Cauchy Distribution
Wikipedia
Source: Wikipedia; Extract taken 01 May 2023
Paper - Abstract

Paper StatisticsBooks / Papers Citing this PaperNotes Citing this PaperText Colour-ConventionsDisclaimer


Introduction

  1. 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: x0, γ) is the distribution of the x-intercept of a ray issuing from (x0, γ) with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.
  2. 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.
  3. In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.
  4. 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.

Comment:

See Wikipedia: Cauchy Distribution.

Text Colour Conventions (see disclaimer)

  1. Blue: Text by me; © Theo Todman, 2024
  2. Mauve: Text by correspondent(s) or other author(s); © the author(s)



© Theo Todman, June 2007 - June 2024. Please address any comments on this page to theo@theotodman.com. File output:
Website Maintenance Dashboard
Return to Top of this Page Return to Theo Todman's Philosophy Page Return to Theo Todman's Home Page