Intensity function
Topic of Poisson processes, part of Stochastic processes
Intensity function
| Type | Topic |
| Taught in | Poisson processes |
| Part | Stochastic processes |
The rate \lambda(t)\ge0 of a non-homogeneous Poisson process, and its integral \Lambda(t)=\int_0^t\lambda(u)\,du, the cumulative intensity. The count in (s,t] is Poisson with mean \Lambda(t)-\Lambda(s). A constant intensity gives the homogeneous process.
In Poisson processes
From Poisson processes
A non-homogeneous Poisson process has a deterministic rate \lambda(u)\ge 0, locally integrable. The cumulative intensity is \Lambda(t)=\int_0^t \lambda(u)\,du. Increments over disjoint intervals stay independent. Their means are differences of \Lambda, and stationarity of increments fails as soon as \lambda is not constant:
N(t)-N(s)\sim\mathrm{Poisson}\Bigl(\int_s^t \lambda(u)\,du\Bigr).
Used in
See also
Nonhomogeneous Poisson process · Homogeneous Poisson process · Forward equations of the Poisson process · Superposition and thinning · Spatial Poisson process · Interarrival time