Poisson process
E559807
The Poisson process is a fundamental stochastic process in probability theory that models random events occurring independently over time or space at a constant average rate.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Poisson distribution | 1 |
| Poisson processes | 1 |
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
counting process
ⓘ
point process ⓘ stochastic process ⓘ |
| alsoKnownAs | homogeneous Poisson process ⓘ |
| assumes |
constant average rate
ⓘ
independent increments ⓘ no simultaneous events with probability 1 ⓘ stationary increments ⓘ |
| field |
probability theory
ⓘ
stochastic processes ⓘ |
| generalizedBy |
compound Poisson process
ⓘ
non‑homogeneous Poisson process ⓘ |
| hasCountingProcessNotation | N(t) ⓘ |
| hasDistributionOfIncrements | Poisson distribution NERFINISHED ⓘ |
| hasIndependentIncrements | true ⓘ |
| hasIndexSet | non‑negative real numbers ⓘ |
| hasInterarrivalDistribution | exponential distribution ⓘ |
| hasInterarrivalTimesIID | true ⓘ |
| hasInterarrivalTimesMean | 1/λ ⓘ |
| hasInterarrivalTimesNotation | T1, T2, … ⓘ |
| hasInterarrivalTimesVariance | 1/λ² ⓘ |
| hasMeanIncrementOnInterval | λt for interval length t ⓘ |
| hasOrderlinessProperty | probability of more than one event in small interval is o(Δt) ⓘ |
| hasParameter | rate λ > 0 ⓘ |
| hasProbabilityGeneratingFunctionOfN(t) | exp(λt(z − 1)) ⓘ |
| hasProperty |
Markov property
NERFINISHED
ⓘ
cadlag sample paths ⓘ memoryless interarrival times ⓘ right‑continuous with left limits ⓘ starts at 0 almost surely ⓘ time‑homogeneous ⓘ |
| hasStateSpace | non‑negative integers ⓘ |
| hasStationaryIncrements | true ⓘ |
| hasSuperpositionProperty | sum of independent Poisson processes is Poisson ⓘ |
| hasThinningProperty | independent thinning yields Poisson subprocesses ⓘ |
| hasVarianceOfIncrementOnInterval | λt for interval length t ⓘ |
| isSpecialCaseOf |
Lévy process
ⓘ
Markov jump process NERFINISHED ⓘ renewal process ⓘ |
| models |
random events in space
ⓘ
random events in time ⓘ |
| satisfies |
N(0) = 0 almost surely
ⓘ
N(t) − N(s) ~ Poisson(λ(t − s)) for t > s ⓘ |
| usedIn |
insurance risk modeling
ⓘ
physics of radioactive decay ⓘ queueing theory ⓘ reliability engineering ⓘ spatial statistics ⓘ telecommunications modeling ⓘ traffic flow modeling ⓘ |
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Poisson processes