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Full Description
Discrete stochastics is the theory of discrete probability spaces. This undergraduate textbook gives a concise introduction into discrete stochastics in general, and into a variety of typical special topics in this field, such as information theory, fluctuation theory, and semigroups of stochastic matrices. The emphasis lies on probability theory rather than on statistical methodology. Motivations, interpretations, and numerous examples and exercises relate the mathematical theory to stochastic experience.
Contents
I. Introduction.- 1. Encountering Random.- 2. Specimens of Stochastic Reasoning.- II. Markovian Dynamics.- 1. Finite-state Markovian dynamical systems.- 2. The convex set of stochastic matrices.- 3. The asymptotic behavior of Pn: some special cases.- 4. Asymptotic behavior of P, P2,...: the method of invariant sets.- III. Discrete Probability Spaces.- 1. The Notion of a Discrete Probability Space (DPS).- 2. Obtaining New Probability Spaces from Given Ones.- 3. Independence.- IV. Independent Identically Distributed (IID) Random Variables.- 1. Addition of independent RVs.- 2. Expectation and Variance.- 3. The Weak Law of Large Numbers (WLLN).- 4. The Central Limit Theorem (CLT) I.- 5. The Central Limit Theorem (CLT) II.- 6. Outlook.- V. Statistics.- 1. Specimens of Statistical Reasoning.- 2. The Game-Theoretical Framework of Statistical Theory.- 3. Tests.- 4. Outlook.- VI. Markov Processes.- 1. Conditional Probabilities.- 2. Markov Processes.- VII. Elements of Information Theory.- 1. Combinatorial and Algebraic Coding Theory.- 2. Source Coding.- 3. Noisy Channels.- VIII. Fluctuation Theory.- 1. The Combinatorial Arcsin Law of Erik Sparre Andersen.- 2. Arcsin.- 3. Symmetrically Distributed Random Variables.- 4. Fluctuations of Random Walks.- 5. The Andersen-Spitzer Formula.- 6. Outlook.- IX. Optimal Strategies in Casinoes: Red and Black.- 1. Strategies and Their Probability of Success.- 2. Some Properties of BOLD.- 3. The Optimality of BOLD for p ? 1/2 ? r.- 4. Non-Optimality of BOLD if p ? 1/2 ? r Fails.- X. Foundational Problems.- 1. The Theory of Randomness.- 2. Subjective Probabilities.- 3. Belief ("bel") Functions.- Appendix A: The Marriage Theorem.- Appendix B: Markovian Semigroups.- Appendix C: One-parameter semigroups of stochastic matrices.
