Actuarial Notes

Introduction to Probability Models

By Sheldon M. Ross · Academic Press · 2019

Study guide: Exam MAS-I

A textbook introducing elementary probability modelling and stochastic processes, and how probability theory can be applied. In the publisher's description those applications run through engineering, computer science, management science, the physical and social sciences and operations research; the twelfth edition keeps the text's hallmark features — its writing style, and exercises and examples across a wide breadth of probability topics — and includes many real-world applications in engineering, science, business and economics.

On the syllabus

  • Exam MAS-I — objectives A1–A6; Sections 4.1–4.8, 5.2.1–5.2.4, 5.3, 5.4.1, 5.4.2, 9.1–9.6, 11.1, 11.2.1 and 11.2.2 (of the 11th, 12th or 13th edition).

Preface

1 Introduction to Probability Theory

2 Random Variables

3 Conditional Probability and Conditional Expectation

  • 3.1 Introduction
  • 3.2 The Discrete Case
  • 3.3 The Continuous Case
  • 3.4 Computing Expectations by Conditioning
    • 3.4.1 Computing Variances by Conditioning
  • 3.5 Computing Probabilities by Conditioning
  • 3.6 Some Applications
    • 3.6.1 A List Model
    • 3.6.2 A Random Graph
    • 3.6.3 Uniform Priors, Polya's Urn Model, and Bose–Einstein Statistics
    • 3.6.4 Mean Time for Patterns
    • 3.6.5 The k-Record Values of Discrete Random Variables
    • 3.6.6 Left Skip Free Random Walks
  • 3.7 An Identity for Compound Random Variables
    • 3.7.1 Poisson Compounding Distribution
    • 3.7.2 Binomial Compounding Distribution
    • 3.7.3 A Compounding Distribution Related to the Negative Binomial
  • Exercises

4 Markov Chains

  • 4.1 Introduction
  • 4.2 Chapman–Kolmogorov Equations
  • 4.3 Classification of States
  • 4.4 Long-Run Proportions and Limiting Probabilities
    • 4.4.1 Limiting Probabilities
  • 4.5 Some Applications
    • 4.5.1 The Gambler's Ruin Problem
    • 4.5.2 A Model for Algorithmic Efficiency
    • 4.5.3 Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem
  • 4.6 Mean Time Spent in Transient States
  • 4.7 Branching Processes
  • 4.8 Time Reversible Markov Chains
  • 4.9 Markov Chain Monte Carlo Methods
  • 4.10 Markov Decision Processes
  • 4.11 Hidden Markov Chains
    • 4.11.1 Predicting the States
  • Exercises
  • References

5 The Exponential Distribution and the Poisson Process

  • 5.1 Introduction
  • 5.2 The Exponential Distribution
    • 5.2.1 Definition
    • 5.2.2 Properties of the Exponential Distribution
    • 5.2.3 Further Properties of the Exponential Distribution
    • 5.2.4 Convolutions of Exponential Random Variables
    • 5.2.5 The Dirichlet Distribution
  • 5.3 The Poisson Process
    • 5.3.1 Counting Processes
    • 5.3.2 Definition of the Poisson Process
    • 5.3.3 Further Properties of Poisson Processes
    • 5.3.4 Conditional Distribution of the Arrival Times
    • 5.3.5 Estimating Software Reliability
  • 5.4 Generalizations of the Poisson Process
  • 5.5 Random Intensity Functions and Hawkes Processes
  • Exercises
  • References

6 Continuous-Time Markov Chains

  • 6.1 Introduction
  • 6.2 Continuous-Time Markov Chains
  • 6.3 Birth and Death Processes
  • 6.4 The Transition Probability Function P_{ij}(t)
  • 6.5 Limiting Probabilities
  • 6.6 Time Reversibility
  • 6.7 The Reversed Chain
  • 6.8 Uniformization
  • 6.9 Computing the Transition Probabilities
  • Exercises
  • References

7 Renewal Theory and Its Applications

  • 7.1 Introduction
  • 7.2 Distribution of N(t)
  • 7.3 Limit Theorems and Their Applications
  • 7.4 Renewal Reward Processes
  • 7.5 Regenerative Processes
    • 7.5.1 Alternating Renewal Processes
  • 7.6 Semi-Markov Processes
  • 7.7 The Inspection Paradox
  • 7.8 Computing the Renewal Function
  • 7.9 Applications to Patterns
    • 7.9.1 Patterns of Discrete Random Variables
    • 7.9.2 The Expected Time to a Maximal Run of Distinct Values
    • 7.9.3 Increasing Runs of Continuous Random Variables
  • 7.10 The Insurance Ruin Problem
  • Exercises
  • References

8 Queueing Theory

  • 8.1 Introduction
  • 8.2 Preliminaries
    • 8.2.1 Cost Equations
    • 8.2.2 Steady-State Probabilities
  • 8.3 Exponential Models
    • 8.3.1 A Single-Server Exponential Queueing System
    • 8.3.2 A Single-Server Exponential Queueing System Having Finite Capacity
    • 8.3.3 Birth and Death Queueing Models
    • 8.3.4 A Shoe Shine Shop
    • 8.3.5 Queueing Systems with Bulk Service
  • 8.4 Network of Queues
    • 8.4.1 Open Systems
    • 8.4.2 Closed Systems
  • 8.5 The System M/G/1
    • 8.5.1 Preliminaries: Work and Another Cost Identity
    • 8.5.2 Application of Work to M/G/1
    • 8.5.3 Busy Periods
  • 8.6 Variations on the M/G/1
    • 8.6.1 The M/G/1 with Random-Sized Batch Arrivals
    • 8.6.2 Priority Queues
    • 8.6.3 An M/G/1 Optimization Example
    • 8.6.4 The M/G/1 Queue with Server Breakdown
  • 8.7 The Model G/M/1
    • 8.7.1 The G/M/1 Busy and Idle Periods
  • 8.8 A Finite Source Model
  • 8.9 Multiserver Queues
    • 8.9.1 Erlang's Loss System
    • 8.9.2 The M/M/k Queue
    • 8.9.3 The G/M/k Queue
    • 8.9.4 The M/G/k Queue
  • Exercises

9 Reliability Theory

  • 9.1 Introduction
  • 9.2 Structure Functions
    • 9.2.1 Minimal Path and Minimal Cut Sets
  • 9.3 Reliability of Systems of Independent Components
  • 9.4 Bounds on the Reliability Function
    • 9.4.1 Method of Inclusion and Exclusion
    • 9.4.2 Second Method for Obtaining Bounds on r(\mathbf{p})
  • 9.5 System Life as a Function of Component Lives
  • 9.6 Expected System Lifetime
    • 9.6.1 An Upper Bound on the Expected Life of a Parallel System
  • 9.7 Systems with Repair
    • 9.7.1 A Series Model with Suspended Animation
  • Exercises
  • References

10 Brownian Motion and Stationary Processes

  • 10.1 Brownian Motion
  • 10.2 Hitting Times, Maximum Variable, and the Gambler's Ruin Problem
  • 10.3 Variations on Brownian Motion
    • 10.3.1 Brownian Motion with Drift
    • 10.3.2 Geometric Brownian Motion
  • 10.4 Pricing Stock Options
    • 10.4.1 An Example in Options Pricing
    • 10.4.2 The Arbitrage Theorem
    • 10.4.3 The Black–Scholes Option Pricing Formula
  • 10.5 The Maximum of Brownian Motion with Drift
  • 10.6 White Noise
  • 10.7 Gaussian Processes
  • 10.8 Stationary and Weakly Stationary Processes
  • 10.9 Harmonic Analysis of Weakly Stationary Processes
  • Exercises
  • References

11 Simulation

  • 11.1 Introduction
  • 11.2 General Techniques for Simulating Continuous Random Variables
    • 11.2.1 The Inverse Transformation Method
    • 11.2.2 The Rejection Method
    • 11.2.3 The Hazard Rate Method
  • 11.3 Special Techniques for Simulating Continuous Random Variables
    • 11.3.1 The Normal Distribution
    • 11.3.2 The Gamma Distribution
    • 11.3.3 The Chi-Squared Distribution
    • 11.3.4 The Beta (n, m) Distribution
    • 11.3.5 The Exponential Distribution—The Von Neumann Algorithm
  • 11.4 Simulating from Discrete Distributions
    • 11.4.1 The Alias Method
  • 11.5 Stochastic Processes
    • 11.5.1 Simulating a Nonhomogeneous Poisson Process
    • 11.5.2 Simulating a Two-Dimensional Poisson Process
  • 11.6 Variance Reduction Techniques
    • 11.6.1 Use of Antithetic Variables
    • 11.6.2 Variance Reduction by Conditioning
    • 11.6.3 Control Variates
    • 11.6.4 Importance Sampling
  • 11.7 Determining the Number of Runs
  • 11.8 Generating from the Stationary Distribution of a Markov Chain
    • 11.8.1 Coupling from the Past
    • 11.8.2 Another Approach
  • Exercises
  • References

12 Coupling

  • 12.1 A Brief Introduction
  • 12.2 Coupling and Stochastic Order Relations
  • 12.3 Stochastic Ordering of Stochastic Processes
  • 12.4 Maximum Couplings, Total Variation Distance, and the Coupling Identity
  • 12.5 Applications of the Coupling Identity
    • 12.5.1 Applications to Markov Chains
  • 12.6 Coupling and Stochastic Optimization
  • 12.7 Chen–Stein Poisson Approximation Bounds
  • Exercises

Solutions to Starred Exercises

Index

Sources