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Topic: Cauchy distribution

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Lorentzian function

Normal distribution

Probability density function

Exponential family

Pareto distribution

Doppler profile

Cauchy product

Cauchy integral theorem

Cauchy sequence

Augustin Louis Cauchy

Pi

Roof

Gary Gordon

Paul Klee

1760

	1.3.6.6.3. Cauchy Distribution
		The likelihood functions for the Cauchy maximum likelihood estimates are given in chapter 16 of Johnson, Kotz, and Balakrishnan.
		The Cauchy distribution is important as an example of a pathological case.
		The mean and standard deviation of the Cauchy distribution are undefined.
	www.itl.nist.gov /div898/handbook/eda/section3/eda3663.htm (401 words)

	Cauchy Distribution
		The Cauchy distribution is specified with two parameters: a and b.
		The Cauchy distribution is a stable Paretian distribution, so a sum of Cauchy random variables is itself Cauchy.
		The standard Cauchy distribution is a special case of the student t distribution with one degree of freedom.
	www.riskglossary.com /articles/cauchy_distribution.htm (341 words)

	PlanetMath: t distribution
		Gossett conducted agricultural experiments and used random numbers to help determine the sampling distribution of the data he collected.
		, the Cauchy distribution with parameters 0 and
		This is version 31 of t distribution, born on 2004-06-24, modified 2006-09-20.
	planetmath.org /encyclopedia/TDistribution.html (244 words)

	NationMaster - Encyclopedia: Cauchy distribution (Site not responding. Last check: )
		As a probability distribution, it is known as the Cauchy distribution while among physicists it is known as the Lorentz distribution or the Breit-Wigner distribution.
		Since it is a distribution function, it integrates to unity: In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total.
		In probability and statistics, the t-distribution or Students distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small.
	www.nationmaster.com /encyclopedia/Cauchy-distribution (2055 words)

	PlanetMath: Cauchy random variable
		Cauchy random variables are used primarily for theoretical purposes, the key point being that the values
		This is version 5 of Cauchy random variable, born on 2001-10-26, modified 2003-10-14.
		I encountered a Cauchy distribution when solving a problem in "Communications Engineering" (3rd Edition) by Proakis and Salehi.
	planetmath.org /encyclopedia/CauchyRandomVariable.html (332 words)

	Cauchy distribution
		The Cauchy distribution is often cited as an example of a distribution which has no mean, variance or higher moments defined, although its mode and median are well defined and both zero.
		The Cauchy distribution is the Student's t-distribution with just one degree of freedom.
		The Cauchy distribution is sometimes called the Lorentz distribution, because it is equivalent to a Lorentzian function whose mean () is zero and whose full width at half maximum (FWHM) is 2.
	www.xasa.com /wiki/en/wikipedia/c/ca/cauchy_distribution.html (485 words)

	Central Limit Theorem (fine print) (Site not responding. Last check: )
		The distribution of an average tends to be Normal, even when the distribution from which the average is computed is decidedly non-Normal.
		The distribution of an average will tend to be Normal as the sample size increases, regardless of the distribution from which the average is taken except when the moments of the parent distribution do not exist.
		The Cauchy has another interesting property - the distribution of the sample average is that same as the distribution of an individual observation, so the scatter never diminishes, regardless of sample size.
	www.statisticalengineering.com /central_limit_theorem_fineprint.htm (574 words)

	1.3.3.14.2. Histogram Interpretation: Symmetric, Non-Normal, Short-Tailed
		For a symmetric distribution, the "body" of a distribution refers to the "center" of the distribution--commonly that region of the distribution where most of the probability resides--the "fat" part of the distribution.
		The classical short-tailed distribution is the uniform (rectangular) distribution in which the probability is constant over a given range and then drops to zero everywhere else--we would speak of this as having no tails, or extremely short tails.
		The common choice of taking N observations and using the calculated sample mean as the best estimate for the center of the distribution is a good choice for the normal distribution (moderate tailed), a poor choice for the uniform distribution (short tailed), and a horrible choice for the Cauchy distribution (long tailed).
	www.6sigma.us /handbook/eda/section3/eda33e2.htm (351 words)

	Augustin Louis Cauchy
		In addition to his heavy workload Cauchy undertook mathematical researches, and he proved in 1811 that the angles of a convex polyhedron are determined by its faces.
		Cauchy was the first to make a rigorous study of the conditions for convergence of infinite series in addition to his rigorous definition of an integral.
		Numerous terms in mathematics bear Cauchy's name: the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy-Riemann equations, the Cauchy distribution in probability, and Cauchy sequences.
	www.stetson.edu /~efriedma/periodictable/html/Cu.html (626 words)

	Springer Online Reference Works
		The definition of an infinitely-divisible distribution is applicable to an equal degree to a distribution on the straight line, on a finite-dimensional Euclidean space and to a number of other, even more general, cases.
		Examples of infinitely-divisible distributions include the normal distribution, the Poisson distribution, the Cauchy distribution, and the "chi-squared" distribution.
		The importance of the role played in the limit theorems of probability theory by infinitely-divisible distributions is due to the fact that these and only these distributions can be the limit distributions for sums of independent random variables subject to the requirement of asymptotic negligibility.
	eom.springer.de /i/i050910.htm (698 words)

	Distribution Fitting
		For predictive purposes it is often desirable to understand the shape of the underlying distribution of the population.
		To determine this underlying distribution, it is common to fit the observed distribution to a theoretical distribution by comparing the frequencies observed in the data to the expected frequencies of the theoretical distribution (i.e., a Chi-square goodness of fit test).
		The major distributions that have been proposed for modeling survival or failure times are the exponential (and linear exponential) distribution, the Weibull distribution of extreme events, and the Gompertz distribution.
	www.statsoft.com /textbook/stdisfit.html (1769 words)

	Ratio Populations
		The median of the distribution of ratios of elements of A divided by elements of B is equal to 90/110 = 0.818, but the pseudo-mean of the distribution is 0.848.
		A plot of the C density distribution for this case is shown in Figure 1.
		Recall that the density distribution of the ratio population q = y/x is
	www.mathpages.com /home/kmath042/kmath042.htm (1808 words)

	[No title]
		The posterior moments of parameters specifying distributions are minimum mean square error Bayesian estimators for the corresponding moments of those parameters, and as such are ubiquitous in the Bayesian approach to statistical inference of distributions.
		The cauchy distribution is most notable for its wide tails, decided absence of high-order moments, and non-existence of less-than-data dimension sufficient statistics.
		In this paper the posterior moments of the position parameter of the Cauchy distribution are found in closed form.
	www.realtime.net /~drwolf/pages/momcauchy93.html (143 words)

	ContinuousDistributions
		is the limiting distribution for the smallest or largest values in large samples drawn from a variety of distributions, including the normal distribution.
		The extreme value distribution is sometimes referred to as the log-Weibull distribution because it describes the distribution of the log of a Weibull distributed random variable.
		Each distribution has a unique characteristic function, which is sometimes used instead of the pdf to define a distribution.
	documents.wolfram.com /v5/Add-onsLinks/StandardPackages/Statistics/ContinuousDistributions.html (588 words)

	R: Posterior mean estimator using quasi-Cauchy prior
		The quasi-Cauchy distribution is used for the nonzero component of the prior
		A value or vector of values of the estimate(s) of the mean(s) of the distribution(s) from which the x are drawn.
		The mean conditional on the mixing parameter is found and is then averaged over the posterior distribution of the mixing parameter, including the atom of probability at zero variance.
	www.stats.ox.ac.uk /~silverma/ebayesthresh/postmean.cauchy.html (94 words)

	Estimators for the Cauchy Distribution - Hanson, Wolf (ResearchIndex)
		Abstract: We discuss the properties of various estimators of the central position of the Cauchy distribution.
		On the contrary, because of the infinite variance of the Cauchy distribution, the average of the measured positions is an extremely poor estimator of the...
		0.1: Posterior Momemts of the Cauchy Distribution - Wolf (1998)
	citeseer.ist.psu.edu /hanson96estimators.html (392 words)

	Newran - random number generator library
		It is particularly appropriate for the situation where one requires sequences of identically distributed random numbers since the set up time for each type of distribution is relatively long but it is efficient when generating each new random number.
		The library includes classes for generating random numbers from a number of distributions and is easily extended to be able to generate random numbers from almost any of the standard distributions.
		However, if you distribute the source, please make it clear which parts are mine and that they are available essentially for free over the Internet.
	www.robertnz.net /nr02doc.htm (2816 words)

	Module scipy.stats.distributions (Site not responding. Last check: )
		cauchy = cauchy_gen(name = 'cauchy', longname = 'Cauchy', extradoc = """ Cauchy distribution cauchy.pdf(x) = 1/(pi(1+x2)) This is the t distribution* with one degree of freedom.
		This is the same as the Levy-stable distribution with a=1/2 and b=1.
		The standard form of this distribution is a standard normal truncated to the range [a,b] --- notice that a and b are defined over the domain of the standard normal.
	www.scipy.org /doc/api_docs/scipy.stats.distributions.html (3305 words)

	Mathematica Documentation: ContinuousDistributions
		The logistic distribution LogisticDistribution[mu, beta] is frequently used in place of the normal distribution when a distribution with longer tails is desired.
		Distributions that are derived from normal distributions with nonzero means are called noncentral distributions.
		This is a pseudorandom array with elements distributed according to the gamma distribution.
	documents.wolfram.com /mathematica/Add-onsLinks/StandardPackages/Statistics/ContinuousDistributions.html (788 words)

	List of Tutorials
		The distributions of two independent binomial variables under the condition of constant sum are hypergeometric.
		Distribution and confidence intervals of the slope and intercept.
		Distribution of the estimator of the variance of the errors.
	www.aiaccess.net /x_tutor_list.htm (763 words)

	Random (JMSL Numerical Library)
		The non-uniform distributions are generated from a uniform distribution.
		If r is an integer, the distribution is often called the Pascal distribution and can be thought of as modeling the length of a sequence of Bernoulli trials until r successes are obtained, where p is the probability of getting a success on any trial.
		The algorithm is an acceptance/rejection method using a wrapped Cauchy distribution as the majorizing distribution.
	www.vni.com /products/imsl/jmsl/v30/api/com/imsl/stat/Random.html (2994 words)

	Functions and CALL Routines : CDF (Site not responding. Last check: )
		The CDF function for the beta distribution returns the probability that an observation from a beta distribution, with shape parameters a and b, is less than or equal to x.
		The CDF function for the chi-squared distribution returns the probability that an observation from a chi-squared distribution, with df degrees of freedom and noncentrality parameter nc, is less than or equal to x.
		The CDF function for the F distribution returns the probability that an observation from an F distribution, with ndf numerator degrees of freedom, ddf denominator degrees of freedom, and noncentrality parameter nc, is less than or equal to x.
	www.asu.edu /sas/sasdoc/sashtml/lgref/z0208980.htm (1101 words)

	Cauchy (a.k.a. Lorentz) Distribution
		The Cauchy distribution is a symmetrical, and to use a technical term, heavy tailed.
		An application of the Cauchy distribution is in software testing where it is necessary to use datasets which contain a few extreme values which might trigger some adverse reaction.
		The range of the Cauchy distribution is from -∞ to +∞.
	www.brighton-webs.co.uk /distributions/cauchy.asp (254 words)

	Project Links \| Concepts \| Cauchy RV
		Due to the lack of a variance for the Cauchy it is said that the Cauchy distribution is "Heavy Tailed".
		The Cauchy RV is a transform of a Uniform RV: If Y is Uniform, and represents the angle of a spinner, X would be the location on a line below the spinner that the ray from the spinner would reach.
		The Cauchy RV is also a transform of two Gaussian RVs: If X is Gaussian N(0,1) and Y is Gaussian N(0,1), then is a Cauchy RV.
	www.ibiblio.org /links/devmodules/probstat/concepts/html/cauchyRV.html (168 words)

	Functions and CALL Routines : PDF (Site not responding. Last check: )
		The PDF function for the beta distribution returns the probability density function of a beta distribution, with shape parameters a and b, which is evaluated at the value x.
		The PDF function for the chi-squared distribution returns the probability density function of a chi-squared distribution, with df degrees of freedom and noncentrality parameter nc, which is evaluated at the value x.
		The PDF function for the F distribution returns the probability density function of an F distribution, with ndf numerator degrees of freedom, ddf denominator degrees of freedom, and noncentrality parameter nc, which is evaluated at the value x.
	www.asu.edu /it/fyi/unix/helpdocs/statistics/sas/sasdoc/sashtml/lgref/z0270634.htm (1051 words)

	Stat 5601 (Geyer, Fall 2003) Parametric Bootstrap
		The theory of the parametric bootstrap is quite similar to that of the nonparametric bootstrap, the only difference is that instead of simulating bootstrap samples that are IID from the empirical distribution (the nonparametric estimate of the distribution of the data) we simulate bootstrap samples that are IID from the estimated parametric model.
		The multinomial distribution is the distribution of categorical measurements on IID individuals.
		when the null distribution of the test statistic depends on the true unknown parameter value and we are plugging in an estimate for the unknown truth).
	www.stat.umn.edu /geyer/old03/5601/examp/parm.html (1063 words)

	Cauchy distribution
		The special case when t = 0 and s = 1 is called the standard Cauchy distribution with the probability density function:
		When U and V are two independent normal random variables with standard normal distributions, then the ratio U/V has the standard Cauchy distribution.
		The Cauchy distribution is sometimes called the Lorentz distribution
	www.ebroadcast.com.au /lookup/encyclopedia/ca/Cauchy_distribution.html (117 words)

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