# Book:Geoffrey Grimmett/Probability: An Introduction

## Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction

Published $\text {1986}$, Oxford Science Publications

ISBN 0-19-853264-4.

### Contents

Preface

#### 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
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 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 Conditional expectation and the 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 Characteristic functions
7.6 Problems
8 The two 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 Convergence in distribution, and characteristic functions
8.5 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
Appendix: Difference equations
Remarks on the problems