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Benford's law arises naturally in a variety of stochastic settings, including products of independent random variables, mixtures of random samples from different distributions, and iterations of random maps. This chapter provides the concepts and tools to analyze significant digits and significands for these basic random processes. Benford's law also arises in many other important fields of stochastics, such as geometric Brownian motion, random matrices, and Bayesian models, and the chapter may serve as a preparation for specialized literature on these advanced topics. By Theorem 4.2 a random variable X is Benford if and only if log ¦X¦ is uniformly distributed modulo one.

This chapter provides a overview of the practical applications of Benford's law. These include fraud detection, detection of natural phenomena, diagnostics and design, computations and computer science, and as a pedagogical tool. In contrast to the rest of the book, this chapter is necessarily expository and informal. It has been organized into a handful of ad hoc categories, which the authors hope will help illuminate the main ideas. None of the conclusions of the experiments or data presented here have been scrutinized or verified by the authors of this book, since the intent here is not to promote or critique any specific application. Rather the goal is to offer a representative cross-section of the related scientific literature, in the hopes that this might continue to facilitate research in both the theory and practical applications of Benford's law.

This chapter provides a brief overview of Benford's law. It states Benford's law of digit bias and describes its history. The chapter then discusses the origins of Benford's law and gives numerous examples of data sets that follow this law, as well as some that do not. From these examples this chapter extracts several explanations as to the prevalence of Benford's law. Finally, the chapter closes with a quick summary of many of the diverse situations in which Benford's law holds, and why an observation that began in looking at the wear and tear in tables of logarithms has become a major tool in subjects as diverse as detecting tax fraud and building efficient computers.

This chapter switches from the traditional analysis of Benford's law using data sets to a search for probability distributions that obey Benford's law. It begins by briefly discussing the origins of Benford's law through the independent efforts of Simon Newcomb (1835–1909) and Frank Benford, Jr. (1883–1948), both of whom made their discoveries through empirical data. Although Benford's law applies to a wide variety of data sets, none of the popular parametric distributions, such as the exponential and normal distributions, agree exactly with Benford's law. The chapter thus highlights the failures of several of these well-known probability distributions in conforming to Benford's law, considers what types of probability distributions might produce data that obey Benford's law, and looks at some of the geometry associated with these probability distributions.

This chapter contains a list of exercises based on the lessons from all the previous chapters as well as their possible solutions. It also contains some supplementary information or references to places where the reader might obtain the supplementary information necessary for a better understanding of the concepts introduced in this volume. In addition to this, the chapter also contains brief bibliographies to further assist the reader or instructor in completing the exercises. It must be noted that some of the problems and exercises listed in this chapter are far more accessible after the later parts of this book have been read.

This introductory chapter provides an overview of Benford' law. Benford's law, also known as the First-digit or Significant-digit law, is the empirical gem of statistical folklore that in many naturally occurring tables of numerical data, the significant digits are not uniformly distributed as might be expected, but instead follow a particular logarithmic distribution. In its most common formulation, the special case of the first significant (i.e., first non-zero) decimal digit, Benford's law asserts that the leading digit is not equally likely to be any one of the nine possible digits 1, 2, … , 9, but is 1 more than 30 percent of the time, and is 9 less than 5 percent of the time, with the probabilities decreasing monotonically in between. The remainder of the chapter covers the history of Benford' law, empirical evidence, early explanations and mathematical framework of Benford' law.

“Act natural, because of Benford’s Law” explains how and why large data sets generated as a result of human behavior concerning health records, population counts, tax returns, stock prices, national debts, election data, and more, have numbers whose first digits are unevenly distributed, with Benford’s Law offering percentages. When an individual tampers with a naturally generated data set, they often introduce fake numbers whose first digits are (more or less) evenly distributed from one to nine. Often, a subsequent investigation reveals that someone has tampered with the data set. Mathematics students and enthusiasts are encouraged to act natural so as to avoid looking like a fraudulent data set that does not observe Benford’s Law. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

This chapter demonstrates the robustness of Benford's law even with regard to basic underlying mathematical hypotheses. It records, without proof, several of the main theorems pertinent to a theory of Benford's law in the context of finitely additive probability, and points the interested reader to further references on that theory in the literature.

This chapter establishes and illustrates three basic invariance properties of the Benford distribution that are instrumental in demonstrating whether or not certain datasets are Benford, and that also prove helpful for predicting which empirical data are likely to follow Benford's law closely. These are the scale-invariance property, base-invariance property, and sum-invariance property.

In science, one-dimensional deterministic (i.e., non-random) systems provide the simplest models for processes that evolve over time. Mathematically, these models take the form of one-dimensional difference or differential equations. This chapter presents the basic theory of Benford's law for them. Specifically, it studies conditions under which these models conform to Benford's law by generating Benford sequences and functions, respectively. The first seven sections of the chapter focus on discrete-time systems (i.e., difference equations) because they are somewhat easier to work with explicitly. Once the Benford properties of discrete-time systems are understood, it is straightforward to establish the analogous properties for continuous-time systems (i.e., differential equations), which is done in the chapter's final section.

This chapter analyzes the initially published results of the 2009 Iranian presidential elections. It applies a small N statistical first-digit frequency test that is as nonparametric as possible in a way that leaves no doubt regarding “how close” the observed system should be to a Benford's law limit. The approach this chapter studies is a local bootstrap model, designed to closely mimic the data in a way that should statistically reproduce its first-digit distributions, given some simple hypotheses about the general behavior of the system. This method was calibrated on several presidential-election first rounds from before 2009 and applied to the 2009 Iranian election.

Benford's law is a statement about the statistical distribution of significant (decimal) digits or, equivalently, about significands, namely fraction parts in floating-point arithmetic. Thus, a natural starting point for any study of Benford's law is the formal definition of significant digits and the significand function. This chapter contains formal definitions, examples, and graphs of significant digits and the significand (mantissa) function, and the probability spaces needed to formulate Benford's law precisely, including the crucial natural domain of “events,” the so-called significand σ‎-algebra.

In order to translate the informal versions of Benford's law into more precise formal statements, it is necessary to specify exactly what the Benford property means in various mathematical contexts. For the purpose of this book, the objects of interest fall mainly into three categories: sequences of real numbers, real-valued functions defined on [0,+ ∞), and probability distributions and random variables. This chapter defines Benford sequences, functions, and random variables, with examples of each.

The uniform distribution characterization of Benford's law is the most basic and powerful of all characterizations, largely because the mathematical theory of uniform distribution modulo one is very well developed for authoritative surveys. This chapter records and develops tools from that theory which will be used throughout this book to establish Benford behavior of sequences, functions, and random variables. Topics discussed include uniform distribution characterization of Benford's law, uniform distribution of sequences and functions, and uniform distribution of random variables.

Chapter 6 studied models based solely on the one-dimensional processes, however, for many applications, these models are often too simple and have to be replaced with or complemented by more sophisticated multi-dimensional models. This chapter studies Benford's law in the simplest deterministic multi-dimensional processes, namely, linear processes in discrete and continuous time. Despite their simplicity, these systems provide important models for many areas of science. Through far-reaching generalizations of results from earlier chapters, they will be shown to very often conform to Benford's law in that their dynamics is an abundant source of Benford sequences and functions. As in the previous chapter, the properties of continuous-time systems (i.e., differential equations) are analogous to those of discrete-time systems, and the chapter focuses on the latter in every but its last section.

One way students learn about data collection is actually to collect some data. This teaches some of the principles of experimental design and sampling and also gives the students a feel for the practical struggles and small decisions needed in real data gathering. This chapter includes several classroom demonstrations and examples to illustrate key ideas, as well as examples of instructions for longer projects. Activities include sampling from the telephone book and Benford’s law; family size and selection bias; aerial photographs to crowd count; experiments with question order and anchoring and measure bias; and taste tests and experimental design.