Probability and Stochastic Modeling

Vladimir I. Rotar is a professor in the Department of Mathematics and Statistics at San Diego State University. Dr. Rotar has authored four books and more than 100 scientific papers on probability theory and its applications in leading mathematical journals.

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.