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

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## Geoffrey Grimmett and Dominic Welsh:

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

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

- ISBN 0-19-870997-8.

### Subject Matter

### 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 CramÃ©r'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**