Knowing the Odds: An Introduction to Probability (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 139)

This book covers in a leisurely manner all the standard material that one would want in a full year probability course with a slant towards applications in financial analysis at the graduate or senior undergraduate honors level. It contains a fair amount of measure theory and real analysis built in but it introduces sigma-fields, measure theory, and expectation in an especially elementary and intuitive way. A large variety of examples and exercises in each chapter enrich the presentation in the text. Readership: Undergraduate and graduate students interested in probability theory. Cover S Title Knowing the Odds: An Introduction to Probability Copyright 2012 by the American Mathematical Society ISBN 978-0-8218-8532-1 QA273.W24 2011 519.2-dc23 LCCN 2012013119 Dedication Contents Preface Introduction Chapter 1 Probability Spaces 1.1. Sets and Sigma-Fields 1.2. Elementary Properties of Probability Spaces 1.3. The Intuition 1.3.1. Symmetry. 1.4. Conditional Probability 1.5. Independence 1.6. Counting: Permutations and Combinations 1.7. The Gambler's Ruin Chapter 2 Random Variables 2.1. Random Variables and Distributions 2.1.1. Elementary Properties of Distribution Functions 2.2. Existence of Random Variables 2.3. Independence of Random Variables 2.4. Types of Distributions 2.5. Expectations I: Discrete Random Variables 2.6. Moments, Means and Variances 2.7. Mean, Median, and Mode 2.8. Special Discrete Distributions Chapter 3 Expectations II: The General Case 3.1. From Discrete to Continuous 3.2. The Expectation as an Integral 3.3. Some Moment Inequalities 3.4. Convex Functions and Jensen's Inequality 3.5. Special Continuous Distribution 3.5.1. Transformation of Densities. 3.6. Joint Distributions and Joint Densities 3.6.1. Covariances and Correlations. 3.6.2. Transformation of Joint Densities 3.7. Conditional Distributions, Densities, and Expectations 3.7.1. Sums of Random Variables 3.7.2. Bivariate and Multivariate Gaussian Chapter 4 Convergence 4.1. Convergence of Random Variables 4.2. Convergence Theorems for Expectations 4.3. Applications Chapter 5 Laws of Large Numbers 5.1. The Weak and Strong Laws 5.2. Normal Numbers 5.3. Sequences of Random Variables: Existence 5.4. Sigma Fields as Information 5.5. Another Look at Independence 5.6. Zero-one Laws Chapter 6 Convergence in Distribution and the CLT 6.1. Characteristic Functions 6.1.1. Levy's Inversion Theorem 6.2. Convergence in Distribution 6.2.1. Weak Convergence* 6.3. Levy's Continuity Theorem 6.4. The Central Limit Theorem 6.4.1. Some Consequences and Extensions 6.5. Stable Laws Chapter 7 Markov Chains and Random Walks 7.1. Stochastic Processes 7.2. Markov Chains 7.2.1. Conditional Independence and the Markov Property 7.3. Classification of States 7.4. Stopping Times 7.5. The Strong Markov Property 7.6. Recurrence and Transience 7.6.1. Examples: Random Walks. One Dimension 7.7. Equilibrium and the Ergodic Theorem for Markov Chains 7.7.1. Stationary Distributions and Equilibrium 7.8. Finite State Markov Chains 7.9. Branching Processes 7.10. The Poisson Process 7.11. Birth and Death Processes Chapter 8 Conditional Expectations 8.1. Conditional Expectations 8.2. Elementary Properties 8.3. Approximations and Projections Chapter 9 Discrete-Parameter Martingales 9.1. Martingales 9.2. System Theorems 9.3. Convergence 9.3.1. Backward Martingales. 9.4. Uniform Integrability 9.5. Applications Application to Conditional Expectations I : Existence Application to Conditional Expectations II : Levy's Theorem Application to the Zero-One Law. The Kolmogorov Strong Law Application to Integration in Infinitely Many Dimensions Application to Sequential Analysis in Statistics. Application to Branching Processes 9.6. Financial Mathematics I: The Martingale Connection 9.6.1. The Basic Financial Market. Chapter 10 Brownian Motion 10.1. Standard Brownian Motion 10.2. Stopping Times and the Strong Markov Property 10.3. The Zero Set of Brownian Motion 10.4. The Reflection Principle 10.5. Recurrence and Hitting Properties 10.6. Path Irregularity 10.7. The Brownian Infinitesimal Generator 10.8. Related Processes 10.9. Higher Dimensional Brownian Motion 10.10. Financial Mathematics II: The Black-Scholes Model 10.11. Skorokhod Embedding 10.11.1. Embedding Sums of Random Variables 10.12. Levy's Construction of Brownian Motion 10.13. The Ornstein-Uhlenbeck Process 10.14. White Noise and the Wiener Integral 10.15. Physical Brownian Motion 10.16. What Brownian Motion Really Does Bibliography Index Back Cover John Walsh, one of the great masters of the subject, has written a superb book on probability. It covers at a leisurely pace all the important topics that students need to know, and provides excellent examples. I regret his book was not available when I taught such a course myself, a few years ago. —Ioannis Karatzas, Columbia University In this wonderful book, John Walsh presents a panoramic view of Probability Theory, starting from basic facts on mean, median and mode, continuing with an excellent account of Markov chains and martingales, and culminating with Brownian motion. Throughout, the author's personal style is apparent; he manages to combine rigor with an emphasis on the key ideas so the reader never loses sight of the forest by being surrounded by too many trees. As noted in the preface, “To teach a course with pleasure, one should learn at the same time.” Indeed, almost all instructors will learn something new from the book (e.g. the potential-theoretic proof of Skorokhod embedding) and at the same time, it is attractive and approachable for students. —Yuval Peres, Microsoft With many examples in each section that enhance the presentation, this book is a welcome addition to the collection of books that serve the needs of advanced undergraduate as well as first year graduate students. The pace is leisurely which makes it more attractive as a text. —Srinivasa Varadhan, Courant Institute, New York This book covers in a leisurely manner all the standard material that one would want in a full year probability course with a slant towards applications in financial analysis at the graduate or senior undergraduate honors level. It contains a fair amount of measure theory and real analysis built in but it introduces sigma-fields, measure theory, and expectation in an especially elementary and intuitive way. A large variety of examples and exercises in each chapter enrich the presentation in the text. Chapter 1. Probability Spaces Chapter 2. Random Variables Chapter 3. Expectations Ii: The General Case Chapter 4. Convergence Chapter 5. Laws Of Large Numbers Chapter 6. Convergence In Distribution And The Clt Chapter 7. Markov Chains And Random Walks Chapter 8. Conditional Expectations Chapter 9. Discrete-parameter Martingales Chapter 10. Brownian Motion John B. Walsh. Includes Bibliographical References And Index.