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On the significands of uniform random variables

  • Arno Berger (a1) and Isaac Twelves (a1)

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.

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Corresponding author

* Postal address: Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada.
** Email address: berger@ualberta.ca
*** Email address: twelves@ualberta.ca

References

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[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, 221227.
[2]Berger, A. and Hill, T. P. (2011). Benford's law strikes back: no simple explanation in sight for mathematical gem. Math. Intelligencer 33, 8591.
[3]Berger, A. and Hill, T. P. (2015). An Introduction to Benford's Law. Princeton University Press.
[4]Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol II, 2nd end. John Wiley, New York.
[5]Fewster, R. M. (2009). A simple explanation of Benford's law. Amer. Statistician 63, 2632.
[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. 5369.
[7]Miller, S. J. (ed.) (2015). Benford's Law: Theory and Applications. Princeton University Press.
[8]Raimi, R. A. (1976). The first digit problem. Amer. Math. Monthly 83, 521538.
[9]Wagon, S. (2009). Benford's law and data spread. Available at http://demonstrations.wolfram.com/BenfordsLawAndDataSpread.

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On the significands of uniform random variables

  • Arno Berger (a1) and Isaac Twelves (a1)

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