PoissonProcess

The counting process of events arriving at rate $\lambda$ per unit time.

PoissonProcess(lambda)an inert process object — carries its rate, unevaluated. SliceDistribution and RandomFunction both read it.

Typed signatures

Type	Package	Applies
(real<0..>) -> expression<PoissonProcess>	statistics	always

Domain: Statistics

Details

• $SliceDistribution(PoissonProcess(\lambda), t) = PoissonDistribution(\lambda t)$ — exact: the event count up to a fixed time $t$ is Poisson by definition.
• RandomFunction(PoissonProcess(lambda), {tmin, tmax, dt}) draws one seeded sample path by summing an independent $Poisson(\lambda\, dt)$ draw per grid step — exact in distribution, since disjoint-interval counts of a Poisson process are themselves independent Poisson draws. Counts are nondecreasing nonnegative integers along the path.

Examples

Basic #

$Mean(PoissonProcess(\lambda)[t]) = \lambda t$

live ↗

In$\mathrm{mean}(\mathrm{SliceDistribution}(\mathrm{PoissonProcess}(3), 5))$Out$15$

live ↗

In$\mathrm{SliceDistribution}(\mathrm{PoissonProcess}(2), 3)$Out$\operatorname{PoissonDistribution}(6)$

Scope #

$Variance(PoissonProcess(\lambda)[t]) = \lambda t$, same as its Mean — a property of the Poisson distribution.

live ↗

In$\mathrm{Variance}(\mathrm{SliceDistribution}(\mathrm{PoissonProcess}(2), 3))$Out$6$

The slice at 4 is $Poisson(12)$, whose median is 12

live ↗

In$\mathrm{median}(\mathrm{PoissonProcess}(3)\lhd4)$Out$12$

About

Examples	4
Checked against	Wolfram Language 1 agree

See also: WienerProcess, SliceDistribution, RandomFunction, PoissonDistribution