Probability An Introduction 2nd Edition by Geoffrey Grimmett, Dominic Welsh 0191019925 9780191019920

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ISBN 10: 0191019925

ISBN 13: 9780191019920

Author: Geoffrey Grimmett, Dominic Welsh

Probability is an area of mathematics of tremendous contemporary importance across all aspects of human endeavour. This book is a compact account of the basic features of probability and random processes at the level of first and second year mathematics undergraduates and Masters’ students in cognate fields. It is suitable for a first course in probability, plus a follow-up course in random processes including Markov chains. A special feature is the authors’ attention to rigorous mathematics: not everything is rigorous, but the need for rigour is explained at difficult junctures. The text is enriched by simple exercises, together with problems (with very brief hints) many of which are taken from final examinations at Cambridge and Oxford. The first eight chapters form a course in basic probability, being an account of events, random variables, and distributions – discrete and continuous random variables are treated separately – together with simple versions of the law of large numbers and the central limit theorem. There is an account of moment generating functions and their applications. The following three chapters are about branching processes, random walks, and continuous-time random processes such as the Poisson process. The final chapter is a fairly extensive account of Markov chains in discrete time. This second edition develops the success of the first edition through an updated presentation, the extensive new chapter on Markov chains, and a number of new sections to ensure comprehensive coverage of the syllabi at major universities.

Probability An Introduction 2nd Table of contents:

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

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Tags: Geoffrey Grimmett, Dominic Welsh, Probability, Introduction