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## Real-valued Random Processes

*Arno Berger and Theodore P. Hill*

### in An Introduction to Benford's Law

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.003.0008
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Applications of Benford's Law

*Arno Berger and Theodore P. Hill*

### in An Introduction to Benford's Law

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.003.0010
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## A Quick Introduction to Benfordʼs Law

*Steven J. Miller*

### in Benford's Law: Theory and Applications

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691147611
- eISBN:
- 9781400866595
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147611.003.0001
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Benfordʼs Law Geometry

*Lawrence Leemis*

### in Benford's Law: Theory and Applications

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691147611
- eISBN:
- 9781400866595
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147611.003.0004
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Chapter Twenty Exercises

*Steven J. Miller (ed.)*

### in Benford's Law: Theory and Applications

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691147611
- eISBN:
- 9781400866595
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147611.003.0020
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Introduction

*Arno Berger and Theodore P. Hill*

### in An Introduction to Benford's Law

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.003.0001
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Act natural, because of Benford’s Law

*Susan D'Agostino*

### in How to Free Your Inner Mathematician: Notes on Mathematics and Life

- Published in print:
- 2020
- Published Online:
- April 2020
- ISBN:
- 9780198843597
- eISBN:
- 9780191879388
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198843597.003.0008
- Subject:
- Mathematics, Educational Mathematics, Applied Mathematics

“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 ... More

## Benford's Law: Theory and Applications

*Steven J. Miller (ed.)*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691147611
- eISBN:
- 9781400866595
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147611.001.0001
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## An Introduction to Benford's Law

*Arno Berger and Theodore P. Hill*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.001.0001
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Finitely Additive Probability and Benford's Law

*Arno Berger and Theodore P. Hill*

### in An Introduction to Benford's Law

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.003.0009
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Scale-, Base-, and Sum-lnvariance

*Arno Berger and Theodore P. Hill*

### in An Introduction to Benford's Law

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.003.0005
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Real-valued Deterministic Processes

*Arno Berger and Theodore P. Hill*

### in An Introduction to Benford's Law

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.003.0006
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Complementing Benfordʼs Law for Small N: A Local Bootstrap

*Boudewijn F. Roukema*

### in Benford's Law: Theory and Applications

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691147611
- eISBN:
- 9781400866595
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147611.003.0010
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Significant Digits and the Significand

*Arno Berger and Theodore P. Hill*

### in An Introduction to Benford's Law

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.003.0002
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## The Benford Property

*Arno Berger and Theodore P. Hill*

### in An Introduction to Benford's Law

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.003.0003
- Subject:
- Mathematics, Probability / Statistics

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. ... More

## The Uniform Distribution and Benford's Law

*Arno Berger and Theodore P. Hill*

### in An Introduction to Benford's Law

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.003.0004
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Multi-dimensional Linear Processes

*Arno Berger and Theodore P. Hill*

### in An Introduction to Benford's Law

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691163062
- eISBN:
- 9781400866588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691163062.003.0007
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Data collection

*Andrew Gelman and Deborah Nolan*

### in Teaching Statistics: A Bag of Tricks

- Published in print:
- 2017
- Published Online:
- September 2017
- ISBN:
- 9780198785699
- eISBN:
- 9780191827518
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198785699.003.0006
- Subject:
- Mathematics, Educational Mathematics

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 ... More

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