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Author: Torben Maack Bisgaard
Publisher: Nova Publishers
ISBN: 9781560728603
Category : Mathematics
Languages : en
Pages : 152
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Book Description
This book contains basic information on characteristic functions and moment sequences that is frequently used in probability theory. Characteristic functions and moment sequences are viewed as special cases of positive definite functions. Positive definite functions occur in diverse parts of mathematics, e.g. in operator theory, moment problems, complex function theory, embedding problems, integral equations, and other areas. However, the area of mathematics in which the largest number of people use positive definite functions (some without knowing it) seems to be that of probability theory.
Author: Torben Maack Bisgaard
Publisher: Nova Publishers
ISBN: 9781560728603
Category : Mathematics
Languages : en
Pages : 152
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Book Description
This book contains basic information on characteristic functions and moment sequences that is frequently used in probability theory. Characteristic functions and moment sequences are viewed as special cases of positive definite functions. Positive definite functions occur in diverse parts of mathematics, e.g. in operator theory, moment problems, complex function theory, embedding problems, integral equations, and other areas. However, the area of mathematics in which the largest number of people use positive definite functions (some without knowing it) seems to be that of probability theory.
Author: Bert E. Fristedt
Publisher: Springer Science & Business Media
ISBN: 1489928375
Category : Mathematics
Languages : en
Pages : 775
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Book Description
Students and teachers of mathematics and related fields will find this book a comprehensive and modern approach to probability theory, providing the background and techniques to go from the beginning graduate level to the point of specialization in research areas of current interest. The book is designed for a two- or three-semester course, assuming only courses in undergraduate real analysis or rigorous advanced calculus, and some elementary linear algebra. A variety of applications—Bayesian statistics, financial mathematics, information theory, tomography, and signal processing—appear as threads to both enhance the understanding of the relevant mathematics and motivate students whose main interests are outside of pure areas.
Author: Marco Taboga
Publisher: Createspace Independent Publishing Platform
ISBN: 9781981369195
Category : Mathematical statistics
Languages : en
Pages : 670
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Book Description
The book is a collection of 80 short and self-contained lectures covering most of the topics that are usually taught in intermediate courses in probability theory and mathematical statistics. There are hundreds of examples, solved exercises and detailed derivations of important results. The step-by-step approach makes the book easy to understand and ideal for self-study. One of the main aims of the book is to be a time saver: it contains several results and proofs, especially on probability distributions, that are hard to find in standard references and are scattered here and there in more specialistic books. The topics covered by the book are as follows. PART 1 - MATHEMATICAL TOOLS: set theory, permutations, combinations, partitions, sequences and limits, review of differentiation and integration rules, the Gamma and Beta functions. PART 2 - FUNDAMENTALS OF PROBABILITY: events, probability, independence, conditional probability, Bayes' rule, random variables and random vectors, expected value, variance, covariance, correlation, covariance matrix, conditional distributions and conditional expectation, independent variables, indicator functions. PART 3 - ADDITIONAL TOPICS IN PROBABILITY THEORY: probabilistic inequalities, construction of probability distributions, transformations of probability distributions, moments and cross-moments, moment generating functions, characteristic functions. PART 4 - PROBABILITY DISTRIBUTIONS: Bernoulli, binomial, Poisson, uniform, exponential, normal, Chi-square, Gamma, Student's t, F, multinomial, multivariate normal, multivariate Student's t, Wishart. PART 5 - MORE DETAILS ABOUT THE NORMAL DISTRIBUTION: linear combinations, quadratic forms, partitions. PART 6 - ASYMPTOTIC THEORY: sequences of random vectors and random variables, pointwise convergence, almost sure convergence, convergence in probability, mean-square convergence, convergence in distribution, relations between modes of convergence, Laws of Large Numbers, Central Limit Theorems, Continuous Mapping Theorem, Slutsky's Theorem. PART 7 - FUNDAMENTALS OF STATISTICS: statistical inference, point estimation, set estimation, hypothesis testing, statistical inferences about the mean, statistical inferences about the variance.
Author: Eric Carlen
Publisher: Springer
ISBN: 1493970054
Category : Mathematics
Languages : en
Pages : 620
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Book Description
This volume presents some of the research topics discussed at the 2014-2015 Annual Thematic Program Discrete Structures: Analysis and Applications at the Institute of Mathematics and its Applications during the Spring 2015 where geometric analysis, convex geometry and concentration phenomena were the focus. Leading experts have written surveys of research problems, making state of the art results more conveniently and widely available. The volume is organized into two parts. Part I contains those contributions that focus primarily on problems motivated by probability theory, while Part II contains those contributions that focus primarily on problems motivated by convex geometry and geometric analysis. This book will be of use to those who research convex geometry, geometric analysis and probability directly or apply such methods in other fields.
Author: Wlodzimierz Bryc
Publisher: Springer Science & Business Media
ISBN: 1461225604
Category : Mathematics
Languages : en
Pages : 142
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Book Description
This book is a concise presentation of the normal distribution on the real line and its counterparts on more abstract spaces, which we shall call the Gaussian distributions. The material is selected towards presenting characteristic properties, or characterizations, of the normal distribution. There are many such properties and there are numerous rel evant works in the literature. In this book special attention is given to characterizations generated by the so called Maxwell's Theorem of statistical mechanics, which is stated in the introduction as Theorem 0.0.1. These characterizations are of interest both intrin sically, and as techniques that are worth being aware of. The book may also serve as a good introduction to diverse analytic methods of probability theory. We use characteristic functions, tail estimates, and occasionally dive into complex analysis. In the book we also show how the characteristic properties can be used to prove important results about the Gaussian processes and the abstract Gaussian vectors. For instance, in Section 5.4 we present Fernique's beautiful proofs of the zero-one law and of the integrability of abstract Gaussian vectors. The central limit theorem is obtained via characterizations in Section 7.3.
Author: Eric W. Weisstein
Publisher: CRC Press
ISBN: 1420035223
Category : Mathematics
Languages : en
Pages : 3253
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Book Description
Upon publication, the first edition of the CRC Concise Encyclopedia of Mathematics received overwhelming accolades for its unparalleled scope, readability, and utility. It soon took its place among the top selling books in the history of Chapman & Hall/CRC, and its popularity continues unabated. Yet also unabated has been the d
Author: Arak M. Mathai
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 311056260X
Category : Mathematics
Languages : en
Pages : 732
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Book Description
This book offers an introduction to concepts of probability theory, probability distributions relevant in the applied sciences, as well as basics of sampling distributions, estimation and hypothesis testing. As a companion for classes for engineers and scientists, the book also covers applied topics such as model building and experiment design. Contents Random phenomena Probability Random variables Expected values Commonly used discrete distributions Commonly used density functions Joint distributions Some multivariate distributions Collection of random variables Sampling distributions Estimation Interval estimation Tests of statistical hypotheses Model building and regression Design of experiments and analysis of variance Questions and answers
Author: George G. Roussas
Publisher: Gulf Professional Publishing
ISBN: 0125990227
Category : Computers
Languages : en
Pages : 463
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Book Description
This book provides in a concise, yet detailed way, the bulk of the probabilistic tools that a student working toward an advanced degree in statistics, probability and other related areas, should be equipped with. The approach is classical, avoiding the use of mathematical tools not necessary for carrying out the discussions. All proofs are presented in full detail. * Excellent exposition marked by a clear, coherent and logical devleopment of the subject * Easy to understand, detailed discussion of material * Complete proofs
Author: Zhidong Bai
Publisher: Springer Science & Business Media
ISBN: 1441906614
Category : Mathematics
Languages : en
Pages : 560
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Book Description
The aim of the book is to introduce basic concepts, main results, and widely applied mathematical tools in the spectral analysis of large dimensional random matrices. The core of the book focuses on results established under moment conditions on random variables using probabilistic methods, and is thus easily applicable to statistics and other areas of science. The book introduces fundamental results, most of them investigated by the authors, such as the semicircular law of Wigner matrices, the Marcenko-Pastur law, the limiting spectral distribution of the multivariate F matrix, limits of extreme eigenvalues, spectrum separation theorems, convergence rates of empirical distributions, central limit theorems of linear spectral statistics, and the partial solution of the famous circular law. While deriving the main results, the book simultaneously emphasizes the ideas and methodologies of the fundamental mathematical tools, among them being: truncation techniques, matrix identities, moment convergence theorems, and the Stieltjes transform. Its treatment is especially fitting to the needs of mathematics and statistics graduate students and beginning researchers, having a basic knowledge of matrix theory and an understanding of probability theory at the graduate level, who desire to learn the concepts and tools in solving problems in this area. It can also serve as a detailed handbook on results of large dimensional random matrices for practical users. This second edition includes two additional chapters, one on the authors' results on the limiting behavior of eigenvectors of sample covariance matrices, another on applications to wireless communications and finance. While attempting to bring this edition up-to-date on recent work, it also provides summaries of other areas which are typically considered part of the general field of random matrix theory.
Author: Alan M. Polansky
Publisher: CRC Press
ISBN: 1439884579
Category : Mathematics
Languages : en
Pages : 353
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Book Description
Helping students develop a good understanding of asymptotic theory, Introduction to Statistical Limit Theory provides a thorough yet accessible treatment of common modes of convergence and their related tools used in statistics. It also discusses how the results can be applied to several common areas in the field.The author explains as much of the
