# Probability Models

Springer Science & Business Media, Sep 23, 2004 - Mathematics - 256 pages
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 analyse 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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### Contents

 Probability Spaces 1 12 The Idea of Probability 2 13 Laws of Probability 3 14 Consequences 5 15 Equally Likely Outcomes 10 16 The Continuous Version 16 17 Intellectual Honesty 21 Conditional Probability and Independence 23
 52 General Random Variables 100 53 Records 113 Convergence and Limit Theorems 117 61 Inequalities 118 62 Convergence 121 63 Limit Theorems 129 64 Summary 137 Stochastic Processes in Discrete Time 139

 22 Bayes Theorem 31 23 Independence 36 24 The BorelCantelli Lemmas 43 Common Probability Distributions 45 32 Probability Generating Functions 53 33 Common Continuous Probability Spaces 54 34 Mixed Probability Spaces 59 Random Variables 61 41 The Definition 62 42 Discrete Random Variables 63 43 Continuous Random Variables 72 44 Jointly Distributed Random Variables 75 45 Conditional Expectation 88 Sums of Random Variables 93
 71 Branching Processes 140 72 Random Walks 145 73 Markov Chains 155 Stochastic Processes in Continuous Time 169 82 Queues 186 83 Renewal Theory 200 The Wiener Process 210 Appendix Common Distributions and Mathematical Facts 223 92 Continuous Distributions 224 93 Miscellaneous Mathematical Facts 225 Bibliography 227 Solutions 229 Index 253 Copyright