The Cauchy distribution is a probability distribution characterized by the absence of a defined mean or variance.

Also known as the Lorentz distribution, the Cauchy distribution is a continuous probability distribution named after the French mathematician Augustin-Louis Cauchy. The probability density function (PDF) of the Cauchy distribution is expressed as:

In this expression, $x_0$ represents the location parameter, and $\gamma$ denotes the scale parameter. The Cauchy distribution exhibits a characteristic “bell” shape; however, it possesses significantly heavier tails compared to the normal distribution. Consequently, extreme values are more likely to occur in the Cauchy distribution than in the normal distribution.

A notable property of the Cauchy distribution is its invariance under translation. This means that if the distribution is shifted by a constant value, its overall shape remains unchanged, although the location parameter $x_0$ will adjust accordingly. Additionally, the Cauchy distribution is classified as a stable distribution, implying that the sum of independent Cauchy-distributed random variables will also yield a Cauchy-distributed result.

The Cauchy distribution finds applications in various fields, including physics, particularly in the analysis of resonance phenomena. It is also utilized in statistics and finance, where it can serve as a model for extreme events or outliers. However, caution is advised when using the Cauchy distribution, as the lack of a defined mean or variance can complicate interpretation in certain contexts.