Superposition and thinning
Section of Poisson processes, part of Stochastic processes
Superposition and thinning
| Type | Section |
| Taught in | Poisson processes |
| Part | Stochastic processes |
Independent Poisson processes of rates \lambda_1 and \lambda_2 superpose to a Poisson process of rate \lambda_1+\lambda_2. The sum of independent Poisson counts is Poisson with the summed mean, and the independent-increments property passes to the sum.
If each event of a rate-\lambda process is kept independently with probability r, the kept process is Poisson of rate r\lambda, and the kept and discarded processes are independent. If the retention probability is a function r(t), the kept intensity is r(t)\lambda(t).
The independence uses an independent mark on each event. A shock that makes many events of the same kind at once is a dependent mark. Ordinary thinning does not describe that stream.
Used in
See also
Homogeneous Poisson process · Forward equations of the Poisson process · Nonhomogeneous Poisson process · Spatial Poisson process · Intensity function · Interarrival time