Research and Applications Towards Mathematics and Computer Science Vol. 2, pp. 76-83.
ISSN/ISBN: Not available at this time. DOI: 10.9734/bpi/ratmcs/v2/5906C
Abstract: It is argued that the three empirical power law distributions: Benford’s law, Zip’s law, and Pareto’s rule together with the central limit theorem are all connected and are the result of the quantization of energy. This argumentation may be considered a physical origin of probability. Benford’s law is the rank’s distribution of the frequencies of digits in numeral random files. Zipf’s law is the rank distribution of the population of cities, the frequency of words in a long text, the bestseller lists, etc. The Pareto rule of thumb, known as the 20-80 rule, is the wealth distribution in developed countries and predicts that about 20% of the population owns 80% of the wealth. Similarly, 20% of the workers are responsible for 80% of the productivity, etc. Hereafter, we derive these laws in two ways: the first, by using conventional probabilistic tools, and the second by calculating the maximum entropy of an ensemble of identical balls randomly distributed in identical boxes. Zipf’s law is obtained for the rank distribution of indistinguishable balls in distinguishable boxes. Benford’s distribution is obtained for the rank distribution of distinguishable balls in distinguishable boxes. I.e. in the distribution of words in long texts, all the words in a given rank are indistinguishable; therefore, the rank distribution is Zipfian. In logarithmic tables, the number of balls with identical 1st digits is distinguishable as there are many different digits in the 2nd, 3rd, etc. places in the mantissa, and therefore the distribution is according to Benford’s law. Pareto 20-80 rule is a specific outcome of Benford’s distribution when the number of ranks is about 10. In this case, the probability of 20% of the high probability ranks is equal to the probability of the rest of 80% of the low probability ranks.
Bibtex:
@article{,
title={Power Law Distributions: A Unified Derivation},
url={https://stm.bookpi.org/RATMCS-V2/article/view/11109}, DOI={10.9734/bpi/ratmcs/v2/5906C},
journal={Research and Applications Towards Mathematics and Computer Science Vol. 2},
author={Kafri, Oded},
year={2023},
month={Jul.},
pages={76–83} }
Reference Type: Book Chapter
Subject Area(s): Probability Theory, Statistics