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

Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (2nd Edition)

Published $\text {2014}$, Oxford Science Publications

ISBN 0-19-870997-8

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

Further Editions

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction