An Introduction To Benford S Law

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An Introduction to Benford's Law

Arno Berger,Theodore P. Hill
An Introduction to Benford s Law Book PDF

Author : Arno Berger,Theodore P. Hill
Publisher : Princeton University Press
Page : 256 pages
File Size : 46,7 Mb
Release : 2015-05-26
Category : Mathematics
ISBN : 9780691163062

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An Introduction to Benford's Law by Arno Berger,Theodore P. Hill Pdf

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.

Benford's Law

Steven J. Miller
Benford s Law Book PDF

Author : Steven J. Miller
Publisher : Princeton University Press
Page : 465 pages
File Size : 44,6 Mb
Release : 2015-06-09
Category : Mathematics
ISBN : 9781400866595

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Benford's Law by Steven J. Miller Pdf

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.

An Introduction to Benford's Law

Arno Berger,Theodore P. Hill
An Introduction to Benford s Law Book PDF

Author : Arno Berger,Theodore P. Hill
Publisher : Princeton University Press
Page : 256 pages
File Size : 44,8 Mb
Release : 2015-05-26
Category : Mathematics
ISBN : 9780691163062

Get Book

An Introduction to Benford's Law by Arno Berger,Theodore P. Hill Pdf

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.

Benford's Law

Mark J. Nigrini
Benford s Law Book PDF

Author : Mark J. Nigrini
Publisher : John Wiley & Sons
Page : 352 pages
File Size : 49,8 Mb
Release : 2012-03-09
Category : Business & Economics
ISBN : 9781118286869

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Benford's Law by Mark J. Nigrini Pdf

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.

The Grapes of Math

Alex Bellos
The Grapes of Math Book PDF

Author : Alex Bellos
Publisher : Simon and Schuster
Page : 352 pages
File Size : 52,9 Mb
Release : 2014-06-10
Category : Mathematics
ISBN : 9781451640120

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The Grapes of Math by Alex Bellos Pdf

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.

Studies in Benford's Law

Alex Kossovsky
Studies in Benford s Law Book PDF

Author : Alex Kossovsky
Publisher : Independently Published
Page : 401 pages
File Size : 55,9 Mb
Release : 2019-04-18
Category : Electronic
ISBN : 172928325X

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Studies in Benford's Law by Alex Kossovsky Pdf

Numbers are written in our digital language system by conveniently utilizing the ten digits 0 to 9, just as books and text are written in English, French, or German, by conveniently utilizing the 26 letters A to Z. Surprisingly, and against all common sense or intuition, the spread of these ten digits within numbers of random data is not equal and uniform, but rather highly uneven. Benford's Law predicts that the first digit on the left-most side of numbers is proportioned between all possible 1 to 9 digits as in LOG(1 + 1/digit) in the approximate, so that occurrences of low digits such as 1, 2, 3 in the first place are much more frequent than occurrences of high digits such as 7, 8, 9. Remarkably, Benford's Law is found to be valid in almost all real-life statistics, such as in data relating to physics, chemistry, astronomy, geology, biology, economics, finance, accounting, engineering, and governmental census information. As such Benford's Law constitutes a unique common thread running through and uniting data sets relating to all scientific disciplines. This book explores the most recent research results and discoveries in the field of Benford's Law with a strong emphasis on relevant real-life physical, financial, accounting, scientific, and demographic data, while tying in diverse and seemingly unrelated areas of mathematics such as prime numbers, quantitative partition models, and exponential growth series. The book also explores the resultant quantitative and digital configurations of an arithmetical mix of random variables, such as random addition processes which are known to yield the symmetric Uniform Distribution as predicated by the Central Limit Theorem, as well as random multiplication processes which are known to yield the skewed Lognormal Distribution. Thus the involvement of various additive and multiplicative terms within a single expression of a stochastic process or a random variable constitutes a tug of war between these two arithmetical operations. Data relating to physical processes in nature as well as to particular financial and accounting transactions are at times found to be modeled after such arithmetical combinations of additions and multiplications of random variables. In addition, the latest methods in forensic digital analysis in the context of data fraud detection are presented in this book as the main application of the Benford phenomenon, including the most recent innovative techniques utilizing Digital Development Pattern.

Forensic Analytics

Mark J. Nigrini
Forensic Analytics Book PDF

Author : Mark J. Nigrini
Publisher : John Wiley & Sons
Page : 549 pages
File Size : 54,8 Mb
Release : 2020-04-20
Category : Business & Economics
ISBN : 9781119585909

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Forensic Analytics by Mark J. Nigrini Pdf

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.

Benford's Law

Alex Ely Kossovsky
Benford s Law Book PDF

Author : Alex Ely Kossovsky
Publisher : World Scientific Publishing Company
Page : 0 pages
File Size : 50,8 Mb
Release : 2015
Category : Distribution (Probability theory)
ISBN : 9814651206

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Benford's Law by Alex Ely Kossovsky Pdf

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.

Probably Not

Lawrence N. Dworsky
Probably Not Book PDF

Author : Lawrence N. Dworsky
Publisher : John Wiley & Sons
Page : 352 pages
File Size : 53,5 Mb
Release : 2019-09-04
Category : Mathematics
ISBN : 9781119518105

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Probably Not by Lawrence N. Dworsky Pdf

A revised edition that explores random numbers, probability, and statistical inference at an introductory mathematical level Written in an engaging and entertaining manner, the revised and updated second edition of Probably Not continues to offer an informative guide to probability and prediction. The expanded second edition contains problem and solution sets. In addition, the book’s illustrative examples reveal how we are living in a statistical world, what we can expect, what we really know based upon the information at hand and explains when we only think we know something. The author introduces the principles of probability and explains probability distribution functions. The book covers combined and conditional probabilities and contains a new section on Bayes Theorem and Bayesian Statistics, which features some simple examples including the Presecutor’s Paradox, and Bayesian vs. Frequentist thinking about statistics. New to this edition is a chapter on Benford’s Law that explores measuring the compliance and financial fraud detection using Benford’s Law. This book: Contains relevant mathematics and examples that demonstrate how to use the concepts presented Features a new chapter on Benford’s Law that explains why we find Benford’s law upheld in so many, but not all, natural situations Presents updated Life insurance tables Contains updates on the Gantt Chart example that further develops the discussion of random events Offers a companion site featuring solutions to the problem sets within the book Written for mathematics and statistics students and professionals, the updated edition of Probably Not: Future Prediction Using Probability and Statistical Inference, Second Edition combines the mathematics of probability with real-world examples. LAWRENCE N. DWORSKY, PhD, is a retired Vice President of the Technical Staff and Director of Motorola’s Components Research Laboratory in Schaumburg, Illinois, USA. He is the author of Introduction to Numerical Electrostatics Using MATLAB from Wiley.

Introductory Statistics

Sheldon M. Ross
Introductory Statistics Book PDF

Author : Sheldon M. Ross
Publisher : Academic Press
Page : 841 pages
File Size : 40,9 Mb
Release : 2010-01-19
Category : Mathematics
ISBN : 9780080922102

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Introductory Statistics by Sheldon M. Ross Pdf

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

An Invitation to Modern Number Theory

Steven J. Miller,Ramin Takloo-Bighash
An Invitation to Modern Number Theory Book PDF

Author : Steven J. Miller,Ramin Takloo-Bighash
Publisher : Princeton University Press
Page : 128 pages
File Size : 48,9 Mb
Release : 2020-08-04
Category : Mathematics
ISBN : 9780691215976

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An Invitation to Modern Number Theory by Steven J. Miller,Ramin Takloo-Bighash Pdf

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.

Chaos and Chance

Arno Berger
Chaos and Chance Book PDF

Author : Arno Berger
Publisher : De Gruyter Textbook
Page : 0 pages
File Size : 55,9 Mb
Release : 2001
Category : Mathematics
ISBN : 3110169908

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Chaos and Chance by Arno Berger Pdf

"An introductory though mathematically reliable text" that brings together the complementary approaches of the phenomenological theory of chaos, with its focus on complexity and unpredictability, and the statistically oriented ergodic theory. It is intended for advanced undergraduates and beginning graduate students from applied sciences; it requires a sound knowledge of calculus, while some basic experience in measure theory and discrete dynamical systems would be helpful. c. Book News Inc.

Benford's Law

Steven J. Miller
Benford s Law Book PDF

Author : Steven J. Miller
Publisher : Princeton University Press
Page : 464 pages
File Size : 40,7 Mb
Release : 2015-05-26
Category : Mathematics
ISBN : 9780691147611

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Benford's Law by Steven J. Miller Pdf

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.

Principles, Process and Practice of Professional Number Juggling

Alan R. Jones
Principles Process and Practice of Professional Number Juggling Book PDF

Author : Alan R. Jones
Publisher : Routledge
Page : 242 pages
File Size : 51,8 Mb
Release : 2018-09-13
Category : Business & Economics
ISBN : 9781351661355

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Principles, Process and Practice of Professional Number Juggling by Alan R. Jones Pdf

Principles, Process and Practice of Professional Number Juggling (Volume 1 of the Working Guides to Estimating & Forecasting series) sets the scene of TRACEability and good estimate practice that is followed in the other volumes in this series of five working guides. It clarifies the difference between an Estimating Process, Procedure, Approach, Method and Technique. It expands on these definitions of Approach (Top-down, Bottom-up and ‘Ethereal’) and Method (Analogy, Parametric and ‘Trusted Source’) and discusses how these form the basis of all other means of establishing an estimate. This volume also underlines the importance of ‘data normalisation’ in any estimating procedure, and demonstrates that the Estimating by Analogy Method, in essence, is a simple extension of Data Normalisation. The author looks at simple measures of assessing the maturity or health of an estimate, and offers a means of assessing a spreadsheet for any inherent risks or errors that may be introduced by failing to follow good practice in spreadsheet design and build. This book provides a taster of the more numerical techniques covered in the remainder of the series by considering how an estimator can potentially exploit Benford’s Law (traditionally used in Fraud Detection) to identify systematic bias from third party contributors. It will be a valuable resource for estimators, engineers, accountants, project risk specialists as well as students of cost engineering.

Nonstandard Analysis in Practice

Francine Diener,Marc Diener
Nonstandard Analysis in Practice Book PDF

Author : Francine Diener,Marc Diener
Publisher : Springer Science & Business Media
Page : 262 pages
File Size : 53,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642577581

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Nonstandard Analysis in Practice by Francine Diener,Marc Diener Pdf

This book introduces the graduate mathematician and researcher to the effective use of nonstandard analysis (NSA). It provides a tutorial introduction to this modern theory of infinitesimals, followed by nine examples of applications, including complex analysis, stochastic differential equations, differential geometry, topology, probability, integration, and asymptotics. It ends with remarks on teaching with infinitesimals.