@MISC{Lebanon_thepoisson, author = {Guy Lebanon}, title = {The Poisson Process}, year = {} }

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Abstract

In this note we summarize the basic definitions and properties involving the Poisson process. For more details see [1]. The Poisson process is a continuous time (t ≥ 0) discrete space Xt ∈ N = {0, 1,..., } Markov process that follows from assuming that the probability of getting a single event in a small time interval [t, t + ∆] is ∆λ (for some λ> 0) and the probability of getting more than a single event in a small time interval is negligible. It is often used to model counting or accumulation of independent events, i.e., Xt is the number of cars arriving in an intersection or the number of calls arriving at a switchboard by time t. Postulates and Differential Equation It is useful to derive the Poisson process from first principles. It exposes its underlying assumptions and points the way for potential generalizations. The basic postulates are 1. The process is a Markov process with stationary transition probabilities i.e., Pij(t) = P (Xt+u = j|Xu = i), t> 0 is independent of u ≥ 0. 2. P (Xt+h − Xt = 1|Xt = x) = λh + o(h) i.e., limh→0 P (Xt+h − Xt = 1|Xt = x)/h = λ 3. X0 = 0 with probability 1 4. P (Xt+h − Xt = 0|Xt = x) = 1 − λh + o(h). Postulates 2 and 4 imply that the probability of multiple events occuring in a short time interval is negligible. Denoting Pm(t) = P (Xt = m) the above postulates imply P0(t + h) = P0(t)P0(h) = P0(t)(1 − λh + o(h)) which in turn imply (P0(t + h) − P0(t))/h = −P0(t)(λh + o(h))/h = −P0(t)(λ + o(1)) and consequentially we get the differential equation P ′ 0(t) = −aP0(t) whose well known solution (subject to initial condition X0 = 0 is P0(t) = e −λt. From the law of total probability we get the following differential equation Pm(t + h) − Pm(t) = Pm(t)P0(h) + Pm−1(t)P1(h) +