Actuarial Notes

Poisson processes (and mixture distributions)

By James W. Daniel · Casualty Actuarial Society · 2008

Study guide: Exam MAS-I

A CAS study note on Poisson processes and mixture distributions that explains the ideas and states the key facts rather than deriving the theory. Written by James W. Daniel of Austin Actuarial Seminars and dated June 26, 2008, it covers the Poisson-process material then needed for Exams MLC/3L of the SoA and CAS; in place of problems of its own, each section ends with a list of practice problems from the SoA Exam MLC/M/3 and CAS Exam 3 archives.

On the syllabus

  • Exam MAS-I — objectives A1–A5; the whole note (an online publication), with the practice problems its Foreword lists available on the CAS website.

Foreword

  • The note covers the material on Poisson processes needed for Exams MLC/3L of the SoA and CAS, concentrating on explaining the ideas and stating the important facts rather than deriving the theory.
  • To save space it lists practice problems to download from the SoA and CAS websites rather than containing problems.

1 Poisson processes

  • 1.1 What's a Poisson process?
    • A Poisson process N with rate (intensity) function \lambda is a counting process with independent increments over non-overlapping intervals (touching at an endpoint is OK), and each increment N(t+h) - N(t) is a Poisson random variable with mean \int_t^{t+h} \lambda(z)\,dz.
    • The mean-value function m(t) = \int_0^t \lambda(z)\,dz — the operational time — gives E[N(t)] = m(t); a constant \lambda makes the process homogeneous, and a rate that varies makes it a non-homogeneous one.
    • 1.1.1 Some important time intervals
      • T_n is the time of the nth event and V_j = T_j - T_{j-1} the jth interevent time; a probability about T_n is best translated into one about N(t) — T_3 > 1.2 exactly when N(1.2) \le 2.
      • The time T(x) from time x to the next event has \Pr[T(x) > t] = e^{-\int_x^{x+t} \lambda(z)\,dz}, the survival-model formula with \lambda in place of the force of mortality \mu.
    • Problems 1.1
      • From the SoA Exam MLC/M/3 archives: Spring 2007 #5, 25, 26; Fall 2006 #10; Spring 2005 #25; Fall 2004 #16, 26; Fall 2003 #26; Fall 2002 #9; Fall 2001 #10.
      • From the CAS Exam 3 archives: Spring 2007 #1; Fall 2006 #28; Spring 2006 #33; Fall 2005 #26, 28; Spring 2005 #14; Fall 2004 #18, 19; Spring 2004 #27.
  • 1.2 Compound Poisson processes
    • A compound Poisson process is S(t) = \sum_{j=1}^{N(t)} X_j, with N a Poisson process and the severities X_j identically distributed and independent of N(t) and of each other.
    • E[S(t)] = E[N(t)]\,E[X] and \text{Var}[S(t)] = E[N(t)]\,\text{Var}[X] + \text{Var}[N(t)]\,E[X]^2 = E[N(t)]\,E[X^2]; when E[N(t)] is large, S(t) is approximated by a Normal random variable with that mean and variance.
    • Problems 1.2
      • From the SoA Exam MLC/M/3 archives: Fall 2005 #40; Fall 2002 #15; Spring 2001 #4.
      • From the CAS Exam 3 archives: Spring 2007 #2; Spring 2004 #26; Fall 2003 #30.
  • 1.3 Creating new processes from existing ones
    • 1.3.1 Counting special types of events
      • Thinning: if an event at time t is of Type j with probability \pi_j(t), the events of each type form independent Poisson processes with rates \pi_j(t)\,\lambda(t).
    • 1.3.2 Sums of Poisson processes
      • The sum of independent Poisson processes with rates \lambda_1 and \lambda_2 is a Poisson process with rate \lambda_1(t) + \lambda_2(t).
    • 1.3.3 Mixture distributions
      • The Mixing Method: a quantity that is linear in the mixture's probability or density function — a probability, the mean, the second moment, but not the variance — is computed as if the case were known, then averaged as the case varies.
      • A Poisson random variable whose mean is gamma with parameters \alpha and \theta is a negative binomial with r = \alpha and \beta = \theta.
      • A mixture of Poisson processes arises when an insurer models each insured's claims as a homogeneous Poisson process but treats the rate \lambda as varying randomly among insureds.
    • Problems 1.3
      • From the SoA Exam MLC/M/3 archives: Spring 2007 #6; Fall 2006 #9; Spring 2005 #5, 6; Fall 2002 #5, 20; Fall 2001 #19, 27; Spring 2001 #3, 15, 37.
      • From the CAS Exam 3 archives: Fall 2005 #27, 31; Spring 2005 #7, 17; Fall 2004 #17; Spring 2004 #31; Fall 2003 #31.
  • 1.4 Homogeneous Poisson processes
    • 1.4.1 Fundamentals
      • With a constant rate each increment N(t+h) - N(t) is Poisson with mean \lambda h, depending only on the interval's length: homogeneous Poisson processes have stationary increments.
      • The interevent times V_j are independent exponential random variables with mean 1/\lambda, and T_n is gamma with \alpha = n and \theta = 1/\lambda; conversely, a counting process with independent exponential interevent times of mean \theta is a homogeneous Poisson process with rate 1/\theta.
    • 1.4.2 Sums of compound Poisson processes
      • Merging two independent compound Poisson processes gives one with rate \lambda_1 + \lambda_2 whose severity is a mixture of X_1 and X_2 with probabilities \lambda_1/(\lambda_1+\lambda_2) and \lambda_2/(\lambda_1+\lambda_2).
    • 1.4.3 Counting special types of events in a homogeneous process
      • Thinning a homogeneous process gives a homogeneous or a non-homogeneous process according to whether \pi_j(t) is constant — one way non-homogeneous processes arise in practice, as with claims whose repair cost exceeds $1,000 under inflation.
    • Problems 1.4
      • From the SoA Exam MLC/M/3 archives: Spring 2007 #5, 6, 26; Fall 2006 #8, 9; Fall 2005 #7,8; Spring 2005 #5, 6, 24; Fall 2003 #11, 20, 26; Fall 2002 #9.
      • From the CAS Exam 3 archives: Spring 2007 #1, 2; Fall 2006 #26, 27; Fall 2005 #29, 31; Spring 2005 #7, 11, 17; Fall 2004 #17; Spring 2004 #31; Fall 2003 #13, #31.

Related readings

  • Introduction to Probability Models (Ross - 2019) — assigned with it for Domain A (Probability Models) of the MAS-I outline
  • Life Contingencies (Struppeck - 2014) — assigned with it for Domain A of the MAS-I outline

Sources