Transcription of 1 IEOR 6711: Notes on the Poisson Process
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1 IEOR 6711: Notes on the Poisson Process
Copyright c 2009 by Karl Sigman 1 IEOR 6711 : Notes on the Poisson Process We present here the essentials of the Poisson point Process with its many interesting properties. As preliminaries, we first define what a point Process is, define the renewal point Process and state and prove the Elementary Renewal Theorem. Point Processes Definition A simple point Process = {tn : n 1} is a sequence of strictly increas- ing points 0 < t1 < t2 < , (1). def with tn as n . With N (0) = 0 we let N (t) denote the number of points that fall in the interval (0, t]; N (t) = max{n : tn t}. {N (t) : t 0} is called the counting Process for . If the tn are random variables then is called a random point Process . def We sometimes allow a point t0 at the origin and define t0 = 0. Xn = tn tn 1 , n 1, is called the nth interarrival time. We view t as time and view tn as the nth arrival time (although there are other kinds of applications in which the points tn denote locations in space as opposed to time).)
n are random variables then is called a random point process. We sometimes allow a point t 0 at the origin and de ne t 0 def= 0. X n= t n t n 1; n 1, is called the nth interarrival time. We view tas time and view t n as the nth arrival time (although there are other kinds of applications in which the points t n denote locations in space as ...
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Stochastic, Random, Probability and stochastic, Random variables, PROBABILITY, Random variables Probability, Chapter 1 Introduction to Econometrics, Variables, SC505 STOCHASTIC PROCESSES Class Notes, Probability, Statistics, and Stochastic Processes, PROBABILITY AND STOCHASTIC PROCESSES, MIT OpenCourseWare, Stochastic Calculus: An Introduction with Applications