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The** Cauchy distribution** (also called the *Lorentz distribution*, *Cauchy–Lorentz distribution*, *Lorentz(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**

**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.