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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


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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