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The Probabilistic Method
by Joel H. Spencer Noga AlonThis Third Edition of The Probabilistic Method reflects the most recent developments in the field while maintaining the standard of excellence that established this book as the leading reference on probabilistic methods in combinatorics. Maintaining its clear writing style, illustrative examples, and practical exercises, this new edition emphasizes methodology, enabling readers to use probabilistic techniques for solving problems in such fields as theoretical computer science, mathematics, and statistical physics. The book begins with a description of tools applied in probabilistic arguments, including basic techniques that use expectation and variance as well as the more recent applications of martingales and correlation inequalities. Next, the authors examine where probabilistic techniques have been applied successfully, exploring such topics as discrepancy and random graphs, circuit complexity, computational geometry, and derandomization of randomized algorithms. Sections labeled "The Probabilistic Lens" offer additional insights into the application of the probabilistic approach, and the appendix has been updated to include methodologies for finding lower bounds for Large Deviations.The Third Edition also features:A new chapter on graph property testing, which is a current topic that incorporates combinatorial, probabilistic, and algorithmic techniques. An elementary approach using probabilistic techniques to the powerful Szemerédi Regularity Lemma and its applications. New sections devoted to percolation and liar games. A new chapter that provides a modern treatment of the Erdös-Rényi phase transition in the Random Graph Process. Written by two leading authorities in the field, The Probabilistic Method, Third Edition is an ideal reference for researchers in combinatorics and algorithm design who would like to better understand the use of probabilistic methods. The book's numerous exercises and examples also make it an excellent textbook for graduate-level courses in mathematics and computer science.
Probabilistic Methods and Distributed Information: Rudolf Ahlswede’s Lectures on Information Theory 5 (Foundations in Signal Processing, Communications and Networking #15)
by Rudolf Ahlswede Alexander Ahlswede Ingo Althöfer Christian Deppe Vladimir Blinovsky Ulrich Tamm Holger Boche Ulrich Krengel Ahmed MansourThe fifth volume of Rudolf Ahlswede’s lectures on Information Theory focuses on several problems that were at the heart of a lot of his research. One of the highlights of the entire lecture note series is surely Part I of this volume on arbitrarily varying channels (AVC), a subject in which Ahlswede was probably the world's leading expert. Appended to Part I is a survey by Holger Boche and Ahmed Mansour on recent results concerning AVC and arbitrarily varying wiretap channels (AVWC). After a short Part II on continuous data compression, Part III, the longest part of the book, is devoted to distributed information. This Part includes discussions on a variety of related topics; among them let us emphasize two which are famously associated with Ahlswede: "multiple descriptions", on which he produced some of the best research worldwide, and "network coding", which had Ahlswede among the authors of its pioneering paper. The final Part IV on "Statistical Inference under Communication constraints" is mainly based on Ahlswede’s joint paper with Imre Csiszar, which received the Best Paper Award of the IEEE Information Theory Society. The lectures presented in this work, which consists of 10 volumes, are suitable for graduate students in Mathematics, and also for those working in Theoretical Computer Science, Physics, and Electrical Engineering with a background in basic Mathematics. The lectures can be used either as the basis for courses or to supplement them in many ways. Ph.D. students will also find research problems, often with conjectures, that offer potential subjects for a thesis. More advanced researchers may find questions which form the basis of entire research programs.
Probabilistic Methods in Telecommunications (Compact Textbooks in Mathematics)
by Benedikt Jahnel Wolfgang KönigProbabilistic modeling and analysis of spatial telecommunication systems have never been more important than they are today. In particular, it is an essential research area for designing and developing next-generation communication networks that are based on multihop message transmission technology. These lecture notes provide valuable insights into the underlying mathematical discipline, stochastic geometry, introducing the theory, mathematical models and basic concepts. They also discuss the latest applications of the theory to telecommunication systems. The text covers several of the most fundamental aspects of quality of service: connectivity, coverage, interference, random environments, and propagation of malware. It especially highlights two important limiting scenarios of large spatial systems: the high-density limit and the ergodic limit. The book also features an analysis of extreme events and their probabilities based on the theory of large deviations. Lastly, it includes a large number of exercises offering ample opportunities for independent self-study.
Probabilistic Metric Spaces
by B. Schweizer A. SklarThis distinctly nonclassical treatment focuses on developing aspects that differ from the theory of ordinary metric spaces, working directly with probability distribution functions rather than random variables. The two-part treatment begins with an overview that discusses the theory's historical evolution, followed by a development of related mathematical machinery. The presentation defines all needed concepts, states all necessary results, and provides relevant proofs.The second part opens with definitions of probabilistic metric spaces and proceeds to examinations of special classes of probabilistic metric spaces, topologies, and several related structures, such as probabilistic normed and inner-product spaces. Throughout, the authors focus on developing aspects that differ from the theory of ordinary metric spaces, rather than simply transferring known metric space results to a more general setting.
Probabilistic Models of Cosmic Backgrounds
by Anatoliy MalyarenkoCombining research methods from various areas of mathematics and physics, Probabilistic Models of Cosmic Backgrounds describes the isotropic random sections of certain fiber bundles and their applications to creating rigorous mathematical models of both discovered and hypothetical cosmic backgrounds.Previously scattered and hard-to-find mathematical and physical theories have been assembled from numerous textbooks, monographs, and research papers, and explained from different or even unexpected points of view. This consists of both classical and newly discovered results necessary for understanding a sophisticated problem of modelling cosmic backgrounds.The book contains a comprehensive description of mathematical and physical aspects of cosmic backgrounds with a clear focus on examples and explicit calculations. Its reader will bridge the gap of misunderstanding between the specialists in various theoretical and applied areas who speak different scientific languages.The audience of the book consists of scholars, students, and professional researchers. A scholar will find basic material for starting their own research. A student will use the book as supplementary material for various courses and modules. A professional mathematician will find a description of several physical phenomena at the rigorous mathematical level. A professional physicist will discover mathematical foundations for well-known physical theories.
Probabilistic Reliability Analysis of Power Systems: A Student’s Introduction
by José L. Rueda Torres Francisco M. Gonzalez-Longatt Mart A. van der Meijden Bart W. Tuinema Alexandru I. StefanovThis textbook provides an introduction to probabilistic reliability analysis of power systems. It discusses a range of probabilistic methods used in reliability modelling of power system components, small systems and large systems. It also presents the benefits of probabilistic methods for modelling renewable energy sources. The textbook describes real-life studies, discussing practical examples and providing interesting problems, teaching students the methods in a thorough and hands-on way.The textbook has chapters dedicated to reliability models for components (reliability functions, component life cycle, two-state Markov model, stress-strength model), small systems (reliability networks, Markov models, fault/event tree analysis) and large systems (generation adequacy, state enumeration, Monte-Carlo simulation). Moreover, it contains chapters about probabilistic optimal power flow, the reliability of underground cables and cyber-physical power systems.After reading this book, engineering students will be able to apply various methods to model the reliability of power system components, smaller and larger systems. The textbook will be accessible to power engineering students, as well as students from mathematics, computer science, physics, mechanical engineering, policy & management, and will allow them to apply reliability analysis methods to their own areas of expertise.
Probabilistic Risk Analysis and Bayesian Decision Theory (SpringerBriefs in Statistics)
by Mark Brewer Marcel van OijenThe book shows how risk, defined as the statistical expectation of loss, can be formally decomposed as the product of two terms: hazard probability and system vulnerability. This requires a specific definition of vulnerability that replaces the many fuzzy definitions abounding in the literature. The approach is expanded to more complex risk analysis with three components rather than two, and with various definitions of hazard. Equations are derived to quantify the uncertainty of each risk component and show how the approach relates to Bayesian decision theory. Intended for statisticians, environmental scientists and risk analysts interested in the theory and application of risk analysis, this book provides precise definitions, new theory, and many examples with full computer code. The approach is based on straightforward use of probability theory which brings rigour and clarity. Only a moderate knowledge and understanding of probability theory is expected from the reader.
Probabilistic Robotics (Intelligent Robotics and Autonomous Agents)
by Sebastian Thrun Wolfram Burgard Dieter Fox<p>An introduction to the techniques and algorithms of the newest field in robotics. <p>Probabilistic robotics is a new and growing area in robotics, concerned with perception and control in the face of uncertainty. Building on the field of mathematical statistics, probabilistic robotics endows robots with a new level of robustness in real-world situations. This book introduces the reader to a wealth of techniques and algorithms in the field. All algorithms are based on a single overarching mathematical foundation. Each chapter provides example implementations in pseudo code, detailed mathematical derivations, discussions from a practitioner's perspective, and extensive lists of exercises and class projects. The book's Web site, www.probabilistic-robotics.org, has additional material. The book is relevant for anyone involved in robotic software development and scientific research. It will also be of interest to applied statisticians and engineers dealing with real-world sensor data.</p>
Probabilistic Search for Tracking Targets
by Eugene Kagan Irad Ben-GalPresents a probabilistic and information-theoretic framework for a search for static or moving targets in discrete time and space.Probabilistic Search for Tracking Targets uses an information-theoretic scheme to present a unified approach for known search methods to allow the development of new algorithms of search. The book addresses search methods under different constraints and assumptions, such as search uncertainty under incomplete information, probabilistic search scheme, observation errors, group testing, search games, distribution of search efforts, single and multiple targets and search agents, as well as online or offline search schemes. The proposed approach is associated with path planning techniques, optimal search algorithms, Markov decision models, decision trees, stochastic local search, artificial intelligence and heuristic information-seeking methods. Furthermore, this book presents novel methods of search for static and moving targets along with practical algorithms of partitioning and search and screening.Probabilistic Search for Tracking Targets includes complete material for undergraduate and graduate courses in modern applications of probabilistic search, decision-making and group testing, and provides several directions for further research in the search theory.The authors:Provide a generalized information-theoretic approach to the problem of real-time search for both static and moving targets over a discrete space.Present a theoretical framework, which covers known information-theoretic algorithms of search, and forms a basis for development and analysis of different algorithms of search over probabilistic space.Use numerous examples of group testing, search and path planning algorithms to illustrate direct implementation in the form of running routines.Consider a relation of the suggested approach with known search theories and methods such as search and screening theory, search games, Markov decision process models of search, data mining methods, coding theory and decision trees.Discuss relevant search applications, such as quality-control search for nonconforming units in a batch or a military search for a hidden target. Provide an accompanying website featuring the algorithms discussed throughout the book, along with practical implementations procedures.
The Probabilistic SIR Model: Project Management in Prevention and Support (essentials)
by Marcus HellwigWith all the insights experienced in the COVID process, one essential remains: "The virus remains a constant companion". In contrast to regularly occurring infection processes, a COVID infection takes a different course. This is characterized by a dynamic that deviates from conventional, well-known processes in that the originators change their identity and develop corresponding variants. Therefore, preventive infection management - supported by statistical-probabilistic analyzes with PSIR - is important for preventive management of resources and infrastructure for the "waves ahead of the wave".
Probabilistic Spiking Neuronal Nets: Neuromathematics for the Computer Era (Lecture Notes on Mathematical Modelling in the Life Sciences)
by Antonio Galves Eva Löcherbach Christophe PouzatThis book provides a self-contained introduction to a new class of stochastic models for systems of spiking neurons. These systems have a large number of interacting components, each one evolving as a stochastic process with a memory of variable length. Several mathematical tools are put to use, such as Markov chains, stochastic chains having memory of variable length, point processes having stochastic intensity, Hawkes processes, random graphs, mean field limits, perfect sampling algorithms, the Context algorithm, and statistical model selection.The book’s focus on mathematically tractable objects distinguishes it from other texts on theoretical neuroscience. The biological complexity of neurons is not ignored, but reduced to some of its main features, such as the intrinsic randomness of neuronal dynamics. This reduction in complexity aims at explaining and reproducing statistical regularities and collective phenomena that are observed in experimental data, an approach that leads to mathematically rigorous results. With an emphasis on a constructive and algorithmic point of view, this book is directed towards mathematicians interested in learning about stochastic network models and their neurobiological underpinning, and neuroscientists interested in learning how to build and prove results with mathematical models that relate to actual experimental settings.
Probabilistic-Statistical Methods for Risk Assessment in Civil Aviation (Springer Aerospace Technology)
by Valery Dmitryevich Sharov Vadim Vadimovich Vorobyov Dmitry Alexandrovich ZatuchnyThis book analyses the models for major risks related to flight safety in the aviation sector and presents risk estimation methods through examples of several known aviation enterprises. The book provides a comprehensive content for professionals engaged in the development of flight safety regulatory framework as well as in the design and operation of ground-based or on-board flight support radio electronic systems. The book is also useful for senior students and postgraduates in aviation specialties, especially those related to air traffic management.
Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games (Probability Theory and Stochastic Modelling #83)
by François Delarue René CarmonaThis two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions.Volume I of the book is entirely devoted to the theory of mean field games without a common noise. The first half of the volume provides a self-contained introduction to mean field games, starting from concrete illustrations of games with a finite number of players, and ending with ready-for-use solvability results. Readers are provided with the tools necessary for the solution of forward-backward stochastic differential equations of the McKean-Vlasov type at the core of the probabilistic approach. The second half of this volume focuses on the main principles of analysis on the Wasserstein space. It includes Lions' approach to the Wasserstein differential calculus, and the applications of its results to the analysis of stochastic mean field control problems. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map through the book allowing different access points for different readers and building up the level of technical detail. The accessible approach and overview will allow interested researchers in the applied sciences to obtain a clear overview of the state of the art in mean field games.
Probabilistic Theory of Mean Field Games with Applications II: Mean Field Games with Common Noise and Master Equations (Probability Theory and Stochastic Modelling #84)
by François Delarue René CarmonaThis two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume II tackles the analysis of mean field games in which the players are affected by a common source of noise. The first part of the volume introduces and studies the concepts of weak and strong equilibria, and establishes general solvability results. The second part is devoted to the study of the master equation, a partial differential equation satisfied by the value function of the game over the space of probability measures. Existence of viscosity and classical solutions are proven and used to study asymptotics of games with finitely many players. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map through the book allowing different access points for different readers and building up the level of technical detail. The accessible approach and overview will allow interested researchers in the applied sciences to obtain a clear overview of the state of the art in mean field games.
Probabilistic Thinking
by Egan J. Chernoff Bharath SriramanThis volume provides a necessary, current and extensive analysis of probabilistic thinking from a number of mathematicians, mathematics educators, and psychologists. The work of 58 contributing authors, investigating probabilistic thinking across the globe, is encapsulated in 6 prefaces, 29 chapters and 6 commentaries. Ultimately, the four main perspectives presented in this volume (Mathematics and Philosophy, Psychology, Stochastics and Mathematics Education) are designed to represent probabilistic thinking in a greater context.
Probabilità, Statistica e Simulazione: Programmi applicativi scritti in R (UNITEXT #131)
by Alberto Rotondi Paolo Pedroni Antonio PievatoloIl libro contiene in forma compatta il programma svolto negli insegnamenti introduttivi di Statistica e tratta alcuni argomenti indispensabili per l'attività di ricerca, come le tecniche di simulazione Monte Carlo, i metodi di inferenza statistica, di best fit e di analisi dei dati di laboratorio. Gli argomenti vengono sviluppati partendo dai fondamenti, evidenziandone gli aspetti applicativi, fino alla descrizione dettagliata di molti casi di particolare rilevanza in ambito scientifico e tecnico. Il testo è rivolto agli studenti universitari dei corsi ad indirizzo scientifico e a tutti quei ricercatori che devono risolvere problemi concreti che coinvolgono l’analisi dei dati e le tecniche di simulazione. In questa edizione, completamente rivista e corretta, sono stati aggiunti alcuni importanti argomenti sul test d’ipotesi (a cui è stato dedicato un capitolo interamente nuovo) e sul trattamento degli errori sistematici. Per la prima volta è stato adottato il software R, con una ricca libreria di programmi originali accessibile al lettore.
Probabilities
by Peter OlofssonPraise for the First Edition"If there is anything you want to know, or remind yourself, about probabilities, then look no further than this comprehensive, yet wittily written and enjoyable, compendium of how to apply probability calculations in real-world situations."- Keith Devlin, Stanford University, National Public Radio's "Math Guy" and author of The Math Gene and The Unfinished GameFrom probable improbabilities to regular irregularities, Probabilities: The Little Numbers That Rule Our Lives, Second Edition investigates the often surprising effects of risk and chance in our lives. Featuring a timely update, the Second Edition continues to be the go-to guidebook for an entertaining presentation on the mathematics of chance and uncertainty. The new edition develops the fundamental mathematics of probability in a unique, clear, and informal way so readers with various levels of experience with probability can understand the little numbers found in everyday life. Illustrating the concepts of probability through relevant and engaging real-world applications, the Second Edition features numerous examples on weather forecasts, DNA evidence, games and gambling, and medical testing. The revised edition also includes:The application of probability in finance, such as option pricingThe introduction of branching processes and the extinction of family namesAn extended discussion on opinion polls and Nate Silver's election predictionsProbabilities: The Little Numbers That Rule Our Lives, Second Edition is an ideal reference for anyone who would like to obtain a better understanding of the mathematics of chance, as well as a useful supplementary textbook for students in any course dealing with probability.
Probabilities: The Little Numbers That Rule Our Lives
by Peter OlofssonWhat are the chances? Find out in this entertaining exploration ofprobabilities in our everyday lives “If there is anything you want to know, or remind yourself, about probabilities, then look no further than this comprehensive, yet wittily written and enjoyable, compendium of how to apply probability calculations in real-world situations.” — Keith Devlin, Stanford University, National Public Radio’s “Math Guy” and author of The Math Gene and The Math Instinct “A delightful guide to the sometimes counterintuitive discipline of probability. Olofsson points out major ideas here, explains classic puzzles there, and everywhere makes free use of witty vignettes to instruct and amuse.” — John Allen Paulos, Temple University, author of Innumeracy and A Mathematician Reads the Newspaper “Beautifully written, with fascinating examples and tidbits of information. Olofsson gently and persuasively shows us how to think clearly about the uncertainty that governs our lives.” — John Haigh, University of Sussex, author of Taking Chances: Winning with Probability From probable improbabilities to regular irregularities, Probabilities: The Little Numbers That Rule Our Lives investigates the often-surprising effects of risk and chance in our everyday lives. With examples ranging from WWII espionage to the O. J. Simpson trial, from bridge to blackjack, from Julius Caesar to Jerry Seinfeld, the reader is taught how to think straight in a world of randomness and uncertainty. Throughout the book, readers learn: Why it is not that surprising for someone to win the lottery twice How a faulty probability calculation forced an innocent woman to spend three years in prison How to place bets if you absolutely insist on gambling How a newspaper turned an opinion poll into one of the greatest election blunders in history Educational, eloquent, and entertaining, Probabilities: The Little Numbers That Rule Our Lives is the ideal companion for anyone who wants to obtain a better understanding of the mathematics of chance.
Probability
by Robert P. DobrowAn introduction to probability at the undergraduate levelChance and randomness are encountered on a daily basis. Authored by a highly qualified professor in the field, Probability: With Applications and R delves into the theories and applications essential to obtaining a thorough understanding of probability.With real-life examples and thoughtful exercises from fields as diverse as biology, computer science, cryptology, ecology, public health, and sports, the book is accessible for a variety of readers. The book's emphasis on simulation through the use of the popular R software language clarifies and illustrates key computational and theoretical results.Probability: With Applications and R helps readers develop problem-solving skills and delivers an appropriate mix of theory and application. The book includes:Chapters covering first principles, conditional probability, independent trials, random variables, discrete distributions, continuous probability, continuous distributions, conditional distribution, and limitsAn early introduction to random variables and Monte Carlo simulation and an emphasis on conditional probability, conditioning, and developing probabilistic intuitionAn R tutorial with example script filesMany classic and historical problems of probability as well as nontraditional material, such as Benford's law, power-law distributions, and Bayesian statisticsA topics section with suitable material for projects and explorations, such as random walk on graphs, Markov chains, and Markov chain Monte CarloChapter-by-chapter summaries and hundreds of practical exercisesProbability: With Applications and R is an ideal text for a beginning course in probability at the undergraduate level.
Probability: Theory and Examples (Cambridge Series in Statistical and Probabilistic Mathematics #49)
by Rick DurrettThis lively introduction to measure-theoretic probability theory covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. Concentrating on results that are the most useful for applications, this comprehensive treatment is a rigorous graduate text and reference. Operating under the philosophy that the best way to learn probability is to see it in action, the book contains extended examples that apply the theory to concrete applications. This fifth edition contains a new chapter on multidimensional Brownian motion and its relationship to partial differential equations (PDEs), an advanced topic that is finding new applications. Setting the foundation for this expansion, Chapter 7 now features a proof of Itô's formula. Key exercises that previously were simply proofs left to the reader have been directly inserted into the text as lemmas. The new edition re-instates discussion about the central limit theorem for martingales and stationary sequences.
Probability: Theory and Examples
by Rick DurrettThis book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.
Probability: An Introduction
by Samuel GoldbergExcellent basic text covers set theory, probability theory for finite sample spaces, binomial theorem, probability distributions, means, standard deviations, probability function of binomial distribution, and other key concepts and methods essential to a thorough understanding of probability. Designed for use by math or statistics departments offering a first course in probability. 360 illustrative problems with answers for half. Only high school algebra needed. Chapter bibliographies.
Probability: A Graduate Course
by Allan GutLike its predecessor, this book starts from the premise that, rather than being a purely mathematical discipline, probability theory is an intimate companion of statistics. The book starts with the basic tools, and goes on to cover a number of subjects in detail, including chapters on inequalities, characteristic functions and convergence. This is followed by a thorough treatment of the three main subjects in probability theory: the law of large numbers, the central limit theorem, and the law of the iterated logarithm. After a discussion of generalizations and extensions, the book concludes with an extensive chapter on martingales. The new edition is comprehensively updated, including some new material as well as around a dozen new references.
Probability: Elements of the Mathematical Theory
by C. R. HeathcoteDesigned for students studying mathematical statistics and probability after completing a course in calculus and real variables, this text deals with basic notions of probability spaces, random variables, distribution functions and generating functions, as well as joint distributions and the convergence properties of sequences of random variables. Includes worked examples and over 250 exercises with solutions.
Probability
by John J. KinneyPraise for the First Edition"This is a well-written and impressively presented introduction to probability and statistics. The text throughout is highly readable, and the author makes liberal use of graphs and diagrams to clarify the theory." - The StatisticianThoroughly updated, Probability: An Introduction with Statistical Applications, Second Edition features a comprehensive exploration of statistical data analysis as an application of probability. The new edition provides an introduction to statistics with accessible coverage of reliability, acceptance sampling, confidence intervals, hypothesis testing, and simple linear regression. Encouraging readers to develop a deeper intuitive understanding of probability, the author presents illustrative geometrical presentations and arguments without the need for rigorous mathematical proofs. The Second Edition features interesting and practical examples from a variety of engineering and scientific fields, as well as:Over 880 problems at varying degrees of difficulty allowing readers to take on more challenging problems as their skill levels increaseChapter-by-chapter projects that aid in the visualization of probability distributionsNew coverage of statistical quality control and quality productionAn appendix dedicated to the use of Mathematica® and a companion website containing the referenced data setsFeaturing a practical and real-world approach, this textbook is ideal for a first course in probability for students majoring in statistics, engineering, business, psychology, operations research, and mathematics. Probability: An Introduction with Statistical Applications, Second Edition is also an excellent reference for researchers and professionals in any discipline who need to make decisions based on data as well as readers interested in learning how to accomplish effective decision making from data.