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

*Any*distribution of a reciprocal of a random variable (Marshall & Olkin, 2007, p.32).- A reciprocal
*continuous*random variable (SciPy, 2009). - A synonym for the
*logarithmic distribution*(Bose & Morin, 2003. p.637).

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

Of all the different versions of the 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 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 (Friar et. al, 2016).

## 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

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

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.

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

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

McLaughlin, M. (1999). Regress+: A Compendium of Common Probability Distributions.

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

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