### Friar, JL, Goldman, T and Pérez-Mercader, J (2016)

#### Ubiquity of Benford’s law and emergence of the reciprocal distribution

Physics Letters A 380(22), pp. 1895–1899.

**ISSN/ISBN:** 0375-9601
**DOI:** 10.1016/j.physleta.2016.03.045

**Abstract:** We apply the Law of Total Probability to the construction of scale-invariant probability distribution functions (pdf's), and require that probability measures be dimensionless and unitless under a continuous change of scales. If the scale-change distribution function is scale invariant then the constructed distribution will also be scale invariant. Repeated application of this construction on an arbitrary set of (normalizable) pdf's results again in scale-invariant distributions. The invariant function of this procedure is given uniquely by the reciprocal distribution, suggesting a kind of universality. We separately demonstrate that the reciprocal distribution results uniquely from requiring maximum entropy for size-class distributions with uniform bin sizes.

**Bibtex:**

```
@article{,
title = "Ubiquity of Benford's law and emergence of the reciprocal distribution",
journal = "Physics Letters A",
volume = "380",
number = "22",
pages = "1895--1899",
year = "2016",
issn = "0375-9601",
doi = "http://dx.doi.org/10.1016/j.physleta.2016.03.045",
url = "http://www.sciencedirect.com/science/article/pii/S0375960116300603",
author = "James L. Friar and Terrance Goldman and Juan P{\'e}rez-Mercader",
}
```

**Reference Type:** Journal Article

**Subject Area(s):** Physics, Probability Theory