leading up to a specified time. This is a counting process where the increments of time are independent of one another (the times do not overlap).

Definition

The counting process known as the Poisson process is defined as:

• N(0) = 0.
• N(t) has independent increments.
• The number of arrivals in any interval of length 𝜏 > 0 follows a Poisson distribution.

Where N(t) is the total number of events that occur by time t.

Images for kids

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  According to one statistical study, the positions of cellular or mobile phone base stations in the Australian city Sydney, pictured above, resemble a realization of a homogeneous Poisson point process, while in many other cities around the world they do not and other point processes are required.

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  An illustration of a marked point process, where the unmarked point process is defined on the positive real line, which often represents time. The random marks take on values in the state space S known as the mark space. Any such marked point process can be interpreted as an unmarked point process on the space [0,\infty]\times S . The marking theorem says that if the original unmarked point process is a Poisson point process and the marks are stochastically independent, then the marked point process is also a Poisson point process on [0,\infty]\times S . If the Poisson point process is homogeneous, then the gaps \tau_i in the diagram are drawn from an exponential distribution.

See also

In Spanish: Proceso de Poisson para niños