# Cauchy Distribution

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## Cauchy distribution

[kō·shē dis·trə′byü·shən]
(statistics)
A distribution function having the form M /[π M 2+ (x-a)2], where x is the variable and M and a are constants. Also known as Cauchy frequency distribution.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Cauchy Distribution

a special type of probability distribution of random variables. Introduced by Cauchy, it is marked by the density The characteristic function is

f(t) = exp (μit − λ ǀ t ǀ)

The Cauchy distribution is unimodal and symmetric with respect to the point x = μ, which is its mode and median. No Figure 1. Cauchy distribution: (a) probability density, (b) distribution function

moments of positive order of a Cauchy distribution exist. Figure 1 depicts a Cauchy distribution for μ = 1.5 and λ = 1.

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When [alpha] =2 and [alpha] =1, the PDF in (3) has a closed-form expression, and the S[alpha]S distribution reduces to a Gaussian distribution and Cauchy distribution, respectively.
The distributions of the error term e and the predictors are contaminated by two types of distributions, t distribution with 5 degrees of freedom ([t.sub.(5)]) and Cauchy distribution with mean equal to 0 and variance equal to 1 (Cauchy (0, 1)).
To address these issues, this paper proposes a novel version of TLBO that is augmented with error correction and Cauchy distribution (ECTLBO) in which the Cauchy distribution is utilized to expand the searching space and error correction to avoid detours to achieve more accurate solutions.
The standard Cauchy distribution (Student's t-distribution with one degree of freedom) has neither a moment-generating function nor finite moments of order greater than or equal to one [Johnson et al., 1994].
Very briefly, the Cauchy distribution has unknown mean and variance, but defined median and mode.
Otherwise the notations [[[lambda].sub.E], RE} are often used in papers for second case due to relation between Cauchy distribution for f(x) and exponential parametrization for Bose-Einstein CF [C.sub.2](q) discussed above.
The newly proposed algorithm uses the concept of Cauchy distribution to follow large steps in global pollination, enhanced local search and dynamic switch probability to control the rate of local and global pollination.
where [u.sub.Cauchy] is the random number between 0 and 1 and the generated value [u.sub.Cauchy] is subjected to the Cauchy distribution.
Distribution Form of distribution Beta distribution f(x; [alpha], [beta]) = [1/B([alpha], [beta])][x.sup.[alpha]-1][(1 - x) .sup.[beta]-1] Cauchy distribution f(x; [x.sub.0], [gamma]) = [1/n] {[gamma]/[[(x - [x.sub.0]).sup.2] + [gamma]]} Chi-square distribution [mathematical expression not reproducible] Continuous uniform f(x) = [1/[b - a]] a [less than or distribution equal to] x [less than or equal to] b Gamma distribution f(x; k, [theta]) = [1/[[theta].sup.k]] [1/[GAMMA](k)][x.sup.k-1][e.sup.-x/ [theta]] Inverse-chi-squared f(x; k) = [[2.sup.-k/2]/[GAMMA] distribution (k/2)][x.sup.-k/2-1][e.sup.-1/2x] Inverse Gamma f(x; [alpha], [beta]) = [[[beta].sup.
For example Cauchy distribution sample TLM were unbiased to the corresponding population quantities and more robust to outliers as reported by (Elamir and Seheult, 2003).
Thus, for the standard Cauchy distribution [[sigma].sup.2] [right arrow] [infinity] but MAD = 1.

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