Friday, January 28, 2005

Looking out for number one

Most of us are familiar with "Zipfian distributions" or "heavy-tailed" distributions, in which the probability of occurrence of the ith most frequent element is roughly proportional to 1/ia, where a is a constant close to 1.

A related "law" is Benford's law, which states that
On a wide variety of statistical data, the first digit is d with the probability log10 ( 1 + 1/d)
An interesting discussion of this law reveals some underlying structure, specifically that if there is any law governing digit distributions, then it is scale invariant iff it is given by Benford's law.

Some other related laws that are more amusing, if less mathematically interesting are:
Lotka's Law:
The number of authors making n contributions is about 1/na of those making one contribution, where a is often nearly 2
Bradford's Law:
Journals in a field can be divided into three parts, each with about one-third of all articles: 1) a core of a few journals, 2) a second zone, with more journals, and 3) a third zone, with the bulk of journals. The number of journals is 1:n:n2


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