Cauchy Distribution


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The Cauchy distribution (also called the Lorentz distribution, Cauchy–Lorentz distributionLorentz(ian) function, or Breit–Wigner distribution) is a family of continuous probably distributions named after Augustin Cauchy. They resemble the normal distribution with a taller peak. Unlike the normal distribution, its fat tails decay much more slowly.

The distribution, which describes resonance behavior, 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].

The following are undefined or do not exist:

  • Coefficient of Variation
  • Mean
  • Standard deviation
  • Kurtosis
  • Moment Generating Function (MGF)
  • Variance

Cauchy Distribution Functions

The Cauchy family. Image: Skbkekas | Wikimedia Commons. CCA by 3.0

Probability Density Function

The general formula for the probability density function is [2]:

Where

  • t = location parameter
  • s = scale parameter

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

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

Cumulative Density 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 in statistics as an example of a pathological distribution, or a distribution that is ill-behaved. However, it does have a few practical applications. For example:

  • Robustness studies.
  • Modeling a ratio of two normal random variables.
  • Modeling polar and non-polar liquids in porous glasses [3].
  • In quantum mechanics, it models the distribution of energy of an unstable state [4].

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

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

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