Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T07:36:38.740Z Has data issue: false hasContentIssue false
  • English
  • Français

An invariant-sum characterization of Benford's law

Part of: Foundations of probability theory

Published online by Cambridge University Press:  14 July 2016

Pieter C. Allaart
Pieter C. Allaart*
Affiliation:
Vrije Universiteit Amsterdam
*
∗Postal address: Department of Mathematics and Computer Science, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081HV Amsterdam, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

The accountant Nigrini remarked that in tables of data distributed according to Benford's law, the sum of all elements with first digit d (d = 1, 2,· ··, 9) is approximately constant. In this note, a mathematical formulation of Nigrini's observation is given and it is shown that Benford's law is the unique probability distribution such that the expected sum of all elements with first digits d1, · ··, dk is constant for every fixed k.

Keywords

FIRST SIGNIFICANT DIGITBENFORD'S LAWMANTISSA FUNCTIONSUM-INVARIANCE

MSC classification

Primary: 60A10: Probabilistic measure theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 34 , Issue 1 , March 1997 , pp. 288 - 291
Copyright
Copyright © Applied Probability Trust 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Hamming, R. W. (1970) On the distribution of numbers. Bell System Tech. J. 49, 1609–1625.Google Scholar
Hill, T. P. (1994) Base-invariance implies Benford's law. Proc. Amer. Math. Soc. 123, 887–895.Google Scholar
Nigrini, M. (1992) The detection of income evasion through an analysis of digital distributions. PhD thesis. Department of Accounting, University of Cincinnati.Google Scholar
Raimi, R. (1976) The first digit problem. Amer. Math. Monthly 83, 521–538.CrossRefGoogle Scholar
Royden, H. L. (1988) Real Analysis. 3rd edn. MacMillan, New York.Google Scholar
Schatte, P. (1988) On mantissa distributions in computing and Benford's law. J. Inf. Process. Cybern. 24, 443–55.Google Scholar
Varian, H. (1972) Benford's law. Amer. Statistician 26, 65–66.Google Scholar