Ubiquity of Benford's law and emergence of the reciprocal distribution
Abstract
In this paper, we apply the Law of Total Probability to the construction of scaleinvariant probability distribution functions (pdf's), and require that probability measures be dimensionless and unitless under a continuous change of scales. If the scalechange 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 scaleinvariant 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 sizeclass distributions with uniform bin sizes.
 Authors:

 Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division
 Harvard Univ., Cambridge, MA (United States). Dept. of Earth and Planetary Sciences; Santa Fe Inst. (SFI), Santa Fe, NM (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 Contributing Org.:
 Harvard Univ., Cambridge, MA (United States)
 OSTI Identifier:
 1321724
 Alternate Identifier(s):
 OSTI ID: 1345436
 Report Number(s):
 LAUR1320486
Journal ID: ISSN 03759601
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physics Letters. A
 Additional Journal Information:
 Journal Volume: 380; Journal Issue: 2223; Journal ID: ISSN 03759601
 Publisher:
 Elsevier
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICS AND COMPUTING; Benford/Reciprocal distribution; Information/Communication theory; Iteration theory
Citation Formats
Friar, James Lewis, Goldman, Terrance J., and PérezMercader, J. Ubiquity of Benford's law and emergence of the reciprocal distribution. United States: N. p., 2016.
Web. doi:10.1016/j.physleta.2016.03.045.
Friar, James Lewis, Goldman, Terrance J., & PérezMercader, J. Ubiquity of Benford's law and emergence of the reciprocal distribution. United States. doi:10.1016/j.physleta.2016.03.045.
Friar, James Lewis, Goldman, Terrance J., and PérezMercader, J. Thu .
"Ubiquity of Benford's law and emergence of the reciprocal distribution". United States. doi:10.1016/j.physleta.2016.03.045. https://www.osti.gov/servlets/purl/1321724.
@article{osti_1321724,
title = {Ubiquity of Benford's law and emergence of the reciprocal distribution},
author = {Friar, James Lewis and Goldman, Terrance J. and PérezMercader, J.},
abstractNote = {In this paper, we apply the Law of Total Probability to the construction of scaleinvariant probability distribution functions (pdf's), and require that probability measures be dimensionless and unitless under a continuous change of scales. If the scalechange 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 scaleinvariant 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 sizeclass distributions with uniform bin sizes.},
doi = {10.1016/j.physleta.2016.03.045},
journal = {Physics Letters. A},
number = 2223,
volume = 380,
place = {United States},
year = {2016},
month = {4}
}
Web of Science