Probability Theory Problems Simulations

Preface -- Probability space -- Random variables and vectors -- Numerical characteristics of random variables and vectors -- Sequences of random variables -- Examples of stochastic processes - Appendix - Bibliography -- Index.

A key pedagogical feature of the textbook is the accessible approach to probability concepts through examples with explanations and problems with solutions. The reader is encouraged to simulate in Matlab random experiments and to explore the theoretical aspects of the probabilistic models behind the studied experiments. By this appropriate balance between simulations and rigorous mathematical approach, the reader can experience the excitement of comprehending basic concepts and can develop the intuitive thinking in solving problems. The current textbook does not contain proofs for the stated theorems, but corresponding references are given. Moreover, the given Matlab codes and detailed solutions make the textbook accessible to researchers and undergraduate students, by learning various techniques from probability theory and its applications in other fields. This book is intended not only for students of mathematics but also for students of natural sciences, engineering, computer science and for science researchers, who possess the basic knowledge of calculus for the mathematical concepts of the textbook and elementary programming skills for the Matlab simulations.

Random processes and Random number generators; Simulation of probability experiments; Gaming, Random Walks, and linear equations; Gambler's ruin with extensions to inventory control; Limiting processes for Random Walks and time series simulation; Monte Carlo integration and solution of differential equations.

This book presents a unified theory of rare event simulation and the variance reduction technique known as importance sampling from the point of view of the probabilistic theory of large deviations. It allows us to view a vast assortment of simulation problems from a unified single perspective.

The purpose of this book is twofold: first, it sets out to equip the reader with a sound understanding of the foundations of probability theory and stochastic processes, offering step-by-step guidance from basic probability theory to advanced topics, such as stochastic differential equations, which typically are presented in textbooks that require a very strong mathematical background. Second, while leading the reader on this journey, it aims to impart the knowledge needed in order to develop algorithms that simulate realistic physical systems. Connections with several fields of pure and applied physics, from quantum mechanics to econophysics, are provided. Furthermore, the inclusion of fully solved exercises will enable the reader to learn quickly and to explore topics not covered in the main text. The book will appeal especially to graduate students wishing to learn how to simulate physical systems and to deepen their knowledge of the mathematical framework, which has very deep connections with modern quantum field theory.

Scientists and engineers often have to deal with systems that exhibit random or unpredictable elements and must effectively evaluate probabilities in each situation. Computer simulations, while the traditional tool used to solve such problems, are limited in the scale and complexity of the problems they can solve. Formalized Probability Theory and Applications Using Theorem Proving discusses some of the limitations inherent in computer systems when applied to problems of probabilistic analysis, and presents a novel solution to these limitations, combining higher-order logic with computer-based theorem proving. Combining practical application with theoretical discussion, this book is an important reference tool for mathematicians, scientists, engineers, and researchers in all STEM fields.

This volume is based on classes in probability for advanced undergraduates held at the IAS/Park City Mathematics Institute. It is derived from both lectures (Chapters 1-10) and computer simulations (Chapters 11-13) that were held during the program. The material is coordinated so that some of the major computer simulations relate to topics covered in the first ten chapters. The goal is to present topics that are accessible to advanced undergraduates, yet are areas of current research in probability. The combination of the lucid yet informal style of the lectures and the hands-on nature of the simulations allows readers to become familiar with some interesting and active areas of probability. The first four chapters discuss random walks and the continuous limit of random walks: Brownian motion. Chapters 5 and 6 consider the fascinating mathematics of card shuffles, including the notions of random walks on a symmetric group and the general idea of random permutations. Chapters 7 and 8 discuss Markov chains, beginning with a standard introduction to the theory. Chapter 8 addresses the recent important application of Markov chains to simulations of random systems on large finite sets: Markov Chain Monte Carlo. Random walks and electrical networks are covered in Chapter 9. Uniform spanning trees, as connected to probability and random walks, are treated in Chapter 10. The final three chapters of the book present simulations. Chapter 11 discusses simulations for random walks. Chapter 12 covers simulation topics such as sampling from continuous distributions, random permutations, and estimating the number of matrices with certain conditions using Markov Chain Monte Carlo. Chapter 13 presents simulations of stochastic differential equations for applications in finance. (The simulations do not require one particular piece of software. They can be done in symbolic computation packages or via programming languages such as $\bold C$.) The volume concludes with a number of problems ranging from routine to very difficult. Of particular note are the problems that are typical of simulation problems given to students by the authors when teaching undergraduate probability.

Some probability problems are so difficult that they stump the smartest mathematicians. This text shows readers how to get numerical answers to difficult probability problems without having to solve complicated mathematical questions.