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Author: Arno Berger
Publisher: Princeton University Press
ISBN: 0691163065
Category : Mathematics
Languages : en
Pages : 256
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Book Description
This book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law’s colorful history, rapidly growing body of empirical evidence, and wide range of applications. An Introduction to Benford’s Law begins with basic facts about significant digits, Benford functions, sequences, and random variables, including tools from the theory of uniform distribution. After introducing the scale-, base-, and sum-invariance characterizations of the law, the book develops the significant-digit properties of both deterministic and stochastic processes, such as iterations of functions, powers of matrices, differential equations, and products, powers, and mixtures of random variables. Two concluding chapters survey the finitely additive theory and the flourishing applications of Benford’s law. Carefully selected diagrams, tables, and close to 150 examples illuminate the main concepts throughout. The text includes many open problems, in addition to dozens of new basic theorems and all the main references. A distinguishing feature is the emphasis on the surprising ubiquity and robustness of the significant-digit law. This text can serve as both a primary reference and a basis for seminars and courses.
Author: Arno Berger
Publisher: Princeton University Press
ISBN: 0691163065
Category : Mathematics
Languages : en
Pages : 256
Get Book Here
Book Description
This book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law’s colorful history, rapidly growing body of empirical evidence, and wide range of applications. An Introduction to Benford’s Law begins with basic facts about significant digits, Benford functions, sequences, and random variables, including tools from the theory of uniform distribution. After introducing the scale-, base-, and sum-invariance characterizations of the law, the book develops the significant-digit properties of both deterministic and stochastic processes, such as iterations of functions, powers of matrices, differential equations, and products, powers, and mixtures of random variables. Two concluding chapters survey the finitely additive theory and the flourishing applications of Benford’s law. Carefully selected diagrams, tables, and close to 150 examples illuminate the main concepts throughout. The text includes many open problems, in addition to dozens of new basic theorems and all the main references. A distinguishing feature is the emphasis on the surprising ubiquity and robustness of the significant-digit law. This text can serve as both a primary reference and a basis for seminars and courses.
Author: Steven J. Miller
Publisher: Princeton University Press
ISBN: 1400866596
Category : Mathematics
Languages : en
Pages : 465
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Book Description
Benford's law states that the leading digits of many data sets are not uniformly distributed from one through nine, but rather exhibit a profound bias. This bias is evident in everything from electricity bills and street addresses to stock prices, population numbers, mortality rates, and the lengths of rivers. Here, Steven Miller brings together many of the world’s leading experts on Benford’s law to demonstrate the many useful techniques that arise from the law, show how truly multidisciplinary it is, and encourage collaboration. Beginning with the general theory, the contributors explain the prevalence of the bias, highlighting explanations for when systems should and should not follow Benford’s law and how quickly such behavior sets in. They go on to discuss important applications in disciplines ranging from accounting and economics to psychology and the natural sciences. The contributors describe how Benford’s law has been successfully used to expose fraud in elections, medical tests, tax filings, and financial reports. Additionally, numerous problems, background materials, and technical details are available online to help instructors create courses around the book. Emphasizing common challenges and techniques across the disciplines, this accessible book shows how Benford’s law can serve as a productive meeting ground for researchers and practitioners in diverse fields.
Author: Mark J. Nigrini
Publisher: John Wiley & Sons
ISBN: 1118286863
Category : Business & Economics
Languages : en
Pages : 352
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Book Description
A powerful new tool for all forensic accountants, or anyone whoanalyzes data that may have been altered Benford's Law gives the expected patterns of the digits in thenumbers in tabulated data such as town and city populations orMadoff's fictitious portfolio returns. Those digits, in unaltereddata, will not occur in equal proportions; there is a large biastowards the lower digits, so much so that nearly one-half of allnumbers are expected to start with the digits 1 or 2. Thesepatterns were originally discovered by physicist Frank Benford inthe early 1930s, and have since been found to apply to alltabulated data. Mark J. Nigrini has been a pioneer in applyingBenford's Law to auditing and forensic accounting, even before hisgroundbreaking 1999 Journal of Accountancy article introducing thisuseful tool to the accounting world. In Benford's Law, Nigrinishows the widespread applicability of Benford's Law and itspractical uses to detect fraud, errors, and other anomalies. Explores primary, associated, and advanced tests, all describedwith data sets that include corporate payments data and electiondata Includes ten fraud detection studies, including vendor fraud,payroll fraud, due diligence when purchasing a business, and taxevasion Covers financial statement fraud, with data from Enron, AIG,and companies that were the target of hedge fund short sales Looks at how to detect Ponzi schemes, including data on Madoff,Waxenberg, and more Examines many other applications, from the Clinton tax returnsand the charitable gifts of Lehman Brothers to tax evasion andnumber invention Benford's Law has 250 figures and uses 50 interestingauthentic and fraudulent real-world data sets to explain boththeory and practice, and concludes with an agenda and directionsfor future research. The companion website adds additionalinformation and resources.
Author: Mark J. Nigrini
Publisher: John Wiley & Sons
ISBN: 1119585902
Category : Business & Economics
Languages : en
Pages : 549
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Book Description
Become the forensic analytics expert in your organization using effective and efficient data analysis tests to find anomalies, biases, and potential fraud—the updated new edition Forensic Analytics reviews the methods and techniques that forensic accountants can use to detect intentional and unintentional errors, fraud, and biases. This updated second edition shows accountants and auditors how analyzing their corporate or public sector data can highlight transactions, balances, or subsets of transactions or balances in need of attention. These tests are made up of a set of initial high-level overview tests followed by a series of more focused tests. These focused tests use a variety of quantitative methods including Benford’s Law, outlier detection, the detection of duplicates, a comparison to benchmarks, time-series methods, risk-scoring, and sometimes simply statistical logic. The tests in the new edition include the newly developed vector variation score that quantifies the change in an array of data from one period to the next. The goals of the tests are to either produce a small sample of suspicious transactions, a small set of transaction groups, or a risk score related to individual transactions or a group of items. The new edition includes over two hundred figures. Each chapter, where applicable, includes one or more cases showing how the tests under discussion could have detected the fraud or anomalies. The new edition also includes two chapters each describing multi-million-dollar fraud schemes and the insights that can be learned from those examples. These interesting real-world examples help to make the text accessible and understandable for accounting professionals and accounting students without rigorous backgrounds in mathematics and statistics. Emphasizing practical applications, the new edition shows how to use either Excel or Access to run these analytics tests. The book also has some coverage on using Minitab, IDEA, R, and Tableau to run forensic-focused tests. The use of SAS and Power BI rounds out the software coverage. The software screenshots use the latest versions of the software available at the time of writing. This authoritative book: Describes the use of statistically-based techniques including Benford’s Law, descriptive statistics, and the vector variation score to detect errors and anomalies Shows how to run most of the tests in Access and Excel, and other data analysis software packages for a small sample of the tests Applies the tests under review in each chapter to the same purchasing card data from a government entity Includes interesting cases studies throughout that are linked to the tests being reviewed. Includes two comprehensive case studies where data analytics could have detected the frauds before they reached multi-million-dollar levels Includes a continually-updated companion website with the data sets used in the chapters, the queries used in the chapters, extra coverage of some topics or cases, end of chapter questions, and end of chapter cases. Written by a prominent educator and researcher in forensic accounting and auditing, the new edition of Forensic Analytics: Methods and Techniques for Forensic Accounting Investigations is an essential resource for forensic accountants, auditors, comptrollers, fraud investigators, and graduate students.
Author: Alex Bellos
Publisher: Simon and Schuster
ISBN: 1451640129
Category : Mathematics
Languages : en
Pages : 352
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Book Description
From triangles, rotations and power laws, to cones, curves and the dreaded calculus, Alex takes you on a journey of mathematical discovery with his signature wit and limitless enthusiasm. He sifts through over 30,000 survey submissions to uncover the world’s favourite number, and meets a mathematician who looks for universes in his garage. He attends the World Mathematical Congress in India, and visits the engineer who designed the first roller-coaster loop. Get hooked on math as Alex delves deep into humankind’s turbulent relationship with numbers, and reveals how they have shaped the world we live in.
Author: Alex Ely Kossovsky
Publisher: World Scientific Publishing Company
ISBN: 9789814651202
Category : Distribution (Probability theory)
Languages : en
Pages : 0
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Book Description
This leads to the key finding that the phenomenon is actually quantitative in nature. Why? The author illustrates that in extreme generality, nature creates many small quantities but very few big quantities, corroborating the motto "small is beautiful", and that therefore all this is applicable just as well to data written in the ancient Roman, Mayan, Egyptian, and other digit-less civilizations. Fraudsters are typically not aware of this digital pattern and tend to invent numbers with approximately equal digital frequencies. The digital analyst can easily check reported data for compliance with this digital law, enabling the detection of tax evasion, Ponzi schemes, and other financial scams.
Author: Sheldon M. Ross
Publisher: Academic Press
ISBN: 0080922104
Category : Mathematics
Languages : en
Pages : 841
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Book Description
Introductory Statistics, Third Edition, presents statistical concepts and techniques in a manner that will teach students not only how and when to utilize the statistical procedures developed, but also to understand why these procedures should be used. This book offers a unique historical perspective, profiling prominent statisticians and historical events in order to motivate learning. To help guide students towards independent learning, exercises and examples using real issues and real data (e.g., stock price models, health issues, gender issues, sports, scientific fraud) are provided. The chapters end with detailed reviews of important concepts and formulas, key terms, and definitions that are useful study tools. Data sets from text and exercise material are available for download in the text website. This text is designed for introductory non-calculus based statistics courses that are offered by mathematics and/or statistics departments to undergraduate students taking a semester course in basic Statistics or a year course in Probability and Statistics. - Unique historical perspective profiling prominent statisticians and historical events to motivate learning by providing interest and context - Use of exercises and examples helps guide the student towards indpendent learning using real issues and real data, e.g. stock price models, health issues, gender issues, sports, scientific fraud. - Summary/Key Terms- chapters end with detailed reviews of important concepts and formulas, key terms and definitions which are useful to students as study tools
Author: Lawrence N. Dworsky
Publisher: John Wiley & Sons
ISBN: 0470282045
Category : Mathematics
Languages : en
Pages : 328
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Book Description
An engaging, entertaining, and informative introduction to probability and prediction in our everyday lives Although Probably Not deals with probability and statistics, it is not heavily mathematical and is not filled with complex derivations, proofs, and theoretical problem sets. This book unveils the world of statistics through questions such as what is known based upon the information at hand and what can be expected to happen. While learning essential concepts including "the confidence factor" and "random walks," readers will be entertained and intrigued as they move from chapter to chapter. Moreover, the author provides a foundation of basic principles to guide decision making in almost all facets of life including playing games, developing winning business strategies, and managing personal finances. Much of the book is organized around easy-to-follow examples that address common, everyday issues such as: How travel time is affected by congestion, driving speed, and traffic lights Why different gambling casino strategies ultimately offer players no advantage How to estimate how many different birds of one species are seen on a walk through the woods Seemingly random events—coin flip games, the Central Limit Theorem, binomial distributions and Poisson distributions, Parrando's Paradox, and Benford's Law—are addressed and treated through key concepts and methods in probability. In addition, fun-to-solve problems including "the shared birthday" and "the prize behind door number one, two, or three" are found throughout the book, which allow readers to test and practice their new probability skills. Requiring little background knowledge of mathematics, readers will gain a greater understanding of the many daily activities and events that involve random processes and statistics. Combining the mathematics of probability with real-world examples, Probably Not is an ideal reference for practitioners and students who would like to learn more about the role of probability and statistics in everyday decision making.
Author: Steven J. Miller
Publisher: Princeton University Press
ISBN: 0691215979
Category : Mathematics
Languages : en
Pages : 526
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Book Description
In a manner accessible to beginning undergraduates, An Invitation to Modern Number Theory introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research. Steven Miller and Ramin Takloo-Bighash introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach's Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory. Providing exercises, references to the background literature, and Web links to previous student research projects, An Invitation to Modern Number Theory can be used to teach a research seminar or a lecture class.
Author: Robert P. Dobrow
Publisher: John Wiley & Sons
ISBN: 1118589440
Category : Mathematics
Languages : en
Pages : 366
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Book Description
An introduction to probability at the undergraduate level Chance 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 limits An early introduction to random variables and Monte Carlo simulation and an emphasis on conditional probability, conditioning, and developing probabilistic intuition An R tutorial with example script files Many classic and historical problems of probability as well as nontraditional material, such as Benford’s law, power-law distributions, and Bayesian statistics A topics section with suitable material for projects and explorations, such as random walk on graphs, Markov chains, and Markov chain Monte Carlo Chapter-by-chapter summaries and hundreds of practical exercises Probability: With Applications and R is an ideal text for a beginning course in probability at the undergraduate level.
