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The Cauchy distribution, named after Augustin-Louis 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 Voigt distribution, or Voigt profile, is the convolution of a normal distribution and a Cauchy distribution. It is found in spectroscopy when spectral line profiles are broadened by a mixture of Lorentzian and Doppler broadening mechanisms. The Chen distribution.
Generally, a Cauchy distribution is any probability distribution belonging to the same location-scale family as this one. Thus, if X has a standard Cauchy distribution and μ is any real number and σ > 0, then Y = μ + σX has a Cauchy distribution whose median is μ and whose first and third quartiles are respectively μ − σ and μ + σ.
The reciprocal 1/X of a random variable X, is a member of the same family of distribution as X, in the following cases: Cauchy distribution, F distribution, log logistic distribution. Examples: If X is a Cauchy (μ, σ) random variable, then 1/X is a Cauchy (μ/C, σ/C) random variable where C = μ 2 + σ 2.
The log-Cauchy distribution is infinitely divisible for some parameters but not for others. [10] Like the lognormal distribution, log-t or log-Student distribution and Weibull distribution, the log-Cauchy distribution is a special case of the generalized beta distribution of the second kind.
The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution. The wrapped Cauchy distribution is often found in the field of spectroscopy where it is used to analyze diffraction patterns (e.g. see Fabry–Pérot interferometer).
This is the characteristic function of the standard Cauchy distribution: thus, the sample mean has the same distribution as the population itself. As a further example, suppose X follows a Gaussian distribution i.e. X ∼ N ( μ , σ 2 ) {\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2})} .
In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density ...