A First Course in Probability with an Emphasis on Stochastic Modeling Probability and Stochastic Modeling not only covers all the topics found in a traditional introductory probability course, but also emphasizes stochastic modeling, including Markov chains, birth-death processes, and reliability models. Unlike most undergraduate-level probability texts, the book also focuses on increasingly important areas, such as martingales, classification of dependency structures, and risk evaluation. Numerous examples, exercises, and models using real-world data demonstrate the practical possibilities and restrictions of different approaches and help students grasp general concepts and theoretical results. The text is suitable for majors in mathematics and statistics as well as majors in computer science, economics, finance, and physics. The author offers two explicit options to teaching the material, which is reflected in "routes" designated by special "roadside" markers. The first route contains basic, self-contained material for a one-semester course. The second provides a more complete exposition for a two-semester course or self-study.
Basic Notions Sample Space and Events Probabilities Counting Techniques Independence and Conditional Probability Independence Conditioning The Borel-Cantelli Theorem Discrete Random Variables Random Variables and Vectors Expected Value Variance and Other Moments. Inequalities for Deviations Some Basic Distributions Convergence of Random Variables. The Law of Large Numbers Conditional Expectation Generating Functions. Branching Processes. Random Walk Revisited Branching Processes Generating Functions Branching Processes Revisited More on Random Walk Markov Chains Definitions and Examples. Probability Distributions of Markov Chains The First Step Analysis. Passage Times Variables Defined on a Markov Chain Ergodicity and Stationary Distributions A Classification of States and Ergodicity Continuous Random Variables Continuous Distributions Some Basic Distributions Continuous Multivariate Distributions Sums of Independent Random Variables Conditional Distributions and Expectations Distributions in the General Case. Simulation Distribution Functions Expected Values On Convergence in Distribution and Probability Simulation Histograms Moment Generating Functions Definitions and Properties Some Examples of Applications Exponential or Bernstein-Chernoff's Bounds The Central Limit Theorem for Independent Random Variables The Central Limit Theorem (CLT) for Independent and Identically Distributed Random Variables The CLT for Independent Variables in the General Case Covariance Analysis. The Multivariate Normal Distribution. The Multivariate Central Limit Theorem Covariance and Correlation Covariance Matrices and Some Applications The Multivariate Normal Distribution Maxima and Minima of Random Variables. Elements of Reliability Theory. Hazard Rate and Survival Probabilities Maxima and Minima of Random Variables. Reliability Characteristics Limit Theorems for Maxima and Minima Hazard Rate. Survival Probabilities Stochastic Processes: Preliminaries A General Definition Processes with Independent Increments Brownian Motion Markov Processes A Representation and Simulation of Markov Processes in Discrete Time Counting and Queuing Processes. Birth and Death Processes: A General Scheme Poisson Processes Birth and Death Processes Elements of Renewal Theory Preliminaries Limit Theorems Some Proofs Martingales in Discrete Time Definitions and Properties Optional Time and Some Applications Martingales and a Financial Market Model Limit Theorems for Martingales Brownian Motion and Martingales in Continuous Time Brownian Motion and Its Generalizations Martingales in Continuous Time More on Dependency Structures Arrangement Structures and the Corresponding Dependencies Measures of Dependency Limit Theorems for Dependent Random Variables Symmetric Distributions. De Finetti's Theorem Comparison of Random Variables. Risk Evaluation Some Particular Criteria Expected Utility Generalizations of the EUM Criterion Appendix References Answers to Exercises Index Exercises appear at the end of each chapter.
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