Probability by Grimmett, Geoffrey -- Read -- Imperial Library of Trantor

Index

Cover Title page Copyright Contents Preface to the second edition Part 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

Part 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

Part 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 of random walks 10.4 The Gambler’s Ruin Problem 10.5 Problems

11. Random processes in continuous time

11.1 Life at a telephone switchboard 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

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12

Remarks on problems

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12

Reading list Index