Stochastic Processes: An Introduction / Edition 2

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Based on a highly popular, well-established course taught by the authors, Stochastic Processes: An Introduction, Second Edition discusses the modeling and analysis of random experiments using the theory of probability. It focuses on the way in which the results or outcomes of experiments vary and evolve over time.

The text begins with a review of relevant fundamental probability. It then covers several basic gambling problems, random walks, and Markov chains. The authors go on to develop random processes continuous in time, including Poisson, birth and death processes, and general population models. While focusing on queues, they present an extended discussion on the analysis of associated stationary processes. The book also explores reliability and other random processes, such as branching processes, martingales, and a simple epidemic. The appendix contains key mathematical results for reference.

Ideal for a one-semester course on stochastic processes, this concise, updated textbook makes the material accessible to students by avoiding specialized applications and instead highlighting simple applications and examples. The associated website contains Mathematica® and R programs that offer flexibility in creating graphs and performing computations.

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Editorial Reviews

From the Publisher
… a clear, easily understandable and rather short overview on stochastic processes. The different topics are motivated very well, there are many graphs and 50—theoretical and practical—examples. … the book is written very carefully … for beginners, one could not imagine a better book.
—Dominik Wied, Statistical Papers (2011) 52

… a good resource as a textbook or as a reference to complement other literature, especially with the examples and problems provided.
Biometrics, 67, September 2011

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Product Details

Meet the Author

Peter W. Jones is a professor and Pro Vice Chancellor for Research and Enterprise at Keele University in Staffordshire, UK.

Peter Smith is a Professor Emeritus in the School of Computing and Mathematics at Keele University in Staffordshire, UK.

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Table of Contents

Some Background on Probability



Conditional probability and independence

Discrete random variables

Continuous random variables

Mean and variance

Some standard discrete probability distributions

Some standard continuous probability distributions

Generating functions

Conditional expectation

Some Gambling Problems

Gambler’s ruin

Probability of ruin

Some numerical simulations

Duration of the game

Some variations of gambler’s ruin

Random Walks


Unrestricted random walks

The probability distribution after n steps

First returns of the symmetric random walk

Markov Chains

States and transitions

Transition probabilities

General two-state Markov chains

Powers of the transition matrix for the m-state chain

Gambler’s ruin as a Markov chain

Classification of states

Classification of chains

Poisson Processes


The Poisson process

Partition theorem approach

Iterative method

The generating function

Variance in terms of the probability generating function

Arrival times

Summary of the Poisson process

Birth and Death Processes


The birth process

Birth process: Generating function equation

The death process

The combined birth and death process

General population processes



The single-server queue

The stationary process

Queues with multiple servers

Queues with fixed service times

Classification of queues

A general approach to the M(λ)/G/1 queue

Reliability and Renewal


The reliability function

Exponential distribution and reliability

Mean time to failure

Reliability of series and parallel systems

Renewal processes

Expected number of renewals

Branching and Other Random Processes


Generational growth

Mean and variance

Probability of extinction

Branching processes and martingales

Stopping rules

The simple epidemic

An iterative solution scheme for the simple epidemic

Computer Simulations and Projects

Answers and Comments on End-of-Chapter Problems


References and Further Reading


Problems appear at the end of each chapter.

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