EE 223 : Stochastic Systems : Estimation and Control Spring 2007 Lecture 20 — April 3

20.1.1 Poisson Process Recall the definition of a Poisson Process with rate γ, depicted in Figure 20.1. This is a stochastic process counting the number of arrivals up to time t starting from time 0, (Nt, t ≥ 0), where the interarrival times are exponentially distributed with mean 1/γ. we can refer to this as a Poisson (γ) process The number of points in an interval (a, b] is denoted N(a, b]. Proposition 20.1. If (a, b] and (c, d] are disjoint then N(a, b] and N(c, d] are independent. Proposition 20.1 allows one to define a Poisson process on R by saying N(a, b] is given by a Poisson distribution with parameter γ(b − a). The number of points in a disjoint intervals are independent. For more details, refer to a first or second course on probability such as EE226A.