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Unformatted text preview: Poisson processes (and mixture distributions) James W. Daniel Austin Actuarial Seminars www.actuarialseminars.com June 26, 2008 c circlecopyrt Copyright 2007 by James W. Daniel; reproduction in whole or in part without the express permission of the author is forbidden. Foreword This document covers the material on Poisson processes needed for Exams MLC/3L of the Society of Actuaries (SoA) and Casualty Actuarial Society (CAS). It concentrates on explaining the ideas and stating the important facts rather than deriving the theory. In order to conserve space, rather than containing problems it instead lists problems for practice that can be downloaded from the SoA website starting at http://www.soa.org/education/resources/edumultiplechoiceessayexaminations.aspx and from the CAS website starting at http://www.casact.org/admissions/studytools/ . 2 Chapter 1 Poisson processes 1.1 What’s a Poisson process? Let’s make our way towards a definition of a Poisson process. First of all, a Poisson process N is a stochastic process—that is, a collection of random variables N ( t ) for each t in some specified set. More specifically, Poisson processes are counting processes: for each t > 0 they count the number of “events” that happen between time 0 and time t . What kind of “events”? It depends on the application. You might want to count the number of insurance claims filed by a particular driver, or the number of callers phoning in to a help line, or the number of people retiring from a particular employer, and so on. Whatever you might mean by an “event”, N ( t ) denotes the number of events that occur after time 0 up through and including time t > 0. N (0) is taken to equal 0—no events can have occurred before you start counting. Since the times at which events occur are assumed to be random, N ( t ) is a random variable for each value of t . Note that N itself is called a random process , distinguishing it from the random variable N ( t ) at each value of t > 0. To understand counting processes, you need to understand the meaning and probability behavior of the increment N ( t + h ) − N ( t ) from time t to time t + h , where h > 0 and of course t ≥ 0. Since N ( t + h ) equals the random number of events up through t + h and N ( t ) equals the random number up through t , the increment is simply the random number of events occurring strictly after time t and up through and including time t + h . Note that N ( t ) itself can be viewed as an increment, namely from time 0 to time t , since N ( t ) = N ( t ) − N (0). A fundamental property of Poisson processes is that increments on nonoverlapping time inter vals are independent of one another as random variables—stated intuitively, knowing something about the number of events in one interval gives you no information about the number in a non overlapping interval. But notice the important modifier “nonoverlapping”. While the increments N (5 . 2) − N (3 . 1) and N (2 . 7)...
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This note was uploaded on 02/29/2012 for the course MATH 472 taught by Professor Zhu during the Spring '08 term at University of Illinois, Urbana Champaign.
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