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Dieses Lehrbuch bietet eine umfassende Einführung in die wichtigsten Gebiete der Wahrscheinlichkeitstheorie und ihre maßtheoretischen Grundlagen. Breite und Auswahl der Themen sind einmalig in der deutschsprachigen Literatur. Die 250 Übungsaufgaben und zahlreichen Abbildungen helfen Lesern den Lernstoff zu vertiefen. Themenschwerpunkte sind u. a. die Maß- und Integrationstheorie, Grenzwertsätze für Summen von Zufallsvariablen, Martingale, Perkolation, Markovketten und elektrische Netzwerke sowie die Konstruktion stochastischer Prozesse.
Most questions in social and biomedical sciences are causal in nature: what would happen to individuals, or to groups, if part of their environment were changed? In this groundbreaking text, two world-renowned experts present statistical methods for studying such questions. This book starts with the notion of potential outcomes, each corresponding to the outcome that would be realized if a subject were exposed to a particular treatment or regime. In this approach, causal effects are comparisons of such potential outcomes. The fundamental problem of causal inference is that we can only observe one of the potential outcomes for a particular subject. The authors discuss how randomized experiments allow us to assess causal effects and then turn to observational studies. They lay out the assumptions needed for causal inference and describe the leading analysis methods, including matching, propensity-score methods, and instrumental variables. Many detailed applications are included, with special focus on practical aspects for the empirical researcher.
This two-volume set on Mathematical Principles of the Internet provides a comprehensive overview of the mathematical principles of Internet engineering. The books do not aim to provide all of the mathematical foundations upon which the Internet is based. Instead, they cover a partial panorama and the key principles. Volume 1 explores Internet engineering, while the supporting mathematics is covered in Volume 2. The chapters on mathematics complement those on the engineering episodes, and an effort has been made to make this work succinct, yet self-contained. Elements of information theory, algebraic coding theory, cryptography, Internet traffic, dynamics and control of Internet congestion, and queueing theory are discussed. In addition, stochastic networks, graph-theoretic algorithms, application of game theory to the Internet, Internet economics, data mining and knowledge discovery, and quantum computation, communication, and cryptography are also discussed. In order to study the structure and function of the Internet, only a basic knowledge of number theory, abstract algebra, matrices and determinants, graph theory, geometry, analysis, optimization theory, probability theory, and stochastic processes, is required. These mathematical disciplines are defined and developed in the books to the extent that is needed to develop and justify their application to Internet engineering.
This self-contained, comprehensive book tackles the principal problems and advanced questions of probability theory and random processes in 22 chapters, presented in a logical order but also suitable for dipping into. They include both classical and more recent results, such as large deviations theory, factorization identities, information theory, stochastic recursive sequences. The book is further distinguished by the inclusion of clear and illustrative proofs of the fundamental results that comprise many methodological improvements aimed at simplifying the arguments and making them more transparent. The importance of the Russian school in the development of probability theory has long been recognized. This book is the translation of the fifth edition of the highly successful Russian textbook. This edition includes a number of new sections, such as a new chapter on large deviation theory for random walks, which are of both theoretical and applied interest. The frequent references to Russian literature throughout this work lend a fresh dimension and make it an invaluable source of reference for Western researchers and advanced students in probability related subjects. Probability Theory will be of interest to both advanced undergraduate and graduate students studying probability theory and its applications. It can serve as a basis for several one-semester courses on probability theory and random processes as well as self-study.
Praise for the First Edition "This is a well-written and impressively presentedintroduction to probability and statistics. The text throughout ishighly readable, and the author makes liberal use of graphs anddiagrams to clarify the theory." - The Statistician Thoroughly updated, Probability: An Introduction withStatistical Applications, Second Edition features acomprehensive exploration of statistical data analysis as anapplication of probability. The new edition provides anintroduction to statistics with accessible coverage of reliability,acceptance sampling, confidence intervals, hypothesis testing, andsimple linear regression. Encouraging readers to develop a deeperintuitive understanding of probability, the author presentsillustrative geometrical presentations and arguments without theneed for rigorous mathematical proofs. The Second Edition features interesting and practicalexamples from a variety of engineering and scientific fields, aswell as: Over 880 problems at varying degrees of difficulty allowingreaders to take on more challenging problems as their skill levelsincrease Chapter-by-chapter projects that aid in the visualization ofprobability distributions New coverage of statistical quality control and qualityproduction An appendix dedicated to the use ofMathematica® and a companion website containing thereferenced data sets Featuring a practical and real-world approach, this textbook isideal for a first course in probability for students majoring instatistics, engineering, business, psychology, operations research,and mathematics. Probability: An Introduction with StatisticalApplications, Second Edition is also an excellent reference forresearchers and professionals in any discipline who need to makedecisions based on data as well as readers interested in learninghow to accomplish effective decision making from data.
This volume is intended for the advanced study of several topics in mathematical statistics. The first part of the book is devoted to sampling theory (from one-dimensional and multidimensional distributions), asymptotic properties of sampling, parameter estimation, sufficient statistics, and statistical estimates. The second part is devoted to hypothesis testing and includes the discussion of families of statistical hypotheses that can be asymptotically distinguished. In particular,the author describes goodness-of-fit and sequential statistical criteria (Kolmogorov, Pearson, Smirnov, and Wald) and studies their main properties. The book is suitable for graduate students and researchers interested in mathematical statistics. It is useful for independent study or supplementaryreading.
The main intended audience for this book is undergraduate students in pure and applied sciences, especially those in engineering. Chapters 2 to 4 cover the probability theory they generally need in their training. Although the treatment of the subject is surely su?cient for non-mathematicians, I intentionally avoided getting too much into detail. For instance, topics such as mixed type random variables and the Dirac delta function are only brie?y mentioned. Courses on probability theory are often considered di?cult. However, after having taught this subject for many years, I have come to the conclusion that one of the biggest problems that the students face when they try to learn probability theory, particularly nowadays, is their de?ciencies in basic di?erential and integral calculus. Integration by parts, for example, is often already forgotten by the students when they take a course on probability. For this reason, I have decided to write a chapter reviewing the basic elements of di?erential calculus. Even though this chapter might not be covered in class, the students can refer to it when needed. In this chapter, an e?ort was made to give the readers a good idea of the use in probability theory of the concepts they should already know. Chapter 2 presents the main results of what is known as elementary probability, including Bayes’ rule and elements of combinatorial analysis.