Book:Geoffrey Grimmett/Probability: An Introduction/Second Edition

Subject Matter

 * Probability Theory

Contents

 * Preface to First Edition
 * Preface to Second Edition

A. BASIC PROBABILITY

 * 1 Events and probabilities
 * 1.1 Experiments with chance
 * 1.2 Outcomes and events
 * 1.3 Probabilities
 * 1.4 Probability spaces
 * 1.5 Discrete sample spaces
 * 1.6 Conditional probabilities
 * 1.7 Independent events
 * 1.8 The partition theorem
 * 1.9 Probability measures are continuous
 * 1.10 Worked problems
 * 1.11 Problems


 * 2 Discrete random variables
 * 2.1 Probability mass functions
 * 2.2 Examples
 * 2.3 Functions of discrete random variables
 * 2.4 Expectation
 * 2.5 Conditional expectation and the partition theorem
 * 2.6 Problems


 * 3 Multivariate discrete distributions and independence
 * 3.1 Bivariate discrete distributions
 * 3.2 Expectation in the multivariate case
 * 3.3 Independence of discrete random variables
 * 3.4 Sums of random variables
 * 3.5 Indicator functions
 * 3.6 Problems


 * 4 Probability generating functions
 * 4.1 Generating functions
 * 4.2 Integer-valued random variables
 * 4.3 Moments
 * 4.4 Sums of independent random variables
 * 4.5 Problems


 * 5 Distribution functions and density functions
 * 5.1 Distribution functions
 * 5.2 Examples of distribution functions
 * 5.3 Continuous random variables
 * 5.4 Some common density functions
 * 5.5 Functions of random variables
 * 5.6 Expectations of continuous random variables
 * 5.7 Geometrical probability
 * 5.8 Problems

B. FURTHER PROBABILITY

 * 6 Multivariate distributions and independence
 * 6.1 Random vectors and independence
 * 6.2 Joint density functions
 * 6.3 Marginal density functions and independence
 * 6.4 Sums of continuous random variables
 * 6.5 Changes of variables
 * 6.6 Conditional density functions
 * 6.7 Expectations of continuous random variables
 * 6.8 Bivariate normal distribution
 * 6.9 Problems


 * 7 Moments, and moment generating functions
 * 7.1 A general note
 * 7.2 Moments
 * 7.3 Variance and covariance
 * 7.4 Moment generating functions
 * 7.5 Two inequalities
 * 7.6 Characteristic functions
 * 7.7 Problems


 * 8 The main limit theorems
 * 8.1 The law of averages
 * 8.2 Chebyshev's inequality and the weak law
 * 8.3 The central limit theorem
 * 8.4 Large deviations and Cramér's theorem
 * 8.5 Convergence in distribution, and characteristic functions
 * 8.6 Problems

C. RANDOM PROCESSES

 * 9 Branching processes
 * 9.1 Random processes
 * 9.2 A model for population growth
 * 9.3 The generating-function method
 * 9.4 An example
 * 9.5 The probability of extinction
 * 9.6 Problems


 * 10 Random walks
 * 10.1 One-dimensional random walks
 * 10.2 Transition probabilities
 * 10.3 Recurrence and transience in random walks
 * 10.4 The Gambler's Ruin problem
 * 10.5 Problems


 * 11 Random processes in continuous time
 * 11.1 Life at a telephone exchange
 * 11.2 Poisson processes
 * 11.3 Inter-arrival times and the exponential distribution
 * 11.4 Population growth and the simple birth process
 * 11.5 Birth and death processes
 * 11.6 A simple queueing model
 * 11.7 Problems


 * 12 Markov Chains
 * 12.1 The Markov property
 * 12.2 Transition probabilities
 * 12.3 Class structure
 * 12.4 Recurrence and transience
 * 12.5 Random walks in one, two and three dimensions
 * 12.6 Hitting times and hitting probabilities
 * 12.7 Stopping times and the strong Markov property
 * 12.8 Classification of states
 * 12.9 Invariant distributions
 * 12.10 Convergence to equilibrium
 * 12.11 Time reversal
 * 12.12 Random walk on a graph
 * 12.13 Problems


 * Appendix A: Elements of combinatorics


 * Appendix B: Difference equations


 * Answers to exercises


 * Remarks on problems


 * Reading list


 * Index