Poisson point process

A Poisson point process of parameter λ is a stochastic process N(t) such that N(O)=0, N(t) is incremented by +1 after a time T distributed according to an exponential law of parameter λ. We are talking about Poisson arrivals if the time between two arrivals is exponential.

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Take the value of N(t) for state, then the continuous-time Markov chain associated with the Poisson process λ is:

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It is possible to know the probability that N is at number k at time t by the formula:

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N(t) is distributed according to a Poisson distribution of parameter λt.

The Poisson process associates and decomposes as follows:

  • The union of n Poisson process is a Poisson process whose parameter is the sum of n parameters
  • A Poisson process that decomposes into n processes with pi probabilities. These n processes are then respective rate Poisson processes λpi

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