por notes - Types of Poisson Processes [edit] Homogeneous...

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Types of Poisson Processes [ edit ] Homogeneous Poisson process A homogeneous Poisson process is characterized by a rate parameter λ such that the number of events in time interval ( t , t + τ] follows a Poisson distribution with associated parameter λτ. This relation is given as where N ( t + τ) − N ( t ) describes the number of events in time interval ( t , t + τ]. Just as a Poisson random variable is characterized by its scalar parameter λ, a homogeneous Poisson process is characterized by its rate parameter λ, which is the expected number of "events" or "arrivals" that occur per unit time. N ( t ) is a sample homogeneous Poisson process, not to be confused with a density or distribution function. [ edit ] Non-Homogeneous Poisson process In general, the rate parameter may change over time. In this case, the generalized rate function is given as λ( t ). Now the expected number of events between time a and time b is Thus, the number of arrivals in the time interval ( a , b ], given as N ( b ) − N ( a ), follows a Poisson distribution with associated parameter λ a , b A homogeneous Poisson process may be viewed as a special case when λ( t ) = λ, a constant rate. [ edit ] Spatial Poisson process A further variation on the Poisson process, called the spatial Poisson process, introduces a spatial dependence on the rate function and is given as where for some vector
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space V (e.g. R 2 or R 3 ). For any set (e.g. a spatial region) with finite measure , the number of events occurring inside this region can be modelled as a Poisson process with associated rate function λ S ( t ) such that In the special case that this generalized rate function is a separable function of time and space, we have: for some function . Without loss of generality, let else we may scale and λ( t ) appropriately. Now, represents the spatial probability density function of these random events in the following sense. The act of sampling this spatial Poisson process is equivalent to sampling a Poisson process with rate function λ( t ), and associating with each event a random vector sampled from the probability density function . A similar result can be shown for the general (non-separable) case. [ edit ] General characteristics of the Poisson process In its most general form, the only two conditions for a stochastic process to be a Poisson process are: Orderliness : which roughly means which implies that arrivals don't occur simultaneously (but this is actually a mathematically-stronger statement). Memorylessness (also called evolution without after-effects): the number of arrivals occurring in any bounded interval of time after time t is independent of the number of arrivals occurring before time t . These seemingly unrestrictive conditions actually impose a great deal of structure in the Poisson
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por notes - Types of Poisson Processes [edit] Homogeneous...

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