poisson_process_notes

# poisson_process_notes - Notes on Poisson Processes October...

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Unformatted text preview: Notes on Poisson Processes October 6, 2010 1 Introduction Depending on the book (or website) you read, a “Poisson Process” can have many different definitions. For me, the key axioms defining it are as follows: First, we fix a time interval [0 , T ], and a certain parameter λ , and we have associated to this interval a certain number X of events that can occur, satisfying: • Associated to any set of DISJOINT subintervals I 1 , ..., I k ⊆ [0 , T ], we have INDEPENDENT random variables X I 1 , ..., X I k , where X I j is the number of events occurring in the time window I j . • Let I := [ x, x + h ] ⊆ [0 , T ]. Then, lim h → P ( X I = 1) λh = 1 . That is to say, as h tends to 0, P ( X I = 1) grows like λh . • Using the same interval I as in the above, we have that the probability that 2 or more events occur in I has size o ( h ); that is, lim h → P ( X I ≥ 2) h = 0 . In the third item we used little-oh notation o ( h ). Let us remind ourselves what this means, since we will use it later throughout the course: Given 1 positive functions...
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poisson_process_notes - Notes on Poisson Processes October...

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