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Cauchy distribution

Index Cauchy distribution

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. [1]

97 relations: Addison-Wesley, Annals of Statistics, Arithmetic mean, Augustin-Louis Cauchy, Binomial distribution, Biometrika, Cauchy principal value, Cauchy process, Central limit theorem, Central moment, Characteristic function (probability theory), Confidence interval, Cumulative distribution function, Cumulative frequency analysis, Differentiable function, Differential entropy, Dirac delta function, Encyclopedia of Statistical Sciences, Expected value, Fat-tailed distribution, Fourier transform, Full width at half maximum, Fundamental solution, Hölder's inequality, Hendrik Lorentz, Homogeneous broadening, Homogeneous function, Hydrology, Hyperbolic distribution, Independence (probability theory), Independent and identically distributed random variables, Indeterminate form, Infinite divisibility (probability), Infinitesimal, Interquartile range, Inverse trigonometric functions, Irénée-Jules Bienaymé, John Wiley & Sons, Laplace's equation, Law of large numbers, Lévy distribution, Lévy flight, Lévy process, Linear map, Location parameter, Location–scale family, Mathematics, Maximum entropy probability distribution, Maximum likelihood estimation, Möbius transformation, ..., McCullagh's parametrization of the Cauchy distributions, Mean, Median, Mode (statistics), Moment (mathematics), Moment-generating function, Multivariate random variable, Multivariate t-distribution, Newton's method, Normal distribution, Nuclear physics, Numerical analysis, Order statistic, Particle physics, Pathological (mathematics), Pearson distribution, Peter McCullagh, Physicist, Pierre-Simon Laplace, Poisson kernel, Probability density function, Probability distribution, Probable error, Q–Q plot, Quantile function, Random variable, Ratio distribution, Real number, Relativistic Breit–Wigner distribution, Resonance, Scale parameter, Siméon Denis Poisson, Slash distribution, Spectral line, Spectroscopy, Stability (probability), Stable distribution, Stable process, Standardized moment, Stigler's law of eponymy, Student's t-distribution, Truncated mean, University of Alabama, Upper half-plane, Variance, Witch of Agnesi, Wrapped Cauchy distribution. Expand index (47 more) »

Addison-Wesley

Addison-Wesley is a publisher of textbooks and computer literature.

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Annals of Statistics

The Annals of Statistics is a peer-reviewed statistics journal published by the Institute of Mathematical Statistics.

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Arithmetic mean

In mathematics and statistics, the arithmetic mean (stress on third syllable of "arithmetic"), or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection.

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Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.

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Binomial distribution

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: a random variable containing a single bit of information: success/yes/true/one (with probability p) or failure/no/false/zero (with probability q.

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Biometrika

Biometrika is a peer-reviewed scientific journal published by Oxford University Press for the Biometrika Trust.

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Cauchy principal value

In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.

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Cauchy process

In probability theory, a Cauchy process is a type of stochastic process.

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Central limit theorem

In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed.

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Central moment

In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable from the mean.

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Characteristic function (probability theory)

In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution.

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Confidence interval

In statistics, a confidence interval (CI) is a type of interval estimate, computed from the statistics of the observed data, that might contain the true value of an unknown population parameter.

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Cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF, also cumulative density function) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. In the case of a continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

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Cumulative frequency analysis

Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value.

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Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

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Differential entropy

Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Shannon to extend the idea of (Shannon) entropy, a measure of average surprisal of a random variable, to continuous probability distributions.

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Dirac delta function

In mathematics, the Dirac delta function (function) is a generalized function or distribution introduced by the physicist Paul Dirac.

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Encyclopedia of Statistical Sciences

The Encyclopedia of Statistical Sciences is an encyclopaedia of statistics published by John Wiley & Sons.

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Expected value

In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents.

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Fat-tailed distribution

A fat-tailed distribution is a probability distribution that has the property, along with the other heavy-tailed distributions, that it exhibits large skewness or kurtosis.

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Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

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Full width at half maximum

Full width at half maximum (FWHM) is an expression of the extent of function given by the difference between the two extreme values of the independent variable at which the dependent variable is equal to half of its maximum value.

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Fundamental solution

In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions).

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Hölder's inequality

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of ''Lp'' spaces.

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Hendrik Lorentz

Hendrik Antoon Lorentz (18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect.

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Homogeneous broadening

Homogeneous broadening is a type of emission spectrum broadening in which all atoms radiating from a specific level under consideration radiate with equal opportunity.

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Homogeneous function

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

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Hydrology

Hydrology is the scientific study of the movement, distribution, and quality of water on Earth and other planets, including the water cycle, water resources and environmental watershed sustainability.

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Hyperbolic distribution

The hyperbolic distribution is a continuous probability distribution characterized by the logarithm of the probability density function being a hyperbola.

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Independence (probability theory)

In probability theory, two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other.

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Independent and identically distributed random variables

In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed (i.i.d. or iid or IID) if each random variable has the same probability distribution as the others and all are mutually independent.

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Indeterminate form

In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is said to take on an indeterminate form.

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Infinite divisibility (probability)

In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed random variables.

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Infinitesimal

In mathematics, infinitesimals are things so small that there is no way to measure them.

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Interquartile range

In descriptive statistics, the interquartile range (IQR), also called the midspread or middle 50%, or technically H-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR.

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Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains).

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Irénée-Jules Bienaymé

Irénée-Jules Bienaymé (28 August 1796 – 19 October 1878), was a French statistician.

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John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

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Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.

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Law of large numbers

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times.

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Lévy distribution

No description.

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Lévy flight

A Lévy flight, named for French mathematician Paul Lévy, is a random walk in which the step-lengths have a probability distribution that is heavy-tailed.

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Lévy process

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random and independent, and statistically identical over different time intervals of the same length.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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Location parameter

In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter x_0, which determines the "location" or shift of the distribution.

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Location–scale family

In probability theory, especially in mathematical statistics, a location–scale family is a family of probability distributions parametrized by a location parameter and a non-negative scale parameter.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Maximum entropy probability distribution

In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions.

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Maximum likelihood estimation

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model, given observations.

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Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

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McCullagh's parametrization of the Cauchy distributions

In probability theory, the "standard" Cauchy distribution is the probability distribution whose probability density function (pdf) is for x real.

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Mean

In mathematics, mean has several different definitions depending on the context.

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Median

The median is the value separating the higher half of a data sample, a population, or a probability distribution, from the lower half.

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Mode (statistics)

The mode of a set of data values is the value that appears most often.

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Moment (mathematics)

In mathematics, a moment is a specific quantitative measure, used in both mechanics and statistics, of the shape of a set of points.

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Moment-generating function

In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution.

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Multivariate random variable

In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value.

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Multivariate t-distribution

In statistics, the multivariate t-distribution (or multivariate Student distribution) is a multivariate probability distribution.

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Newton's method

In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.

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Normal distribution

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution.

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Nuclear physics

Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions.

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Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

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Order statistic

In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value.

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Particle physics

Particle physics (also high energy physics) is the branch of physics that studies the nature of the particles that constitute matter and radiation.

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Pathological (mathematics)

In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved.

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Pearson distribution

The Pearson distribution is a family of continuous probability distributions.

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Peter McCullagh

Peter McCullagh (born 8 January 1952) is an Irish statistician and John D. MacArthur Distinguished Service Professor in the Department of Statistics at the University of Chicago.

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Physicist

A physicist is a scientist who has specialized knowledge in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.

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Pierre-Simon Laplace

Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French scholar whose work was important to the development of mathematics, statistics, physics and astronomy.

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Poisson kernel

In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc.

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Probability density function

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function, whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

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Probability distribution

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

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Probable error

In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.

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Q–Q plot

In statistics, a Q–Q (quantile-quantile) plot is a probability plot, which is a graphical method for comparing two probability distributions by plotting their quantiles against each other.

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Quantile function

In probability and statistics, the quantile function specifies, for a given probability in the probability distribution of a random variable, the value at which the probability of the random variable is less than or equal to the given probability.

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Random variable

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.

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Ratio distribution

A ratio distribution (or quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Relativistic Breit–Wigner distribution

The relativistic Breit–Wigner distribution (after the 1936 nuclear resonance formula of Gregory Breit and Eugene Wigner) is a continuous probability distribution with the following probability density function, See (page 98 onwards) for a discussion of the widths of particles in the PYTHIA manual.

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Resonance

In physics, resonance is a phenomenon in which a vibrating system or external force drives another system to oscillate with greater amplitude at specific frequencies.

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Scale parameter

In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions.

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Siméon Denis Poisson

Baron Siméon Denis Poisson FRS FRSE (21 June 1781 – 25 April 1840) was a French mathematician, engineer, and physicist, who made several scientific advances.

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Slash distribution

In probability theory, the slash distribution is the probability distribution of a standard normal variate divided by an independent standard uniform variate.

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Spectral line

A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies.

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Spectroscopy

Spectroscopy is the study of the interaction between matter and electromagnetic radiation.

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Stability (probability)

In probability theory, the stability of a random variable is the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters.

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Stable distribution

No description.

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Stable process

In probability theory, a stable process is a type of stochastic process.

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Standardized moment

In probability theory and statistics, the standardized moment of a probability distribution is a moment (normally a higher degree central moment) that is normalized.

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Stigler's law of eponymy

Stigler's law of eponymy is a process proposed by University of Chicago statistics professor Stephen Stigler in his 1980 publication "Stigler’s law of eponymy".

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Student's t-distribution

In probability and statistics, Student's t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.

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Truncated mean

A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median.

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University of Alabama

The University of Alabama (Alabama or UA) is a public research university located in Tuscaloosa, Alabama, United States, and the flagship of the University of Alabama System.

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Upper half-plane

In mathematics, the upper half-plane H is the set of complex numbers with positive imaginary part: The term arises from a common visualization of the complex number x + iy as the point (x,y) in the plane endowed with Cartesian coordinates.

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Variance

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean.

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Witch of Agnesi

In mathematics, the Witch of Agnesi is a cubic plane curve defined from two diametrically opposite points of a circle.

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Wrapped Cauchy distribution

No description.

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Cauchy Distribution, Cauchy Random Variable, Cauchy noise, Cauchy-Lorentz distribution, Cauchy–Lorentz distribution, Lorentz distribution, Lorentz function, Lorentz profile, Lorentzian Function, Lorentzian Lineshape, Lorentzian distribution, Lorentzian function, Lorentzian lineshape, Lorentzian profile, Lorenz distribution, Multivariate Cauchy distribution.

References

[1] https://en.wikipedia.org/wiki/Cauchy_distribution

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