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.
Highlights
Examples are generated by applying these general methods to several families of Benford random variables
As has been pointed out before, the theory of anomalous numbers is really the theory of phenomena and events, and the numbers but play the poor part of lifeless symbols for living things.”. This argument seems compelling, and it might seem to apply to Benford random variables as well as to geometric sequences and exponential functions
The best way to define Benford random variables is via the significand function
Summary
The physicist Frank Benford for whom Benford’s Law is named, considered his “law of anomalous numbers” as evidence of a “real world” phenomenon He realized that geometric sequences and exponential functions are generally base 10. As has been pointed out before, the theory of anomalous numbers is really the theory of phenomena and events, and the numbers but play the poor part of lifeless symbols for living things.” This argument seems compelling, and it might seem to apply to Benford random variables as well as to geometric sequences and exponential functions. The base b first digit law is introduced, and several examples of random variables are presented that are Benford relative to one base but not to another. Stats 2021, 4 discusses Berger and Hill’s concept of “base-invariant significant digits.” Section 9 is a summary and a look ahead
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
