@MISC{Birth_poissonprocesses, author = {Age Dependent Birth}, title = {Poisson processes}, year = {} }

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Abstract

A Poisson process is a model for a series of random observations occurring in time. x x x x x x x x t Let Y (t) denote the number of observations by time t. Note that for t < s, Y (s) − Y (t) is the number of observations in the time interval (t, s]. We assume the following: 1) Observations occur one at a time. 2) Numbers of observations in disjoint time intervals are independent random variables, i.e., if t0 < t1 < · · · < tm, then Y (tk) − Y (tk−1), k = 1,..., m are independent random variables. 3) The distribution of Y (t + a) − Y (t) does not depend on t.•First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 3 Characterization of a Poisson process Theorem 1 Under assumptions 1), 2), and 3), there is a constant λ> 0 such that, for t < s, Y (s) − Y (t) is Poisson distributed with parameter λ(s − t), that is, P {Y (s) − Y (t) = k} = (λ(s − t))k e k!