Cauchy Distribution

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What is the Cauchy distribution?

The Cauchy distribution (also called the Lorentz distribution, Cauchy–Lorentz distributionLorentz(ian) function, or Breit–Wigner distribution) is a family of continuous probably distributions that resemble normal distributions with taller peaks. However, their fat tails decay much more slowly than the normal distribution.

The Cauchy distribution, named after 18th century mathematician Augustin-Louis Cauchy, is well known for the fact that it’s expected value and other moments do not exist.  This is one reason why this ill-behaved distribution is “…best known as a pathological case.” [1] It is not stable under convolution and it doesn’t belong to any of the common parametric distribution families. In addition, the following are undefined or do not exist:

Despite its limitations, the Cauchy’s properties — such as robustness — make it useful in some areas of study.

Cauchy Distribution Functions

cauchy distribution pdf
Plots of the PDFs for several members of the Cauchy family of probability distributions. [2].

Probability Density Function

The general formula for the probability density function (PDF) is [3]:

cauchy distribution formula pdf

Where

When t = 0, s = 1, the equation reduces to the standard Cauchy distribution:

Support (range) for the PDF is on (-∞, ∞)

Cumulative Distribution Function (CDF):

The Cauchy percent point function is

The Cauchy hazard function is

The Cauchy cumulative hazard function is

The Cauchy survival function is defined as

Inverse survival function:

Quantile Function

Applications

The Cauchy distribution is often used as an example of a “pathological” distribution — a distribution that is ill-behaved. However, it does have a few practical applications. For example:

  • Robustness studies. The Cauchy is robust because it has heavy tails, which means is has a high probability of producing extreme values — thus making it a good choice for robustness studies.
  • Modeling a ratio of two normal random variables.
  • Modeling polar and non-polar liquids in porous glasses [4].
  • In quantum mechanics, it models the distribution of energy of an unstable state [5].

References

Cauchy Image: https://creativecommons.org/licenses/by-sa/3.0/

[1] Segura et. al (2004). A Guide to Laws and Theorems Named After Economists. Edward Elgar publishing.

[2] Skbkekas, CC BY 3.0 https://creativecommons.org/licenses/by/3.0, via Wikimedia Commons

[3] Engineering Statistics Handbook. Online: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm

[4] Stapf et. al (1996). Proton and deuteron field-cycling. Colloids and Surfaces: A Physicochemical and Engineering Aspects 115, 107-114.

[5] Grewel and Andrews (2015). Kalman Filtering. John Wiley & Sons.

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