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Poisson process
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(Definition)
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A counting process
 is called a simple Poisson, or simply a Poisson process with parameter  , also known as the intensity, if
 ,-
 has stationary independent increments,
-
 ,
-
 ,
where  is the O notation.
Remarks.
- The intensity
 is assumed to be a constant in terms of  .
- Condition 3 above says that the rate in which the an event occurs once in time interval
 , as  approaches 0, is  . Condition 4 says that the event occurs more than once is very unlikely (the rate approaches zero as the time interval shrinks to zero).
- It can be shown that
 has a Poisson distribution (hence the name of the stochastic process) with parameter  :
- Therefore,
![$ \operatorname{E}[X(t)]=\lambda t$](http://images.planetmath.org:8080/cache/objects/6733/l2h/img16.png "$ \operatorname{E}[X(t)]=\lambda t$") .
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"Poisson process" is owned by CWoo. [ full author list (2) ]
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(view preamble)
| Other names: |
homogeneous Poisson process |
| Also defines: |
simple Poisson process, intensity |
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Cross-references: stochastic process, Poisson distribution, interval, event, terms, O notation, stationary independent increments, parameter, counting process
There are 7 references to this entry.
This is version 5 of Poisson process, born on 2005-02-09, modified 2006-10-04.
Object id is 6733, canonical name is PoissonProcess.
Accessed 8534 times total.
Classification:
| AMS MSC: | 60G51 (Probability theory and stochastic processes :: Stochastic processes :: Processes with independent increments) |
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Pending Errata and Addenda
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