Cauchy distribution

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 ) {displaystyle f(x;x_{0},gamma )} is the distribution of the x-intercept of a ray issuing from ( x 0 , γ ) {displaystyle (x_{0},gamma )} with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables if the denominator distribution has mean zero. 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 ) {displaystyle f(x;x_{0},gamma )} is the distribution of the x-intercept of a ray issuing from ( x 0 , γ ) {displaystyle (x_{0},gamma )} with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables if the denominator distribution has 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 the section 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. In spectroscopy, it is the description of the shape of spectral lines which are subject to homogeneous broadening in which all atoms interact in the same way with the frequency range contained in the line shape. Many mechanisms cause homogeneous broadening, most notably collision broadening. 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. Functions with the form of the density function of the Cauchy distribution were studied by mathematicians in the 17th century, but in a different context and under the title of the witch of Agnesi. Despite its name, the first explicit analysis of the properties of the Cauchy distribution was published by the French mathematician Poisson in 1824, with Cauchy only becoming associated with it during an academic controversy in 1853. As such, the name of the distribution is a case of Stigler's Law of Eponymy. Poisson noted that if the mean of observations following such a distribution were taken, the mean error did not converge to any finite number. As such, Laplace's use of the Central Limit Theorem with such a distribution was inappropriate, as it assumed a finite mean and variance. Despite this, Poisson did not regard the issue as important, in contrast to Bienaymé, who was to engage Cauchy in a long dispute over the matter. The Cauchy distribution has the probability density function (PDF) where x 0 {displaystyle x_{0}} is the location parameter, specifying the location of the peak of the distribution, and γ {displaystyle gamma } is the scale parameter which specifies the half-width at half-maximum (HWHM), alternatively 2 γ {displaystyle 2gamma } is full width at half maximum (FWHM). γ {displaystyle gamma } is also equal to half the interquartile range and is sometimes called the probable error. Augustin-Louis Cauchy exploited such a density function in 1827 with an infinitesimal scale parameter, defining what would now be called a Dirac delta function. The maximum value or amplitude of the Cauchy PDF is 1 π γ {displaystyle {frac {1}{pi gamma }}} , located at x = x 0 {displaystyle x=x_{0}} . It is sometimes convenient to express the PDF in terms of the complex parameter ψ = x 0 + i γ {displaystyle psi =x_{0}+igamma }