Logarithmic(θ)

The logarithmic distribution (sometimes known as the Logarithmic Series distribution) is a discrete, positive distribution, peaking at x = 1, with one parameter and a long right tail. The figures below show two examples of the Logarithmic distribution.

#### Uses

The logarithmic distribution is not very commonly used in risk analysis. However, it has been used to describe, for example: the number of items purchased by a consumer in a particular period; the number of bird and plant species in an area; and the number of parasites per host. There is some theory that relates the latter two to an observation by Newcomb (1881) that the frequency of use of different digits in natural numbers followed a Logarithmic distribution:

Adapted from: Survival Distributions Satisfying Benford's Law. Lawrence M. LEEMIS, Bruce W. SCHMEISER, and Diane L. EVANS

Astronomer and mathematician Simon Newcomb noticed how much faster the first pages of tables of logarithms wear out than the last ones leading to the counter-intuitive conclusion that the first significant digit in the values in a logarithm table is not uniformly distributed between 1 and 9. Using a heuristic argument, he found that ones occur most often (more than 30% of the time) and nines least often (less than 5% of the time). More specifically, if the random variable X denotes the first significant digit, then Pr(X = x) = log10 (1 + 1=x) ; for x = 1,2..,9. He published this ’logarithm law’ in the American Journal of Mathematics in 1881.

General Electric physicist Frank Benford (1938) arrived at the same conclusion as Newcomb concerning logarithm tables. He collected data from fields to see if natural and sociological datasets would also obey the logarithm law. He often found good agreement, including datasets as diverse as the areas of rivers, American League baseball statistics, atomic weights of elements, death rates, and numbers appearing in Reader's Digest.

The observation has become known as Benford's law.

The links to the Logarithmic software specific models are provided here: