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This book tells the story of a newly discovered and apparently mysterious digital phenomenon of Benford's Law. This phenomenon manifests itself by the empirical finding that not all digits are created equal, but rather that low digits such as 1, 2, 3 occur much more frequently than high digits such as 7, 8, 9, in accounting, financial, scientific, and almost all other data types. This work represents the first ever published work giving a comprehensive and in depth account of all the theoretical aspects and applications of Benford's Law. The reader is subsequently led into a fascinating intellectual journey through the interacting worlds of digits, numbers, and quantities, a journey that ends with the compelling conclusion that the entire phenomenon is truly quantitative in nature, and applicable just as well to the ancient Roman, Mayan, and Egyptian digit-less civilizations. The second section covers the applications of the law in forensic data analysis for the purpose of fraud detection. It is concise, reader-friendly, and can be understood without deep knowledge in statistical theory or difficult mathematics. This fraud detection section gathers all known methods, results, and standards in the accounting and auditing industry, from quite a wide variety of articles on this issue, summarizes and fuses them into a singular coherent whole. In addition, a newly invented (patent-pending) digital algorithm is presented, enabling the auditor to detect such fraud even when the sophisticated and well-educated cheater is aware of the law and attempts to appear as if he or she is innocently complying with the digital pattern. A large portion of the book is devoted to understanding the variety of causes explanations of the phenomenon. Seeing Benford's Law in this bird's eye view enables the reader to see the forest in all its glory and beauty instead of tiring one's self repeatedly checking individual trees.
Contrary to common intuition that all digits should occur randomly with equal chances in real data, empirical examinations consistently show that not all digits are created equal, but rather that low digits such as {1, 2, 3} occur much more frequently than high digits such as {7, 8, 9} in almost all data types, such as those relating to geology, chemistry, astronomy, physics, and engineering, as well as in accounting, financial, econometrics, and demographics data sets. This intriguing digital phenomenon is known as Benford's Law. This book gives a comprehensive and in-depth account of all the theoretical aspects, results, causes and explanations of Benford's Law, with a strong emphasis on the connection to real-life data and the physical manifestation of the law. In addition to such a bird's eye view of the digital phenomenon, the conceptual distinctions between digits, numbers, and quantities are explored; leading to the key finding that the phenomenon is actually quantitative in nature; originating from the fact 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. The forensic fraud detection section in this book is written in a very concise and reader-friendly style; gathering all known methods and standards in the accounting and auditing industry; summarizing and fusing them into a singular coherent whole; and can be understood without deep knowledge in statistical theory or advanced mathematics. In addition, a digital algorithm is presented, enabling the auditor to detect fraud even when the sophisticated cheater is aware of the law and invents numbers accordingly. The algorithm employs a subtle inner digital pattern within the Benford's pattern itself. This newly discovered pattern is deemed to be nearly universal, being even more prevalent than the Benford phenomenon, as it is found in all random data sets, Benford as well as non-Benford types. Contents:Benford's LawForensic Digital AnalysisFraud DetectionData Compliance TestsConceptual and Mathematical FoundationsBenford's Law in the Physical SciencesTopics in Benford's LawThe Law of Relative Quantities Readership: Professionals, researchers and serious students of financial and data analysis, forensic accounting, fraud investigation, auditing, mathematics and probability and statistics. Key Features:The book is a concise account of practical applications of the phenomenon of fraud detection and it corrects several errors committed in the field where mistaken applications are usedThe perceptive reader interested in knowing about the use of this digital law in fraud detection, would be able to learn about it with a minimal amount of effort and time, without searching through literally hundreds of various small articles on the topicThe book provides numerous new theoretical points-of-view of the phenomenon, new methods for testing data for compliance, and fuses many different aspects of the law into a singular explanationKeywords:Benford's Law;Digits;Quantities;Relative Quantities;Numbers;Fraud;Fraud Detection;Data;Data Analysis;Forensic Analysis;Pattern;Physics;Chemistry;Geology;Astronomy
This new edition includes the latest advances and developments in computational probability involving A Probability Programming Language (APPL). The book examines and presents, in a systematic manner, computational probability methods that encompass data structures and algorithms. The developed techniques address problems that require exact probability calculations, many of which have been considered intractable in the past. The book addresses the plight of the probabilist by providing algorithms to perform calculations associated with random variables. Computational Probability: Algorithms and Applications in the Mathematical Sciences, 2nd Edition begins with an introductory chapter that contains short examples involving the elementary use of APPL. Chapter 2 reviews the Maple data structures and functions necessary to implement APPL. This is followed by a discussion of the development of the data structures and algorithms (Chapters 3–6 for continuous random variables and Chapters 7–9 for discrete random variables) used in APPL. The book concludes with Chapters 10–15 introducing a sampling of various applications in the mathematical sciences. This book should appeal to researchers in the mathematical sciences with an interest in applied probability and instructors using the book for a special topics course in computational probability taught in a mathematics, statistics, operations research, management science, or industrial engineering department.
Jeder kennt p = 3,14159..., viele kennen e = 2,71828..., einige i. Und dann? Die "viertwichtigste" Konstante ist die Eulersche Zahl g = 0,5772156... - benannt nach dem genialen Leonhard Euler (1707-1783). Bis heute ist unbekannt, ob g eine rationale Zahl ist. Das Buch lotet die "obskure" Konstante aus. Die Reise beginnt mit Logarithmen und der harmonischen Reihe. Es folgen Zeta-Funktionen und Eulers wunderbare Identität, Bernoulli-Zahlen, Madelungsche Konstanten, Fettfinger in Wörterbüchern, elende mathematische Würmer und Jeeps in der Wüste. Besser kann man nicht über Mathematik schreiben. Was Julian Havil dazu zu sagen hat, ist spektakulär.
Das vorliegende Buch handelt von Benfords Gesetz über die Verteilung signifikanter Ziffern von realen Zahlen und dessen Anwendung in der Marktforschung. Benfords Gesetz besagt kurzgefasst, dass die Anfangsziffern bestimmter Datenmengen nicht gleichverteilt sind, sondern einer logarithmischen Verteilung folgen. Es werden ein Wahrscheinlichkeitsraum für Benfords Gesetz und Formeln für die Verteilung der ersten, zweiten und n-ten Ziffer sowie die gemeinsame Verteilung der ersten n Ziffern eingeführt. Ferner werden die besonderen Eigenschaften der Benford-Verteilung wie die Skalen- und Baseninvarianz betrachtet. Als Hauptresultat wird ein Grenzwertsatz für signifikante Ziffern angegeben und bewiesen. Als besondere Anwendungsmöglichkeit wird die Aufdeckung von Fälschungen bei Interviews in der Marktforschung betrachtet. Dazu werden die Prozesse der Datenerhebung beleuchtet und Ergebnisse bisheriger Studien vorgestellt. Die verschiedenen in der Marktforschung auftauchenden Datentypen werden analysiert und ihre Eignung als Prüfgrößen untersucht. Darauf aufbauend wird ein Programm zum Test auf die Benford-Verteilung vorgestellt und eine mögliche Testfrage auf Tauglichkeit untersucht.

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