The Theory of Probability Explorations and Applications

From classical foundations to advanced modern theory, this self-contained and comprehensive guide to probability weaves together mathematical proofs, historical context and richly detailed illustrative applications. A theorem discovery approach is used throughout, setting each proof within its historical setting and is accompanied by a consistent emphasis on elementary methods of proof. Each topic is presented in a modular framework, combining fundamental concepts with worked examples, problems and digressions which, although mathematically rigorous, require no specialised or advanced mathematical background. Augmenting this core material are over 80 richly embellished practical applications of probability theory, drawn from a broad spectrum of areas both classical and modern, each tailor-made to illustrate the magnificent scope of the formal results. Providing a solid grounding in practical probability, without sacrificing mathematical rigour or historical richness, this insightful book is a fascinating reference and essential resource, for all engineers, computer scientists and mathematicians.

Part I. Elements: 1. Probability spaces; 2. Conditional probability; 3. A first look at independence; 4. Probability sieves; 5. Numbers play a game of chance; 6. The normal law; 7. Probabilities on the real line; 8. The Bernoulli schema; 9. The essence of randomness; 10. The coda of the normal; Part II. Foundations: 11. Distribution functions and measure; 12. Random variables; 13. Great expectations; 14. Variations on a theme of integration; 15. Laplace transforms; 16. The law of large numbers; 17. From inequalities to concentration; 18. Poisson approximation; 19. Convergence in law, selection theorems; 20. Normal approximation; Part III. Appendices: 21. Sequences, functions, spaces.

Santosh S. Venkatesh is an Associate Professor of Electrical and Systems Engineering at the University of Pennsylvania, whose research interests include probability, information, communication and learning theory, pattern recognition, computational neuroscience, epidemiology and computer security. He is a member of the David Mahoney Institute for Neurological Sciences and has been awarded the Lindback Award for Distinguished Teaching.