Base Dependence of Benford Random Variables

Abstract

A random variable X that is base b Benford will not in general be base c Benford when c≠b. This paper builds on two of my earlier papers and is an attempt to cast some light on the issue of base dependence. Following some introductory material, the “Benford spectrum” of a positive random variable is introduced and known analytic results about Benford spectra are summarized. Some standard machinery for a “Benford analysis” is introduced and combined with my method of “seed functions” to yield tools to analyze the base c Benford properties of a base b Benford random variable. Examples are generated by applying these general methods to several families of Benford random variables. Berger and Hill’s concept of “base-invariant significant digits” is discussed. Some potential extensions are sketched.

Keywords

• significand function
• Benford random variable
• base b first digit law
• Benford spectrum
• Benford analysis
• seed functions