Probability and Stochastic Modeling (Hardcover)

Basic NotionsSample Space and EventsProbabilitiesCounting Techniques Independence and Conditional ProbabilityIndependenceConditioningThe Borel-Cantelli Theorem Discrete Random VariablesRandom Variables and VectorsExpected ValueVariance and Other Moments. Inequalities for DeviationsSome Basic DistributionsConvergence of Random Variables. The Law of Large NumbersConditional Expectation Generating Functions. Branching Processes. Random Walk RevisitedBranching Processes Generating Functions Branching Processes Revisited More on Random Walk Markov ChainsDefinitions and Examples. Probability Distributions of Markov ChainsThe First Step Analysis. Passage TimesVariables Defined on a Markov ChainErgodicity and Stationary DistributionsA Classification of States and Ergodicity Continuous Random VariablesContinuous DistributionsSome Basic Distributions Continuous Multivariate Distributions Sums of Independent Random Variables Conditional Distributions and Expectations Distributions in the General Case. SimulationDistribution Functions Expected Values On Convergence in Distribution and Probability Simulation Histograms Moment Generating FunctionsDefinitions and PropertiesSome Examples of ApplicationsExponential or Bernstein-Chernoff’s Bounds The Central Limit Theorem for Independent Random VariablesThe Central Limit Theorem (CLT) for Independent and Identically Distributed Random VariablesThe CLT for Independent Variables in the General Case Covariance Analysis. The Multivariate Normal Distribution. The Multivariate Central Limit TheoremCovariance and CorrelationCovariance Matrices and Some ApplicationsThe 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 CharacteristicsLimit Theorems for Maxima and Minima Hazard Rate. Survival Probabilities Stochastic Processes: PreliminariesA 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 ProcessesBirth and Death Processes Elements of Renewal Theory Preliminaries Limit Theorems Some Proofs Martingales in Discrete Time Definitions and PropertiesOptional Time and Some ApplicationsMartingales and a Financial Market ModelLimit Theorems for Martingales Brownian Motion and Martingales in Continuous Time Brownian Motion and Its GeneralizationsMartingales in Continuous Time More on Dependency Structures Arrangement Structures and the Corresponding DependenciesMeasures of Dependency Limit Theorems for Dependent Random Variables Symmetric Distributions. De Finetti’s Theorem Comparison of Random Variables. Risk Evaluation Some Particular CriteriaExpected UtilityGeneralizations of the EUM Criterion Appendix References Answers to Exercises Index Exercises appear at the end of each chapter.