## Probability ModelsProbability 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 |

253 | |

### Common terms and phrases

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