Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T04:34:08.407Z Has data issue: false hasContentIssue false
  • English
  • Français

On the significands of uniform random variables

Part of: Arithmetic, number theory Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  26 July 2018

Arno Berger  and
Isaac Twelves
Arno Berger*
Affiliation:
University of Alberta
Isaac Twelves*
Affiliation:
University of Alberta
*
* Postal address: Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada.
* Postal address: Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

For all α > 0 and real random variables X, we establish sharp bounds for the smallest and the largest deviation of αX from the logarithmic distribution also known as Benford's law. In the case of uniform X, the value of the smallest possible deviation is determined explicitly. Our elementary calculation puts into perspective the recurring claims that a random variable conforms to Benford's law, at least approximately, whenever it has large spread.

Keywords

SignificandBenford random variableKolmogorov distancespreaduniform random variable

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62E15: Exact distribution theory 97F50: Real numbers, complex numbers
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 353 - 367
Copyright
Copyright © Applied Probability Trust 2018 

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

[1]Aldous, D. and Phan, T. (2010). When can one test an explanation? Compare and contrast Benford's law and the fuzzy CLT. Amer. Statistician 64, 221–227. 10.1198/tast.2010.09098Google Scholar
[2]Berger, A. and Hill, T. P. (2011). Benford's law strikes back: no simple explanation in sight for mathematical gem. Math. Intelligencer 33, 85–91. 10.1007/s00283-010-9182-3Google Scholar
[3]Berger, A. and Hill, T. P. (2015). An Introduction to Benford's Law. Princeton University Press. Google Scholar
[4]Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol II, 2nd end. John Wiley, New York. Google Scholar
[5]Fewster, R. M. (2009). A simple explanation of Benford's law. Amer. Statistician 63, 26–32. 10.1198/tast.2009.0005Google Scholar
[6]Gauvrit, N. and Delahaye, J.-P. (2011). Scatter and regularity imply Benford's law... and more. In Randomness Through Complexity, ed. H. Zenil, World Scientific, Singapore, pp. 53–69. 10.1142/9789814327756_0004Google Scholar
[7]Miller, S. J. (ed.) (2015). Benford's Law: Theory and Applications. Princeton University Press. Google Scholar
[8]Raimi, R. A. (1976). The first digit problem. Amer. Math. Monthly 83, 521–538. 10.1080/00029890.1976.11994162Google Scholar
[9]Wagon, S. (2009). Benford's law and data spread. Available at http://demonstrations.wolfram.com/BenfordsLawAndDataSpread. Google Scholar