The reciprocal distribution, also known as the log-uniform distribution, is a continuous probability distribution characterized by its proportional probability density function (PDF), which is reciprocal to the variable within its two bounding parameters (lower and upper limits of its support).
Although it is commonly used to describe the log-uniform distribution, the term reciprocal distribution is not universally defined. The literature includes various definitions, such as any distribution of a reciprocal of a random variable ), a reciprocal continuous random variable , or as a synonym for the logarithmic distribution . However, most probability density functions (PDFs) for a “reciprocal distribution” involve a logarithm, in one form or another, such as for the under-workings of Benford’s law, pink noise, and distributions of mantissas. The mantissa comprises the part of a logarithm following the decimal point or the part of a floating point number following the decimal point. For instance, in .12345678 * 102, the mantissa is 12345678. The log-uniform / reciprocal distribution is widely used in numerical analysis because a computer’s arithmetic operations transform mantissas with initial arbitrary distributions into the reciprocal distribution as a limiting distribution.
Definition of the reciprocal distribution (log-uniform distribution)
The log-uniform distribution PDF is
- a = parameter for the lower bound,
- b = parameter for the upper bound,
- loge = the natural log function (logarithm to base e).
Relationship to other distributions
The log-uniform distribution is a type of inverse distribution; an inverse distribution is the distribution of the reciprocal of a random variable. The reciprocal (inverse) of a log-uniform random variable also has a reciprocal distribution. For example, if X is a random variable with a log-uniform distribution, then the inverse distribution of X is the distribution of 1/X.
The uniform distribution, a continuous probability distribution with a constant probability density function over a specified interval, is related to the log-uniform distribution. The probability density function of the uniform distribution is equal to 1/(b-a), where a and b are the lower and upper limits of the interval. The log-uniform distribution can be thought of as the distribution of the logarithm of a uniform random variable. In other words, if X is a uniform random variable with lower limit a and upper limit b, then Y = log(X) is a log-uniform random variable with lower limit log(a) and upper limit log(b). The relationship between the log-uniform distribution and the uniform distribution can also be used to simulate log-uniform random variables. For example, if we want to simulate a log-uniform random variable with lower limit a and upper limit b, we can first simulate a uniform random variable X with lower limit a and upper limit b, and then take the logarithm of X.
Other “Reciprocal distribution” definitions
- Pink noise: The reciprocal distribution, with a PDF of 1 / (y log(B/A) , is used for describing pink (1/f) noise and as an uninformed prior distribution for scale parameters in Bayesian inference. SciPy stats uses this PDF.
- Benford’s Law: The reciprocal distribution, as used in Benford’s Law, is a continuous probability distribution on the open interval (a, b). Its PDF is defined by r(x) ≡ c/x, where ‘x’ is a random variable, and ‘c’ is a normalization constant of 1/ ln b (when 1/b ≤ x < 1) .
- General use: The term “reciprocal distribution,” outside of probability and statistics, doesn’t refer to a probability distribution at all. Instead, it means “You scratch my back and I’ll scratch yours,” as in the case of “Reciprocal distribution of raw materials is only fair.”
 Marshall, A. & Olkin, L. (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families.
 SciPy Stats (2009). scipy.stats.reciprocal. Retrievd December 11, 2017 from: https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats.reciprocal.html
 Bose, P. & Morin, P. (2003). Algorithms and Computation: 13th International Symposium, ISAAC 2002 Vancouver, BC, Canada, November 21-23, 2002, Proceedings.
 McLaughlin, M. (1999). Regress+: A Compendium of Common Probability Distributions. Retrieved 5/18/23 from http://www.ub.edu/stat/docencia/Diplomatura/Compendium.pdf
 Friar et al., (2016). Ubiquity of Benford’s law and emergence of the reciprocal distribution. Physics Letters A, Volume 380, Issue 22-23, p. 1895-1899.