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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 process. In particular, they imply that the time between consecutive events (called interarrival times) are independent random variables. For the homogeneous Poisson process, these inter- arrival times are exponentially -distributed with parameter λ. Also, the memorylessness property
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