An Introduction To Benfords Law

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An Introduction to Benford's Law

Author : Arno Berger
Publisher : Princeton University Press
ISBN 13 : 0691163065
Total Pages : 256 pages
Book Rating : 4.6/5 (911 download)

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Book Synopsis An Introduction to Benford's Law by : Arno Berger

Download or read book An Introduction to Benford's Law written by Arno Berger and published by Princeton University Press. This book was released on 2015-05-26 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Author : Steven J. Miller
Publisher : Princeton University Press
ISBN 13 : 1400866596
Total Pages : 465 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Benford's Law by : Steven J. Miller

Download or read book Benford's Law written by Steven J. Miller and published by Princeton University Press. This book was released on 2015-06-09 with total page 465 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Benford's Law

Author : Mark J. Nigrini
Publisher : John Wiley & Sons
ISBN 13 : 1118286863
Total Pages : 352 pages
Book Rating : 4.1/5 (182 download)

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Book Synopsis Benford's Law by : Mark J. Nigrini

Download or read book Benford's Law written by Mark J. Nigrini and published by John Wiley & Sons. This book was released on 2012-03-09 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

An Introduction to Benford's Law

Author : Arno Berger
Publisher : Princeton University Press
ISBN 13 : 0691163065
Total Pages : 256 pages
Book Rating : 4.6/5 (911 download)

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Book Synopsis An Introduction to Benford's Law by : Arno Berger

The Grapes of Math

Author : Alex Bellos
Publisher : Simon and Schuster
ISBN 13 : 1451640129
Total Pages : 352 pages
Book Rating : 4.4/5 (516 download)

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Book Synopsis The Grapes of Math by : Alex Bellos

Download or read book The Grapes of Math written by Alex Bellos and published by Simon and Schuster. This book was released on 2014-06-10 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Forensic Analytics

Author : Mark J. Nigrini
Publisher : John Wiley & Sons
ISBN 13 : 1119585902
Total Pages : 549 pages
Book Rating : 4.1/5 (195 download)

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Book Synopsis Forensic Analytics by : Mark J. Nigrini

Download or read book Forensic Analytics written by Mark J. Nigrini and published by John Wiley & Sons. This book was released on 2020-04-20 with total page 549 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Studies in Benford's Law

Author : Alex Kossovsky
Publisher : Independently Published
ISBN 13 : 9781729283257
Total Pages : 401 pages
Book Rating : 4.2/5 (832 download)

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Book Synopsis Studies in Benford's Law by : Alex Kossovsky

Download or read book Studies in Benford's Law written by Alex Kossovsky and published by Independently Published. This book was released on 2019-04-18 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Introductory Statistics

Author : Sheldon M. Ross
Publisher : Academic Press
ISBN 13 : 0080922104
Total Pages : 841 pages
Book Rating : 4.0/5 (89 download)

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Book Synopsis Introductory Statistics by : Sheldon M. Ross

Download or read book Introductory Statistics written by Sheldon M. Ross and published by Academic Press. This book was released on 2010-01-19 with total page 841 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Benford's Law

Author : Alex Ely Kossovsky
Publisher : World Scientific Publishing Company
ISBN 13 : 9789814651202
Total Pages : 0 pages
Book Rating : 4.6/5 (512 download)

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Book Synopsis Benford's Law by : Alex Ely Kossovsky

Download or read book Benford's Law written by Alex Ely Kossovsky and published by World Scientific Publishing Company. This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Author : Lawrence N. Dworsky
Publisher : John Wiley & Sons
ISBN 13 : 1119518105
Total Pages : 352 pages
Book Rating : 4.1/5 (195 download)

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Book Synopsis Probably Not by : Lawrence N. Dworsky

Download or read book Probably Not written by Lawrence N. Dworsky and published by John Wiley & Sons. This book was released on 2019-09-04 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Probability

Author : Robert P. Dobrow
Publisher : John Wiley & Sons
ISBN 13 : 1118589440
Total Pages : 366 pages
Book Rating : 4.1/5 (185 download)

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Book Synopsis Probability by : Robert P. Dobrow

Download or read book Probability written by Robert P. Dobrow and published by John Wiley & Sons. This book was released on 2013-10-16 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Far Futures

Author : Gregory Benford
Publisher : Macmillan
ISBN 13 : 9780312863791
Total Pages : 352 pages
Book Rating : 4.8/5 (637 download)

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Book Synopsis Far Futures by : Gregory Benford

Download or read book Far Futures written by Gregory Benford and published by Macmillan. This book was released on 1997-08-15 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: Collection of five hard science fiction novellas, all set at least ten thousand years in the future that confront the issues of cosmology, astronomy, evolution, and biology.

An Invitation to Modern Number Theory

Author : Steven J. Miller
Publisher : Princeton University Press
ISBN 13 : 9780691120607
Total Pages : 532 pages
Book Rating : 4.1/5 (26 download)

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Book Synopsis An Invitation to Modern Number Theory by : Steven J. Miller

Download or read book An Invitation to Modern Number Theory written by Steven J. Miller and published by Princeton University Press. This book was released on 2006-03-26 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: PART 1. BASIC NUMBER THEORY -- 1. Mod p Arithmetic, Group Theory and Cryptography -- 2. Arithmetic Functions -- 3. Zeta and L-Functions -- 4. Solutions to Diophantine Equations -- PART 2. CONTINUED FRACTIONS AND APPROXIMATIONS -- 5. Algebraic and Transcendental Numbers -- 6. The Proof of Roth's Theorem -- 7. Introduction to Continued Fractions -- PART 3. PROBABILISTIC METHODS AND EQUIDISTRIBUTION -- 8. Introduction to Probability -- 9. Applications of Probability: Benford's Law and Hypothesis Testing -- 10. Distribution of Digits of Continued Fractions -- 11. Introduction to Fourier Analysis -- 12. f n k g and Poissonian Behavior -- PART 4. THE CIRCLE METHOD -- 13. Introduction to the Circle Method -- 14. Circle Method: Heuristics for Germain Primes -- PART 5. RANDOM MATRIX THEORY AND L-FUNCTIONS -- 15. From Nuclear Physics to L-Functions -- 16. Random Matrix Theory: Eigenvalue Densities -- 17. Random Matrix Theory: Spacings between Adjacent Eigenvalues -- 18. The Explicit Formula and Density Conjectures -- Appendix A. Analysis Review -- Appendix B. Linear Algebra Review -- Appendix C. Hints and Remarks on the Exercises -- Appendix D. Concluding Remarks.

Single Digits

Author : Marc Chamberland
Publisher : Princeton University Press
ISBN 13 : 0691175691
Total Pages : 240 pages
Book Rating : 4.6/5 (911 download)

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Book Synopsis Single Digits by : Marc Chamberland

Download or read book Single Digits written by Marc Chamberland and published by Princeton University Press. This book was released on 2017-05-30 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The numbers one through nine have remarkable mathematical properties and characteristics. For instance, why do eight perfect card shuffles leave a standard deck of cards unchanged? Are there really "six degrees of separation" between all pairs of people? And how can any map need only four colors to ensure that no regions of the same color touch? In Single Digits, Marc Chamberland takes readers on a fascinating exploration of small numbers, from one to nine, looking at their history, applications, and connections to various areas of mathematics, including number theory, geometry, chaos theory, numerical analysis, and mathematical physics."--Jacket.

Introduction to Probability

Author : David F. Anderson
Publisher : Cambridge University Press
ISBN 13 : 110824498X
Total Pages : 447 pages
Book Rating : 4.1/5 (82 download)

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Book Synopsis Introduction to Probability by : David F. Anderson

Download or read book Introduction to Probability written by David F. Anderson and published by Cambridge University Press. This book was released on 2017-11-02 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: This classroom-tested textbook is an introduction to probability theory, with the right balance between mathematical precision, probabilistic intuition, and concrete applications. Introduction to Probability covers the material precisely, while avoiding excessive technical details. After introducing the basic vocabulary of randomness, including events, probabilities, and random variables, the text offers the reader a first glimpse of the major theorems of the subject: the law of large numbers and the central limit theorem. The important probability distributions are introduced organically as they arise from applications. The discrete and continuous sides of probability are treated together to emphasize their similarities. Intended for students with a calculus background, the text teaches not only the nuts and bolts of probability theory and how to solve specific problems, but also why the methods of solution work.

Forensic Analytics

Author : Mark J. Nigrini
Publisher : John Wiley & Sons
ISBN 13 : 9781118087633
Total Pages : 480 pages
Book Rating : 4.0/5 (876 download)

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Book Synopsis Forensic Analytics by : Mark J. Nigrini

Download or read book Forensic Analytics written by Mark J. Nigrini and published by John Wiley & Sons. This book was released on 2011-05-12 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discover how to detect fraud, biases, or errors in your data using Access or Excel With over 300 images, Forensic Analytics reviews and shows how twenty substantive and rigorous tests can be used to detect fraud, errors, estimates, or biases in your data. For each test, the original data is shown with the steps needed to get to the final result. The tests range from high-level data overviews to assess the reasonableness of data, to highly focused tests that give small samples of highly suspicious transactions. These tests are relevant to your organization, whether small or large, for profit, nonprofit, or government-related. Demonstrates how to use Access, Excel, and PowerPoint in a forensic setting Explores use of statistical techniques such as Benford's Law, descriptive statistics, correlation, and time-series analysis to detect fraud and errors Discusses the detection of financial statement fraud using various statistical approaches Explains how to score locations, agents, customers, or employees for fraud risk Shows you how to become the data analytics expert in your organization Forensic Analytics shows how you can use Microsoft Access and Excel as your primary data interrogation tools to find exceptional, irregular, and anomalous records.

Election Fraud

Author : R. Michael Alvarez
Publisher : Brookings Institution Press
ISBN 13 : 0815701608
Total Pages : 272 pages
Book Rating : 4.8/5 (157 download)

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Book Synopsis Election Fraud by : R. Michael Alvarez

Download or read book Election Fraud written by R. Michael Alvarez and published by Brookings Institution Press. This book was released on 2009-11-01 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: Allegations of fraud have marred recent elections around the world, from Russia and Italy to Mexico and the United States. Such charges raise fundamental questions about the quality of democracy in each country. Yet election fraud and, more broadly, electoral manipulation remain remarkably understudied concepts. There is no consensus on what constitutes election fraud, let alone how to detect and deter it. E lection Fraud: Detecting and Deterring Electoral Manipulation brings together experts on election law, election administration, and U.S. and comparative politics to address these critical issues. The first part of the book, which opens with an essay by Craig Donsanto of the U.S. Department of Justice, examines the U.S. understanding of election fraud in comparative perspective. In the second part of the book, D. Roderick Kiewiet, Jonathan N. Katz, and other scholars of U.S. elections draw on a wide variety of sources, including survey data, incident reports, and state-collected fraud allegations, to measure the extent and nature of election fraud in the United States. Finally, the third part of the book analyzes techniques for detecting and potentially deterring fraud. These strategies include both statistical analysis, as Walter R. Mebane, Jr. and Peter Ordeshook explain, and the now widespread practice of election monitoring, which Alberto Simpser examines in an intriguing essay.