There isn’t a universal definition for the “reciprocal distribution.” Definitions from the literature include:

*Any*distribution of a reciprocal of a random variable [1]).- A reciprocal
*continuous*random variable [2]. - A synonym for the
*logarithmic distribution*[3].

That said, *most *PDFs for a reciprocal distribution involve a logarithm in one form or another: for pink noise, distributions of mantissas (the part of a logarithm that follows the decimal point), or the under-workings of Benford’s law.

Of all the different versions of the probability distribution function (PDF) for the reciprocal distribution, the pink noise/Bayesian inference one is by far the most common.

## 1. Pink Noise / Bayesian Inference

The **reciprocal distribution** is used to describe pink (1/f) noise, or as an uninformed prior distribution for scale parameters in Bayesian inference.

SciPy stats also uses this PDF.

## 2. Distribution of Mantissas

The *mantissa* is the part of the logarithm following the decimal point, or the part of the floating point number (closely related to scientific notation) following the decimal point. For example, .12345678 * 102, .12345678 is the mantissa.

In his book, *Numerical Methods for Scientists and Engineers*, Richard Hamming uses a reciprocal distribution to describe the probability of finding the number *x *in the base *b *(The base in logarithmic calculation is the subscript to the right of “log”; the base in log_{3}(x) is “3”). The pdf for this probability is:

## 3. Reciprocal Distribution (Benford’s Law)

The **reciprocal distribution** is a continuous probability distribution defined on the open interval (*a*, *b*). The Probability Density Function (PDF) is *r(x) ≡ c/x*,

Where:

*x = a*random variable,*c*= the normalization constant*c = 1/ ln b*(when*x*ranges from*1/b*to*1*).

This PDF is the underpinnings of Benford’s Law [6].

## 4. Other Uses and Meanings

Outside of probability and statistics, the term “reciprocal distribution” doesn’t involve a probability distribution at all; it refers to *You scratch my back and I’ll scratch yours*. For example “Reciprocal distribution of raw materials is only fair—”.

## References

[1] Marshall, A. & Olkin, L. (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families.

[2] 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

[3] Bose, P. & Morin, P. (2003). Algorithms and Computation: 13th International Symposium, ISAAC 2002 Vancouver, BC, Canada, November 21-23, 2002, Proceedings.

[4] McLaughlin, M. (1999). Regress+: A Compendium of Common Probability Distributions. Retrieved 5/18/23 from http://www.ub.edu/stat/docencia/Diplomatura/Compendium.pdf

[5] Hamming, R. (2012). Numerical Methods for Scientists and Engineers. Courier Corporation.

[6] 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.