Related papers: Few more Comments on Benford's Law
Benford's law is a famous law in statistics which states that the leading digits of random variables in diverse data sets appear not uniformly from 1 to 9; the probability that d (d=1,...,9) appears as a leading digit is given by…
Benford's law is an empirical edict stating that the lower digits appear more often than higher ones as the first few significant digits in statistics of natural phenomena and mathematical tables. A marked proportion of such analyses is…
Benford's Law predicts that the first significant digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently…
Feller's classic text 'An Introduction to Probability Theory and its Applications' contains a derivation of the well known significant-digit law called Benford's law. More specifically, Feller gives a sufficient condition ("large spread")…
The law of likelihood underlies a general framework, known as the likelihood paradigm, for representing and interpreting statistical evidence. As stated, the law applies only to simple hypotheses, and there have been reservations about…
Prime numbers seem to distribute among the natural numbers with no other law than that of chance, however its global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated sci-…
Benford's law is an empirical ``law'' governing the frequency of leading digits in numerical data sets. Surprisingly, for mathematical sequences the predictions derived from it can be uncannily accurate. For example, among the first billion…
Quantitative linguistics has been allowed, in the last few decades, within the admittedly blurry boundaries of the field of complex systems. A growing host of applied mathematicians and statistical physicists devote their efforts to…
We show the leading digits of a variety of systems satisfying certain conditions follow Benford's Law. For each system proving this involves two main ingredients. One is a structure theorem of the limiting distribution, specific to the…
Some formulas and speculations are presented relative to integrable systems and quantum mechanics.
There are now many theoretical explanations for why Benford's law of digit bias surfaces in so many diverse fields and data sets. After briefly reviewing some of these, we discuss in depth recurrence relations. As these are discrete…
Nature and our world have a bias! Roughly $30\%$ of the time the number $1$ occurs as the leading digit in many datasets base $10$. This phenomenon is known as Benford's law and it arrises in diverse fields such as the stock market,…
For arbitrary Borel probability measures with compact support on the real line, characterizations are established of the best finitely supported approximations, relative to three familiar probability metrics (Levy, Kantorovich, and…
Zipf's law is just one out of many universal laws proposed to describe statistical regularities in language. Here we review and critically discuss how these laws can be statistically interpreted, fitted, and tested (falsified). The modern…
The intriguing law of anomalous numbers, also named Benford's law, states that the significant digits of data follow a logarithmic distribution favoring the smallest values. In this work, we test the compliance with this law of the atomic…
Benford's law states that for many random variables X > 0 its leading digit D = D(X) satisfies approximately the equation P(D = d) = log_{10}(1 + 1/d) for d = 1,2,...,9. This phenomenon follows from another, maybe more intuitive fact,…
Further formulas are presented involving quantum mechanics, thermodynamics, and integrable systems. Modifications of dispersionless theory are developed.
In this paper the Zipf-Mandelbrot law is revisited in the context of linguistics. Despite its widespread popularity the Zipf--Mandelbrot law can only describe the statistical behaviour of a rather restricted fraction of the total number of…
The following work is written in easy language for college level students. It shows how the first digit probabilities of a group of continuous real-valued functions can be calculated. Thus, examples explaining how the probabilities are…
There are two fair ways to distribute particles in boxes. The first way is to divide the particles equally between the boxes. The second way, which is calculated here, is to score fairly the particles between the boxes. The obtained power…