Introduction to Probability Models

Citation

Ross, S. M. (2023). Introduction to Probability Models (13th ed.). Academic Press (Elsevier).

Chapter Summary

The 13th edition has 13 chapters in about 870 pages. The publisher says it keeps the organization that students and professors have relied on since 1972, and lists new coverage of martingales. Chapters 1 to 11 carry the titles used in earlier editions, Chapter 12 (Coupling) was added in the 12th edition, and Chapter 13 (Martingales) is new. The section-level details below follow the contents and preface of the tenth edition, which the publisher has posted; the current edition has been revised, so individual sections may differ.

Chapter 1: Introduction to Probability Theory

Presents the axiomatic framework: sample spaces and events, probabilities defined on events, conditional probability, independent events and Bayes' formula.

Chapter 2: Random Variables

Introduces discrete random variables (Bernoulli, binomial, geometric, Poisson) and continuous ones (uniform, exponential, gamma, normal), expectation, jointly distributed random variables and covariance, moment generating functions, limit theorems and a first look at stochastic processes.

Chapter 3: Conditional Probability and Conditional Expectation

Develops conditioning, which the preface calls one of the key tools of the subject and stresses throughout. It covers the discrete and continuous cases, computing expectations, variances and probabilities by conditioning, and applications such as a list model, a random graph and the Polya urn.

Chapter 4: Markov Chains

The first stochastic process: Chapman–Kolmogorov equations, classification of states, limiting probabilities, the gambler's ruin problem, branching processes, time reversibility, Markov chain Monte Carlo methods, Markov decision processes and hidden Markov chains.

Chapter 5: The Exponential Distribution and the Poisson Process

Studies the exponential distribution and its properties, then the Poisson process (definition, interarrival and waiting times, conditional distribution of arrival times) and its generalizations: nonhomogeneous, compound and mixed Poisson processes.

Chapter 6: Continuous-Time Markov Chains

Markov chains in continuous time with an emphasis on birth and death models, the transition probability function, limiting probabilities, time reversibility and uniformization.

Chapter 7: Renewal Theory and Its Applications

Counting processes more general than the Poisson: the distribution of N(t), limit theorems, renewal reward and regenerative processes, semi-Markov processes, the inspection paradox, patterns and the insurance ruin problem.

Chapters 8–9: Queueing Theory; Reliability Theory

Queueing covers cost identities, exponential models, networks of queues, the M/G/1 and G/M/1 systems and multiserver queues. Reliability covers structure functions, minimal path and cut sets, reliability bounds, system life as a function of component lives and systems with repair.

Chapters 10–11: Brownian Motion and Stationary Processes; Simulation

Chapter 10 covers Brownian motion, its variations, option pricing and the arbitrage theorem leading to the Black–Scholes formula, white noise, Gaussian and stationary processes. Chapter 11 covers generating random variables (inverse transformation, rejection, alias methods), simulating stochastic processes, variance reduction (antithetic variables, control variates, importance sampling) and generating from a stationary distribution.

Chapters 12–13: Coupling; Martingales

Chapter 12 was added in the 12th edition; a review in The Mathematical Gazette describes it as a short chapter on coupling and stochastic order. Chapter 13, on martingales, is the new coverage the publisher lists for the 13th edition.

Key Concepts

Chapters 1–3: Probability and Conditioning

• Conditional expectation: the average of a variable given information, used to compute expectations, variances and probabilities in stages.
• Moment generating function: a tool for sums of independent random variables and for limit theorems.

Chapters 4–7: Markov Chains, Poisson and Renewal Processes

• Markov chain: a process whose next state depends only on the present state, studied through classification of states and limiting probabilities.
• Poisson process: random arrivals with exponential interarrival times.
• Renewal process: a counting process with iid interarrival times, more general than the Poisson.

Chapters 8–11: Models for Applications

• Queueing system: customers, servers and waiting lines analysed with cost identities and steady-state probabilities.
• Structure function: the mapping from component states to system state in reliability.
• Brownian motion and the Black–Scholes formula: a continuous-time model and its use in option pricing.
• Simulation and variance reduction: generating random variables and making estimates more efficient.

Chapters 12–13: Coupling and Martingales

• Coupling: building random variables on one probability space to compare their distributions, with stochastic order.
• Martingale: a process whose expected next value, given the past, equals its present value.

Critical Analysis

Strengths

• Breadth in one volume. The sequence runs from elementary probability through Markov chains, Poisson processes, renewal theory, queueing, reliability, Brownian motion, simulation and now coupling and martingales. The publisher presents it as a single source for courses from undergraduate to graduate level.
• Conditioning as an organizing idea. The tenth edition's preface stresses conditioning as a key tool throughout, and Chapter 3 is devoted to it before the stochastic processes begin.
• Flexible for several courses. The preface suggests a one-year course in probability models, a one-semester introduction to probability from Chapters 1 to 3 and parts of others, or a course in elementary stochastic processes, and describes using Chapters 5 and 8 with parts of Chapters 4 and 6 for queueing theory.
• Large exercise sets with partial solutions. The Mathematical Gazette's review of the 12th edition notes that chapters end with lengthy exercise sets, usually well over thirty questions and sometimes over eighty, with a small number of starred exercises solved at the back and a free instructors' manual for all solutions.

Limitations

• Informal level of rigour. The tenth edition's preface says the book follows the heuristic, nonrigorous approach rather than measure theory. The Gazette adds that convergence theorems are largely ignored, sigma algebras are not mentioned and the modelling assumptions are not discussed. Students who want a rigorous foundation need a measure-theoretic text.
• Size. At about 870 pages the book is long for one course, and the same review of the 12th edition calls its length and bulk somewhat intimidating; courses must pick chapters.
• Index centred on mathematics. According to the Gazette, the index is mainly about the mathematics, not the applications, so finding an application by name can be hard.
• Limited solutions. Only the starred exercises have solutions in the book; the complete solutions manual goes to instructors who adopt it.
• Late additions. Coupling and martingales were added in the last two editions, and Chapter 12 is described as short, so readers wanting depth on either topic will need a dedicated book.

Where It Fits in the Course

Among the stochastic processes books on this site, Ross is the broad, mostly calculus-based survey of applied probability models. Dobrow's Introduction to Stochastic Processes with R is a closer undergraduate alternative that pairs Markov chains, Poisson processes and Brownian motion with simulation in R. Resnick's Adventures in Stochastic Processes treats similar material with more mathematical rigour.

The finance-oriented books pick up where Chapter 10 stops. Shreve's Stochastic Calculus for Finance I develops conditional expectation, martingales and change of measure in discrete time without measure theory, Lamberton and Lapeyre's Introduction to Stochastic Calculus Applied to Finance, Björk's Arbitrage Theory in Continuous Time and Øksendal's Stochastic Differential Equations treat continuous-time stochastic calculus and arbitrage pricing at a more advanced level.

How to Use This Book

The publisher describes the readership as undergraduate students of all backgrounds in probability modelling courses, typically in mathematics or statistics departments. The site lists a calculus-based probability course as the prerequisite, although the tenth edition's preface treats Chapters 1 to 3 as a possible one-semester introduction to probability itself.

A practical plan:

• Core: Chapters 1 to 3 (skim if probability is familiar), then Chapters 4 to 7 on Markov chains, the Poisson process, continuous-time chains and renewal theory.
• By interest: Chapter 8 for queueing, Chapter 9 for reliability, Chapter 10 for finance and Brownian motion, Chapter 11 for simulation, and Chapters 12 and 13 for coupling and martingales.
• Exercises: work the starred exercises first and check them against the solutions at the back, then take on the remaining problems.