Based on a well-established and popular course taught by the authors over many years, Stochastic Processes: An Introduction, Third Edition, discusses the modelling and analysis of random experiments, where processes evolve over time. The text begins with a review of relevant fundamental probability. It then covers gambling problems, random walks, and Markov chains. The authors go on to discuss random processes continuous in time, including Poisson, birth and death processes, and general population models, and present an extended discussion on the analysis of associated stationary processes in queues.
The book also explores reliability and other random processes, such as branching, martingales, and simple epidemics. A new chapter describing Brownian motion, where the outcomes are continuously observed over continuous time, is included. Further applications, worked examples and problems, and biographical details have been added to this edition. Much of the text has been reworked. The appendix contains key results in probability for reference.
This concise, updated book makes the material accessible, highlighting simple applications and examples. A solutions manual with fully worked answers of all end-of-chapter problems, and Mathematica® and R programs illustrating many processes discussed in the book, can be downloaded from crcpress.com.
Author(s): Peter W. Jones, Peter Smith
Series: Texts in Statistical Science
Publisher: CRC Press
Year: 2018
Language: English
Pages: 271
Tags: Probability, Stochasti Processes, Random Walks, Markov Chains, Poisson Processes, Queues, Brownian Motion, Mathematica, R
cover
......Page 1
Half Title
......Page 2
Series Editors......Page 3
Title
......Page 8
Copyright......Page 9
Contents......Page 10
Preface to the Third Edition......Page 14
1.2 Probability......Page 16
1.3 Conditional probability and independence......Page 20
1.4 Discrete random variables......Page 23
1.5 Continuous random variables......Page 24
1.6 Mean and variance......Page 26
1.7 Some standard discrete probability distributions......Page 27
1.8 Some standard continuous probability distributions......Page 30
1.9 Generating functions......Page 33
1.10 Conditional expectation......Page 38
1.11 Problems......Page 42
2.2 Probability of ruin......Page 48
2.3 Some numerical simulations......Page 52
2.4 Duration of the game......Page 54
2.5.1 The infinitely rich opponent......Page 56
2.5.3 Changing the stakes......Page 58
2.6 Problems......Page 59
3.1 Introduction......Page 64
3.2 Unrestricted random walks......Page 65
3.3 The exact probability distribution of a random walk......Page 67
3.4 First returns of the symmetric random walk......Page 69
3.5 Problems......Page 72
4.1 States and transitions......Page 80
4.2 Transition probabilities......Page 81
4.3 General two-state Markov chains......Page 85
4.4 Powers of the general transition matrix......Page 87
4.5 Gambler’s ruin as a Markov chain......Page 95
4.6 Classification of states......Page 98
4.7 Classification of chains......Page 105
4.8 A wildlife Markov chain model......Page 109
4.9 Problems......Page 111
5.2 The Poisson process......Page 120
5.3 Partition theorem approach......Page 123
5.4 Iterative method......Page 124
5.5 The generating function......Page 125
5.6 Arrival times......Page 127
5.8 Problems......Page 130
6.2 The birth process......Page 134
6.3 Birth process: Generating function equation......Page 137
6.4 The death process......Page 139
6.5 The combined birth and death process......Page 142
6.6 General population processes......Page 147
6.7 Problems......Page 150
7.1 Introduction......Page 160
7.2 The single-server queue......Page 161
7.3 The limiting process......Page 163
7.4 Queues with multiple servers......Page 169
7.5 Queues with fixed service times......Page 174
7.7 Problems......Page 177
8.2 The reliability function......Page 184
8.3 Exponential distribution and reliability......Page 186
8.4 Mean time to failure......Page 187
8.5 Reliability of series and parallel systems......Page 188
8.6 Renewal processes......Page 191
8.7 Expected number of renewals......Page 193
8.8 Problems......Page 194
9.2 Generational growth......Page 198
9.3 Mean and variance......Page 201
9.4 Probability of extinction......Page 203
9.5 Branching processes and martingales......Page 206
9.6 Stopping rules......Page 210
9.7 A continuous time epidemic......Page 212
9.8 A discrete time epidemic model......Page 214
9.9 Deterministic epidemic models......Page 217
9.10 An iterative solution scheme for the simple epidemic......Page 219
9.11 Problems......Page 222
10.2 Brownian motion......Page 228
10.3 Wiener process as a limit of a random walk......Page 230
10.5 Scaling......Page 232
10.6 First visit times......Page 234
10.7 Other Brownian motions in one dimension......Page 236
10.8 Brownian motion in more than one dimension......Page 238
10.9 Problems......Page 239
chapter
11 Computer Simulations and Projects......Page 242
Answers and Comments on End-of-Chapter Problems......Page 252
Appendix......Page 260
References and Further Reading......Page 264
Index......Page 266
