Stochastic Processes: An Introduction, Second Edition(Chapman & Hall/CRC Texts in Statistical Science)

Using Mathematica(R) and R, this updated text discusses the modeling and analysis of random experiments using the theory of probability. It illustrates discrete random processes through the classical gambler's ruin problem and its variants. It also covers continuous random processes, such as Poisson and general population models. With over 50 worked examples and more than 200 end-of-chapter problems, the text describes applications of probability to modeling problems in engineering, medicine, and biology. The book's website includes the Mathematica and R programs as well as a solutions manual for instructors upon qualfying course adoption.

Table of Contents:

Some Background on Probability

Introduction

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

Introduction

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

Introduction

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

Introduction

The birth process

Birth process: Generating function equation

The death process

The combined birth and death process

General population processes

Queues

Introduction

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

Introduction

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

Introduction

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

Appendix

References and Further Reading

Index

Problems appear at the end of each chapter.