Probability Theory, Random Processes and Mathematical Statistics - Y. Rozanov

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Résumé :
The study of random phenomena encountered in the real world is based on probability theory, mathematical statistics and the theory of random processes. The choice of the most suitable mathematical model is made on the basis of statistical data collected by observations. These models provide numerous tools for the analysis, prediction and, ultimately, control of random phenomena. br/ The first part of the present volume (Chapters 1-3) can serve as a self-contained, elementary introduction to probability, random processes and statistics. It contains a number of relatively simple and typical examples of random phenomena which allow a natural introduction of general structures and methods. Some basic knowledge of elements of real/complex analysis, linear algebra and ordinary differential equations is required. br/ The second part (Chapters 4-6) provides a foundation for stochastic analysis, gives information on basic models of random processes and tools to study them. A certain familiarity with elements of functional analysis is necessary. Important material is presented in the form of examples to keep readers involved. br/ emAudience:/em A concise textbook for a graduate level course, with carefully selected topics representing the most important areas of modern probability, random processes and statistics.

Sommaire:
1. Introductory Probability Theory.- 1. The Notion of Probability.- 2. Some Probability Models.- 3. Random Variables.- 4. Mathematical Expectation.- 5. Correlation.- 6. Characteristic Functions.- 7. The Central Limit Theorem.- 2. Random Processes.- 1. Random Processes with Discrete State Space.- 2. Random Processes with Continuous States.- 3. An Introduction to Mathematical Statistics.- 1. Some Examples of Statistical Problems and Methods.- 2. Optimality of Statistical Decisions.- 4. Basic Elements of Probability Theory.- 1. General Probability Distributions.- 2. Conditional Probabilities and Expectations.- 3. Conditional Expectations and Martingales.- 5. Elements of Stochastic Analysis and Stochastic Differential Equations.- 1. Stochastic Series.- 2. Stochastic Integrals.- 3. Stochastic Integral Representations.- 4. Stochastic Differential Equations.