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Title: Ubiquity of Benford's law and emergence of the reciprocal distribution

Journal Article · · Physics Letters. A
 [1];  [1]; ORCiD logo [2]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division
  2. Harvard Univ., Cambridge, MA (United States). Dept. of Earth and Planetary Sciences; Santa Fe Inst. (SFI), Santa Fe, NM (United States)
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In this paper, we apply the Law of Total Probability to the construction of scale-invariant probability distribution functions (pdf's), and require that probability measures be dimensionless and unitless under a continuous change of scales. If the scale-change distribution function is scale invariant then the constructed distribution will also be scale invariant. Repeated application of this construction on an arbitrary set of (normalizable) pdf's results again in scale-invariant distributions. The invariant function of this procedure is given uniquely by the reciprocal distribution, suggesting a kind of universality. Finally, we separately demonstrate that the reciprocal distribution results uniquely from requiring maximum entropy for size-class distributions with uniform bin sizes.

Research Organization:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA)
Contributing Organization:
Harvard Univ., Cambridge, MA (United States)
Grant/Contract Number:
AC52-06NA25396
OSTI ID:
1321724
Alternate ID(s):
OSTI ID: 1345436
Report Number(s):
LA-UR-13-20486
Journal Information:
Physics Letters. A, Vol. 380, Issue 22-23; ISSN 0375-9601
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 3 works
Citation information provided by
Web of Science

References (16)

Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection book January 2012
Note on the Frequency of Use of the Different Digits in Natural Numbers journal January 1881
A Statistical Derivation of the Significant-Digit Law journal November 1995
Benford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem journal February 2011
A basic theory of Benford’s Law journal January 2011
Benford’s law and cross-sections of A(n, $ \alpha$ )B reactions journal June 2011
Benford’s law and complex atomic spectra journal January 2008
Newcomb-Benford law in Astrophysical Sources journal November 2006
Benford's law in the natural sciences journal November 2010
Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf journal April 2001
Benford’s law, its applicability and breakdown in the IR spectra of polymers journal February 2016
Genome Sizes and the Benford Distribution journal May 2012
On the Distribution of Numbers journal October 1970
On the Distribution of First Significant Digits journal December 1961
A Mathematical Theory of Communication journal July 1948
A note on the roundoff error in the Numerov algorithm journal September 1978

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Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
97 MATHEMATICS AND COMPUTING
Benford/Reciprocal distribution
Information/Communication theory
Iteration theory