# 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

### 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