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On the significands of uniform random variables

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

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.
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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 

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References

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