Introduction to Probability Theory and Its Applications

Chapter I The Exponential and the Uniform Densities1. Introduction2. Densities. Convolutions3. The Exponential Density4. Waiting Time Paradoxes. The Poisson Process5. The Persistence of Bad Luck6. Waiting Times and Order Statistics7. The Uniform Distribution8. Random Splittings9. Convolutions and Covering Theorems10. Random Directions11. The Use of Lebesgue Measure12. Empirical Distributions13. Problems for SolutionChapter II Special Densities. Randomization1. Notations and Conventions2. Gamma Distributions3. Related Distributions of Statistics4. Some Common Densities5. Randomization and Mixtures6. Discrete Distributions7. Bessel Functions and Random Walks8. Distributions on a Circle9. Problems for SolutionChapter III Densities in Higher Dimensions. Normal Densities and Processes1. Densities2. Conditional Distributions3. Return to the Exponential and the Uniform Distributions4. A Characterization of the Normal Distribution5. Matrix Notation. The Covariance Matrix6. Normal Densities and Distributions7. Stationary Normal Processes8. Markovian Normal Densities9. Problems for SolutionChapter IV Probability Measures and Spaces1. Baire Functions2. Interval Functions and Integrals in Rr3. σAlgebras. Measurability4. Probability Spaces. Random Variables5. The Extension Theorem6. Product Spaces. Sequences of Independent Variables7. Null Sets. CompletionChapter V Probability Distributions in Rr1. Distributions and Expectations2. Preliminaries3. Densities4. Convolutions5. Symmetrization6. Integration by Parts. Existence of Moments7. Chebyshev's Inequality8. Further Inequalities. Convex Functions9. Simple Conditional Distributions. Mixtures10. Conditional Distributions11. Conditional Expectations12. Problems for SolutionChapter VI A Survey of Some Important Distributions and Processes1. Stable Distributions in R12. Examples3. Infinitely Divisible Distributions in R14. Processes with Independent Increments5. Ruin Problems in Compound Poisson Processes6. Renewal Processes7. Examples and Problems8. Random Walks9. The Queuing Process10. Persistent and Transient Random Walks11. General Markov Chains12. Martingales13. Problems for SolutionChapter VII Laws of Large Numbers. Applications in Analysis1. Main Lemma and Notations2. Bernstein Polynomials. Absolutely Monotone Functions3. Moment Problems4. Application to Exchangeable Variables5. Generalized Taylor Formula aFeller, William is the author of 'Introduction to Probability Theory and Its Applications' with ISBN 9780471257097 and ISBN 0471257095.