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Probability and Stochastic Modeling

Vladimir I. Rotar
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
We do not plan to review this book.

Basic Notions
Sample Space and Events
Counting Techniques


Independence and Conditional Probability
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


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
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




Answers to Exercises