Probability Models

by Haigh, John

• ISBN13: 9781852334314
• ISBN10: 1852334312
• Format: Paperback
• Copyright: 2002-03-01
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Probability Spaces	1	(22)
Introduction	1	(1)
The Idea of Probability	2	(1)
Laws of Probability	3	(2)
Consequences	5	(5)
Equally Likely Outcomes	10	(6)
The Continuous Version	16	(5)
Intellectual Honesty	21	(2)
Conditional Probability and Independence	23	(22)
Conditional Probability	23	(8)
Bayes' Theorem	31	(5)
Independence	36	(7)
The Borel-Cantelli Lemmas	43	(2)
Common Probability Distributions	45	(16)
Common Discrete Probability Spaces	45	(8)
Probability Generating Functions	53	(1)
Common Continuous Probability Spaces	54	(5)
Mixed Probability Spaces	59	(2)
Random Variables	61	(32)
The Definition	62	(1)
Discrete Random Variables	63	(9)
Continuous Random Variables	72	(3)
Jointly Distributed Random Variables	75	(13)
Conditional Expectation	88	(5)
Sums of Random Variables	93	(24)
Discrete Variables	93	(7)
General Random Variables	100	(13)
Records	113	(4)
Convergence and Limit Theorems	117	(22)
Inequalities	118	(3)
Convergence	121	(8)
Limit Theorems	129	(8)
Summary	137	(2)
Stochastic Processes in Discrete Time	139	(30)
Branching Processes	140	(5)
Random Walks	145	(10)
Markov Chains	155	(14)
Stochastic Processes in Continuous Time	169	(54)
Markov Chains in Continuous Time	169	(17)
Queues	186	(14)
Renewal Theory	200	(10)
Brownian Motion: The Wiener Process	210	(13)
Appendix: Common Distributions and Mathematical Facts	223	(4)
Discrete Distributions	223	(1)
Continuous Distributions	224	(1)
Miscellaneous Mathematical Facts	225	(2)
Bibliography	227	(2)
Solutions	229	(24)
Index	253