Cauchy Distribution |
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Wikipedia |

Source: Wikipedia; Extract taken 01 May 2023 |

Paper - Abstract |

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__Introduction__

- The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution f(x: x
_{0}, γ) is the distribution of the x-intercept of a ray issuing from (x_{0}, γ) with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero. - The Cauchy distribution is often used in statistics as the canonical example of a "pathological" distribution since both its expected value and its variance are undefined (but see § Explanation of undefined moments below). The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist. The Cauchy distribution has no moment generating function.
- In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.
- It is one of the few distributions that is stable and has a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.

See Wikipedia: Cauchy Distribution.

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