Probability Models / Edition 1

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Probability Models is designed to aid students studying probability as part of an undergraduate course on mathematics or mathematics and statistics. It describes how to set up and analyze models of real-life phenomena that involve elements of chance. Motivation comes from everyday experiences of probability via dice and cards, the idea of fairness in games of chance, and the random ways in which, say, birthdays are shared or particular events arise. Applications include branching processes, random walks, Markov chains, queues, renewal theory, and Brownian motion. No specific knowledge of the subject is assumed, only a familiarity with the notions of calculus, and the summation of series. Where the full story would call for a deeper mathematical background, the difficulties are noted and appropriate references given. The main topics arise naturally, with definitions and theorems supported by fully worked examples and some 200 set exercises, all with solutions.
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Editorial Reviews

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"…gives a concise non-measure theoretic introduction to the basics of probability theory and shastic processes. The overall level is that of a first university course on the subject, given that the students have had introductory courses on linear algebra and real analysis. Numerous examples make the text fairly light reading…the mathematically less trained reader will find the language (and terminology) used pleasant: no unnecessary pedantic notation is wasted. The more mathematically inclined reader will learn form both the examples and pedagogic line of approach. In summary, I find [this book] a useful text to have on my shelves and would consider using it as a textbook for science and engineering students. Mathematics students would benefit from it as a first contact with the world of randomness before delving deeper using a measure theoretic approach."


"What makes this book so interesting is the fact that, in only two hundred and fifty pages, the reader is brought from the very beginning to a fairly high level in the knowledge of probability theory. ... There is a wealth of about two hundred exercisis, with solutions, which makes the book useful for teaching."

ISI Short Book Reviews, Vol. 22/3, December 2002

"‘The purpose of this book is to provide a sound introduction to the study of real-world phenomena that possess random variation’. … this one is a nice choice, written in a broad and lively style." (P. Schmitt, Monatshefte für Mathematik, Vol. 143 (1), 2004)

"A clear treatment of probability theory … it has a great deal to recommend it. … there is a real attempt to provide a readable account of the material without getting too bogged down in analytical detail and … without ignoring the issues or resorting to over-simplification. … It is good to have such specialized material in what is essentially a text for undergraduates, and I can recommend this stimulating book to anybody who is looking for a way to spice up their knowledge." (Gerry Leversha, The Mathematical Gazette, Vol. 88 (512), 2004)

"What makes this book so interesting is the fact that, in only two hundred and fifty pages, the reader is brought from the very beginning to a fairly high level in the knowledge of probability theory. … The author deals in an elegant way with important theorems such as the central limit theorem and the laws of large numbers. … There is a wealth of about two hundred exercises, with solutions, which makes the book useful for teaching." (N.D.C. Veraverbeke, Short Book Reviews, Vol. 22 (3), 2002)

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Product Details

  • ISBN-13: 9781852334314
  • Publisher: Springer London
  • Publication date: 9/24/2004
  • Series: Springer Undergraduate Mathematics Series
  • Edition description: 1st ed. 2002. Corr. 2nd printing 2004
  • Edition number: 1
  • Pages: 256
  • Product dimensions: 6.84 (w) x 9.28 (h) x 0.64 (d)

Table of Contents

1 Probability Spaces 1
The Idea of Probability 2
Laws of Probability 3
Consequences 5
Equally Likely Outcomes 10
The Continuous Version 16
Intellectual Honesty 21
2 Conditional Probability and Independence 23
Conditional Probability 23
Bayes' Theorem 31
Independence 36
The Borel-Cantelli Lemmas 43
3 Common Probability Distributions 45
Common Discrete Probability Spaces 45
Probability Generating Functions 53
Common Continuous Probability Spaces 54
Mixed Probability Spaces 59
4 Random Variables 61
The Definition 62
Discrete Random Variables 63
Continuous Random Variables 72
Jointly Distributed Random Variables 75
Conditional Expectation 88
5 Sums of Random Variables 93
Discrete Variables 93
General Random Variables 100
Records 113
6 Convergence and Limit Theorems 117
Inequalities 118
Convergence 121
Limit Theorems 129
7 Stochastic Processes in Discrete Time 139
Branching Processes 140
Random Walks 145
Markov Chains 155
8 Stochastic Processes in Continuous Time 169
Markov Chains in Continuous Time 169
Queues 186
Renewal Theory 200
Brownian Motion: The Wiener Process 210
9 Appendix: Common Distributions and Mathematical Facts 223
Discrete Distributions 223
Continuous Distributions 224
Miscellaneous Mathematical Facts 225
Bibliography 227
Solutions 229
Index 253
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