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| In [[probability theory]], a '''Poisson process''' is a [[stochastic process]] which counts the number of events<ref group="note">The word ''event'' used here is not an instance of the concept of [[event (probability theory)|''event'']] as frequently used in probability theory.</ref> and the time that these events occur in a given time interval. The time between each pair of consecutive events has an [[exponential distribution]] with parameter ''λ'' and each of these inter-arrival times is assumed to be independent of other inter-arrival times. | | In probability theory, a '''Poisson process''' is a stochastic process which counts the number of events<ref group="note">The word ''event'' used here is not an instance of the concept of event (probability theory)|''event'' as frequently used in probability theory.</ref> and the time that these events occur in a given time interval. The time between each pair of consecutive events has an exponential distribution with parameter ''λ'' and each of these inter-arrival times is assumed to be independent of other inter-arrival times. |
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| The Poisson process is a [[continuous-time process]]; the sum of a [[Bernoulli process]] can be thought of as its discrete-time counterpart. A Poisson process is a pure-birth process, the simplest example of a [[birth-death process]]. It is also a [[point process]] on the real half-line. | | The Poisson process is a continuous-time process; the sum of a Bernoulli process can be thought of as its discrete-time counterpart. A Poisson process is a pure-birth process, the simplest example of a birth-death process. It is also a point process on the real half-line. |
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| ==Definition== | | ==Definition== |
| {{Probability distribution
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| | name = Poisson
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| | type = mass
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| | pdf_image = [[File:poisson pmf.svg|325px|Plot of the Poisson PMF]]<br />The horizontal axis is the index ''k'', the number of occurrences. The function is only defined at integer values of ''k''. The connecting lines are only guides for the eye.
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| | cdf_image = [[File:poisson cdf.svg|325px|Plot of the Poisson CDF]]<br />The horizontal axis is the index ''k'', the number of occurrences. The CDF is discontinuous at the integers of ''k'' and flat everywhere else because a variable that is Poisson distributed only takes on integer values.
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| | notation = <math>\mathrm{Pois}(\lambda)\,</math>
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| | parameters = ''λ'' > 0 ([[real number|real]])
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| | support = ''k'' ∈ { 0, 1, 2, 3, ... }
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| | pdf = <math>\frac{\lambda^k}{k!}\cdot e^{-\lambda}</math>
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| | cdf = <math>\frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}\!</math> --or-- <math>e^{-\lambda} \sum_{i=0}^{\lfloor k\rfloor} \frac{\lambda^i}{i!}\ </math>
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| (for <math>k\ge 0</math> where <math>\Gamma(x, y)\,\!</math> is the [[Incomplete gamma function]] and <math>\lfloor k\rfloor</math> is the [[floor function]])
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| | mean = <math>\lambda\,\!</math>
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| | median = <math>\approx\lfloor\lambda+1/3-0.02/\lambda\rfloor</math>
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| | mode = <math>\lfloor\lambda\rfloor,\,\lceil\lambda\rceil - 1</math>
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| | variance = <math>\lambda\,\!</math>
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| | skewness = <math>\lambda^{-1/2}\,</math>
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| | kurtosis = <math>\lambda^{-1}\,</math>
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| | entropy = <math>\lambda[1\!-\!\log(\lambda)]\!+\!e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k\log(k!)}{k!}</math>
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| (for large <math>\lambda</math>)
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| <math>\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -</math><br> <math> \frac{19}{360 \lambda^3} + O\left(\frac{1}{\lambda^4}\right)</math><!--formula split with spaces-->
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| | pgf = <math> \exp(\lambda(z - 1))\,</math>
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| | mgf = <math>\exp(\lambda (e^{t}-1))\,</math>
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| | char = <math>\exp(\lambda (e^{it}-1))\,</math>
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| }}
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| The basic form of Poisson process, often referred to simply as "the Poisson process", is a continuous-time [[counting process]] {''N''(''t''), ''t'' ≥ 0} that possesses the following properties: | | [[File:Poisson Math.png|right]] |
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| | The basic form of Poisson process, often referred to simply as "the Poisson process", is a continuous-time counting process {''N''(''t''), ''t'' ≥ 0} that possesses the following properties: |
| * ''N''(0) = 0 | | * ''N''(0) = 0 |
| * [[Independent increments]] (the numbers of occurrences counted in disjoint intervals are independent from each other) | | * Independent increments (the numbers of occurrences counted in disjoint intervals are independent from each other) |
| * [[Stationary increments]] (the probability distribution of the number of occurrences counted in any time interval only depends on the length of the interval) | | * Stationary increments (the probability distribution of the number of occurrences counted in any time interval only depends on the length of the interval) |
| * The [[probability distribution]] of ''N''(''t'') is a [[Poisson distribution]]. | | * The probability distribution of ''N''(''t'') is a Poisson distribution. |
| * No counted occurrences are simultaneous. | | * No counted occurrences are simultaneous. |
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| Consequences of this definition include: | | Consequences of this definition include: |
| * The probability distribution of the waiting time until the next occurrence is an [[exponential distribution]]. | | * The probability distribution of the waiting time until the next occurrence is an exponential distribution. |
| * The occurrences are [[Uniform distribution (continuous)|distributed uniformly]] on any interval of time. (Note that ''N''(''t''), the total number of occurrences, has a Poisson distribution over (0, ''t''], whereas the location of an individual occurrence on {{math|''t'' ∈ (''a'', ''b'']}} is uniform.) | | * The occurrences are Uniform distribution (continuous)|distributed uniformly on any interval of time. (Note that ''N''(''t''), the total number of occurrences, has a Poisson distribution over (0, ''t''], whereas the location of an individual occurrence on {{math|''t'' ∈ (''a'', ''b'']}} is uniform.) |
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| Other types of Poisson process are described below. | | Other types of Poisson process are described below. |
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| ==Types==
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| ===Homogeneous===
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| [[Image:SampleProcess.png|frame|right|Sample Path of a counting Poisson process ''N''(''t'')]]
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| The ''homogeneous'' Poisson process is one of the most well-known [[Lévy process]]es. This process is characterized by a rate parameter λ, also known as ''intensity'', such that the number of events in time [[Interval (mathematics)|interval]] (''t'', ''t'' + ''τ''] follows a [[Poisson distribution]] with associated parameter ''λτ''. This relation is given as
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| :<math> P [(N(t+ \tau) - N(t)) = k] = \frac{e^{-\lambda \tau} (\lambda \tau)^k}{k!} \qquad k= 0,1,\ldots,</math>
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| where ''N''(''t'' + ''τ'') − ''N''(''t'') = ''k'' is the number of events in time interval (''t'', ''t'' + ''τ''].
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| 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 value|expected]] number of "events" or "arrivals" that occur per unit time.
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| ''N''(''t'') is a sample homogeneous Poisson process, not to be confused with a density or distribution function.
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| ===Non-homogeneous===
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| {{Main|Non-homogeneous Poisson process}}
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| In general, the rate parameter may change over time; such a process is called a '''non-homogeneous Poisson process''' or '''inhomogeneous Poisson process.'''
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| 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
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| :<math>\lambda_{a,b} = \int_a^b \lambda(t)\,dt.</math>
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| 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<sub>''a'',''b''</sub>
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| :<math> P [(N(b) - N(a)) = k] = \frac{e^{-\lambda_{a,b}} (\lambda_{a,b})^k}{k!} \qquad k= 0,1,\ldots.</math>
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| A rate function λ(''t'') in a non-homogeneous Poisson process can be either a deterministic function of time or an independent stochastic process, giving rise to a [[Cox process]]. A homogeneous Poisson process may be viewed as a special case when λ(''t'') = λ, a constant rate.
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| ===Spatial===
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| An important variation on the (notionally time-based) Poisson process is the spatial Poisson process. In the case of a one-dimension space (a line) the theory differs from that of a time-based Poisson process only in the interpretation of the index variable. For higher dimension spaces, where the index variable (now ''x'') is in some [[vector space]] ''V'' (e.g. '''R'''<sup>2</sup> or '''R'''<sup>3</sup>), a spatial Poisson process can be defined by the requirement that the random variables defined as the counts of the number of "events" inside each of a number of non-overlapping finite sub-regions of ''V'' should each have a Poisson distribution and should be independent of each other.
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| ===Space-time===
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| A further variation on the Poisson process, the space-time Poisson process, allows for separately distinguished space and time variables. Even though this can theoretically be treated as a pure spatial process by treating "time" as just another component of a vector space, it is convenient in most applications to treat space and time separately, both for modeling purposes in practical applications and because of the types of properties of such processes that it is interesting to study.
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| In comparison to a time-based inhomogeneous Poisson process, the extension to a space-time Poisson process can introduce a spatial dependence into the rate function, such that it is defined as <math>\lambda(x,t)</math>, where <math>x \in V</math> for some [[vector space]] ''V'' (e.g. '''R'''<sup>2</sup> or '''R'''<sup>3</sup>). However a space-time Poisson process may have a rate function that is constant with respect to either or both of ''x'' and ''t''. For any set <math>S \subset V</math> (e.g. a spatial region) with finite [[measure (mathematics)|measure]] <math>\mu(S)</math>, the number of events occurring inside this region can be modeled as a Poisson process with associated rate function λ<sub>''S''</sub>(''t'') such that
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| :<math>\lambda_S(t) = \int_S \lambda(x,t)\,d\mu(x).</math>
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| ====Separable space-time processes====
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| In the special case that this generalized rate function is a separable function of time and space, we have:
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| :<math>\lambda(x,t) = f(x) \lambda(t) \,</math>
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| for some function <math>f(x)</math>. Without loss of generality, let
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| :<math>\int_V f(x) \, d\mu(x)=1.</math>
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| (If this is not the case, ''λ''(''t'') can be scaled appropriately.) Now, <math>f(x)</math> 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 <math>X</math> sampled from the probability density function <math>f(x)</math>. A similar result can be shown for the general (non-separable) case.
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| ==Characterisation==
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| In its most general form, the only two conditions for a [[counting process]] to be a Poisson process are:{{Citation needed|date=May 2011}}
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| * '''Orderliness''': which roughly means
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| ::<math>\lim_{\Delta t\to 0} P(N(t+\Delta t) - N(t) > 1 \mid N(t+\Delta t) - N(t) \geq 1)=0 </math>
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| :which implies that arrivals don't occur simultaneously (but this is actually a mathematically stronger statement).
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| * '''[[Memorylessness]]''' (also called evolution without after-effects): the number of arrivals occurring in any bounded interval of time after time ''t'' is [[statistical independence|independent]] of the number of arrivals occurring before time ''t''.
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| 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 [[Statistical independence|independent]] random variables. For the homogeneous Poisson process, these inter-arrival times are [[Exponential distribution|exponentially distributed]] with parameter ''λ'' (mean 1/''λ'').
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| Also, the memorylessness property entails that the number of events in any time interval is independent of the number of events in any other interval that is disjoint from it. This latter property is known as the ''independent increments'' property of the Poisson process.
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| ==Properties==
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| As defined above, the stochastic process {''N''(''t'')} is a [[Markov process]], or more specifically, a [[continuous-time Markov process]].{{Citation needed|date=May 2011}}
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| To illustrate the [[exponential distribution|exponentially distributed]] inter-arrival times property, consider a homogeneous Poisson process ''N''(''t'') with rate parameter λ, and let ''T''<sub>''k''</sub> be the time of the ''k''th arrival, for ''k'' = 1, 2, 3, ... . Clearly the number of arrivals before some fixed time ''t'' is less than ''k'' ''if and only if'' the waiting time until the ''k''th arrival is more than ''t''. In symbols, the event [''N''(''t'') < ''k''] occurs ''if and only if'' the event [''T''<sub>''k''</sub> > ''t''] occurs. Consequently the probabilities of these events are the same:
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| :<math>P(T_k>t) = P(N(t)<k). \, </math>
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| In particular, consider the waiting time until the first arrival. Clearly that time is more than ''t'' ''if and only if'' the number of arrivals before time ''t'' is 0. Combining this latter property with the above probability distribution for the number of homogeneous Poisson process events in a fixed interval gives
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| :<math>P(T_1>t)=P(N(t)=0)=P [(N(t) - N(0)) = 0] = \frac{e^{-\lambda t} (\lambda t)^0}{0!} = e^{-\lambda t}.</math>
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| Consequently, the waiting time until the first arrival ''T''<sub>1</sub> has an [[exponential distribution]], and is thus [[memorylessness|memoryless]]. One can similarly show that the other interarrival times ''T''<sub>''k''</sub> − ''T''<sub>''k''−1</sub> share the same distribution. Hence, they are independent, identically distributed ([[i.i.d.]]) random variables with parameter ''λ'' > 0; and expected value 1/λ. For example, if the average rate of arrivals is 5 per minute, then the average waiting time between arrivals is 1/5 minute.
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| ==Applications==
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| The classic example of phenomena well modelled by a Poisson process is deaths due to horse kick in the Prussian army, as shown by [[Ladislaus Bortkiewicz]] in 1898.<ref>Ladislaus von Bortkiewicz, ''Das Gesetz der kleinen Zahlen'' [The law of small numbers] (Leipzig, Germany: B.G. Teubner, 1898). On [http://books.google.com/books?id=o_k3AAAAMAAJ&pg=PA1#v=onepage&q&f=false page 1], Bortkiewicz presents the Poisson distribution. On [http://books.google.com/books?id=o_k3AAAAMAAJ&pg=PA23#v=onepage&q&f=false pages 23-25], Bortkiewicz presents his famous analysis of "4. Beispiel: Die durch Schlag eines Pferdes im preussischen Heere Getöteten." (4. Example: Those killed in the Prussian army by a horse's kick.).</ref><ref>{{cite book|title=Statistical Methods for Groundwater Monitoring|page=72|first1=Robert D.|last1=Gibbons|first2=Dulal|last2=Bhaumik|first3=Subhash|last3=Aryal|year=2009|publisher=John Wiley and Sons|isbn=0-470-16496-4}}</ref> The following examples are also well-modeled by the Poisson process:
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| * Requests for telephone calls at a switchboard.
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| * Goals scored in a [[soccer]] match.<ref>{{cite doi|10.1209/0295-5075/89/38007}}</ref>
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| * Requests for individual documents on a web server.<ref name="ArlittMartin" />
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| * Particle emissions due to [[radioactive decay]] by an unstable substance. In this case the Poisson process is non-homogeneous in a predictable manner - the emission rate declines as particles are emitted.
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| * [[L. F. Richardson]] showed that the outbreak of war followed a Poisson process from 1820 to 1950.<ref>{{cite doi|10.1511/2002.1.10}}</ref>
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| * Photons landing on a photodiode, in particular in low light environments. This phenomena is related to [[shot noise]].
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| In [[queueing theory]], the times of customer/job arrivals at queues are often assumed to be a Poisson process.
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| ==Occurrence==
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| The [[Palm–Khintchine theorem]] provides a result that shows that the superposition of many low intensity non-Poisson point processes will be close to a Poisson process.
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| ==See also==
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| *[[Compound Poisson distribution]]
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| *[[Compound Poisson process]]
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| *[[Fractional Poisson process]]
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| *[[Renewal process]]
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| *[[Gamma distribution]]
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| *[[Markovian arrival processes]]
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| *[[Poisson sampling]]
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| *[[Non-homogeneous Poisson process]] where ''λ'' can be function of time ''λ''(''t'')
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| *[[Cox process]] where ''λ''(''t'') can be a stochastic process
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| *[[Bartlett's theorem]]
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| ==Notes==
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| {{reflist|group="note"}}
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| *{{cite book |last=Cox |first=D. R. |last2=Isham |first2=V. I. |year=1980 |title=Point Processes |location= |publisher=Chapman & Hall |isbn=0-412-21910-7 }}
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| *{{cite book |last=Ross |first=S. M. |year=1995 |title=Stochastic Processes |location= |publisher=Wiley |isbn=978-0-471-12062-9 }}
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| *{{cite book |last=Snyder |first=D. L. |last2=Miller |first2=M. I. |year=1991 |title=Random Point Processes in Time and Space |location= |publisher=Springer-Verlag |isbn=0-387-97577-2 }}
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| *{{cite book |last=Kingman |last2=Charles |first2=John Frank |year=1992 |title=Poisson processes. Vol. 3. |location= |publisher=Clarendon Press }}
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| {{Stochastic processes}}
| | </references> |
| | <references group="note"/> |
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| [[Category:Stochastic processes]] | | [[Category:ReDiClus]][[Category:Neurobiology]] |
| [[Category:Markov processes]] | |
| [[Category:Poisson processes| ]]
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| [[Category:Spatial processes]]
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