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Chebyshev's inequality

Index Chebyshev's inequality

In probability theory, Chebyshev's inequality (also spelled as Tchebysheff's inequality, Нера́венство Чебышёва, also called Bienaymé-Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. [1]

85 relations: Absolute value, American Statistical Association, Andrey Markov, Annals of Applied Probability, Annals of Human Genetics, Banach space, Cantelli's inequality, Carl Friedrich Gauss, Cengage, Chebyshev's sum inequality, Chebyshev–Markov–Stieltjes inequalities, Concave function, Concentration inequality, Convex conjugate, Convex function, Cornish–Fisher expansion, Covariance matrix, Cumulant, Deviation risk measure, Dimension, Dirac delta function, Eaton's inequality, Eli Upfal, Exchangeable random variables, Expected value, Extended real number line, Francesco Paolo Cantelli, Fréchet space, Function (mathematics), Geoffrey Grimmett, Hermite–Hadamard inequality, Hilbert space, Indicator function, Interval (mathematics), Irénée-Jules Bienaymé, J. B. S. Haldane, Jensen's inequality, John Wiley & Sons, Journal of the American Statistical Association, Kantorovich inequality, Kolmogorov's inequality, Kurtosis, Law of large numbers, Le Cam's theorem, Logarithmically concave function, Mahalanobis distance, Markov's inequality, Maurice Kendall, Mean, Measurable function, ..., Measure (mathematics), Measure space, Median, Michael Mitzenmacher, Mizar system, Mode (statistics), Monotonic function, Multidimensional Chebyshev's inequality, Multivariate random variable, Normal distribution, Pafnuty Chebyshev, Paley–Zygmund inequality, Pareto distribution, Pettis integral, Probability distribution, Probability theory, Rademacher distribution, Random variable, Real number, Samuelson's inequality, Sankhya (journal), Semidefinite programming, Semivariance, Skewness, Standard deviation, Standard score, Statistics and Computing, Symmetric probability distribution, The American Statistician, Transpose, Unimodality, United States Environmental Protection Agency, Variance, Vysochanskij–Petunin inequality, 68–95–99.7 rule. Expand index (35 more) »

Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

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American Statistical Association

The American Statistical Association (ASA) is the main professional organization for statisticians and related professionals in the United States.

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Andrey Markov

Andrey (Andrei) Andreyevich Markov (Андре́й Андре́евич Ма́рков, in older works also spelled Markoff) (14 June 1856 N.S. – 20 July 1922) was a Russian mathematician.

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Annals of Applied Probability

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

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Annals of Human Genetics

The Annals of Human Genetics is a bimonthly peer-reviewed scientific journal covering human genetics.

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Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

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Cantelli's inequality

In probability theory, Cantelli's inequality, named after Francesco Paolo Cantelli, is a generalization of Chebyshev's inequality in the case of a single "tail".

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Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

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Cengage

Cengage is an educational content, technology, and services company for the higher education, K-12, professional, and library markets worldwide.

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Chebyshev's sum inequality

In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if and then Similarly, if and then.

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Chebyshev–Markov–Stieltjes inequalities

In mathematical analysis, the Chebyshev–Markov–Stieltjes inequalities are inequalities related to the problem of moments that were formulated in the 1880s by Pafnuty Chebyshev and proved independently by Andrey Markov and (somewhat later) by Thomas Jan Stieltjes.

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Concave function

In mathematics, a concave function is the negative of a convex function.

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Concentration inequality

In probability theory, concentration inequalities provide bounds on how a random variable deviates from some value (typically, its expected value).

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Convex conjugate

In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions.

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Convex function

In mathematics, a real-valued function defined on an ''n''-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions.

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Cornish–Fisher expansion

The Cornish–Fisher expansion is an asymptotic expansion used to approximate the quantiles of a probability distribution based on its cumulants.

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Covariance matrix

In probability theory and statistics, a covariance matrix (also known as dispersion matrix or variance–covariance matrix) is a matrix whose element in the i, j position is the covariance between the i-th and j-th elements of a random vector.

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Cumulant

In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution.

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Deviation risk measure

In financial mathematics, a deviation risk measure is a function to quantify financial risk (and not necessarily downside risk) in a different method than a general risk measure.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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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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Eaton's inequality

In probability theory, Eaton's inequality is a bound on the largest values of a linear combination of bounded random variables.

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Eli Upfal

Eli Upfal is a computer science researcher, currently the Rush C. Hawkins Professor of Computer Science at Brown University.

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Exchangeable random variables

In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence such that future observations behave like earlier observations.

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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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Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system by adding two elements: and (read as positive infinity and negative infinity respectively).

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Francesco Paolo Cantelli

Francesco Paolo Cantelli (20 December 1875, Palermo21 July 1966, Rome) was an Italian mathematician.

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Fréchet space

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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Geoffrey Grimmett

Geoffrey Richard Grimmett FRS (born 20 December 1950) is a mathematician known for his work on the mathematics of random systems arising in probability theory and statistical mechanics, especially percolation theory and the contact process.

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Hermite–Hadamard inequality

In mathematics, the Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called Hadamard's inequality, states that if a function ƒ: → R is convex, then the following chain of inequalities hold.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.

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Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

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Irénée-Jules Bienaymé

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

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J. B. S. Haldane

John Burdon Sanderson Haldane (5 November 18921 December 1964) was an English scientist known for his work in the study of physiology, genetics, evolutionary biology, and in mathematics, where he made innovative contributions to the fields of statistics and biostatistics.

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Jensen's inequality

In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.

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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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Journal of the American Statistical Association

The Journal of the American Statistical Association (JASA) is the primary journal published by the American Statistical Association, the main professional body for statisticians in the United States.

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Kantorovich inequality

In mathematics, the Kantorovich inequality is a particular case of the Cauchy–Schwarz inequality, which is itself a generalization of the triangle inequality.

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Kolmogorov's inequality

In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound.

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Kurtosis

In probability theory and statistics, kurtosis (from κυρτός, kyrtos or kurtos, meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable.

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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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Le Cam's theorem

In probability theory, Le Cam's theorem, named after Lucien le Cam (1924 – 2000), states the following.

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Logarithmically concave function

In convex analysis, a non-negative function is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality for all and.

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Mahalanobis distance

The Mahalanobis distance is a measure of the distance between a point P and a distribution D, introduced by P. C. Mahalanobis in 1936.

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Markov's inequality

In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant.

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Maurice Kendall

Sir Maurice George Kendall, FBA (6 September 1907 – 29 March 1983) was a British statistician, widely known for his contribution to statistics.

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Mean

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

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Measurable function

In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open.

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Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

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Measure space

A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes.

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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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Michael Mitzenmacher

Michael David Mitzenmacher is an American computer scientist working in algorithms.

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Mizar system

The Mizar system consists of a formal language for writing mathematical definitions and proofs, a proof assistant, which is able to mechanically check proofs written in this language, and a library of formalized mathematics, which can be used in the proof of new theorems.

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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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Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.

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Multidimensional Chebyshev's inequality

In probability theory, the multidimensional Chebyshev's inequality is a generalization of Chebyshev's inequality, which puts a bound on the probability of the event that a random variable differs from its expected value by more than a specified amount.

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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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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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Pafnuty Chebyshev

Pafnuty Lvovich Chebyshev (p) (–) was a Russian mathematician.

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Paley–Zygmund inequality

In mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of its mean and variance (i.e., its first two moments).

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Pareto distribution

No description.

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Pettis integral

In mathematics, the Pettis integral or Gelfand–Pettis integral, named after I. M. Gelfand and B. J. Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality.

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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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Probability theory

Probability theory is the branch of mathematics concerned with probability.

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Rademacher distribution

No description.

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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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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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Samuelson's inequality

In statistics, Samuelson's inequality, named after the economist Paul Samuelson, also called the Laguerre–Samuelson inequality, after the mathematician Edmond Laguerre, states that every one of any collection x1,..., xn, is within uncorrected sample standard deviations of their sample mean.

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Sankhya (journal)

Sankhyā: The Indian Journal of Statistics is a quarterly peer-reviewed scientific journal on statistics published by the Indian Statistical Institute (ISI).

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Semidefinite programming

Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron.

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Semivariance

In spatial statistics, the empirical semivariance is described by semivariance.

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Skewness

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean.

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Standard deviation

In statistics, the standard deviation (SD, also represented by the Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.

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Standard score

In statistics, the standard score is the signed number of standard deviations by which the value of an observation or data point differs from the mean value of what is being observed or measured.

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Statistics and Computing

Statistics and Computing is a peer-reviewed academic journal that deals with statistics and computing.

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Symmetric probability distribution

In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function or probability mass function is reflected around a vertical line at some value of the random variable represented by the distribution.

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The American Statistician

The American Statistician is a quarterly peer-reviewed scientific journal covering statistics published by Taylor & Francis on behalf of the American Statistical Association.

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Transpose

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).

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Unimodality

In mathematics, unimodality means possessing a unique mode.

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United States Environmental Protection Agency

The Environmental Protection Agency is an independent agency of the United States federal government for environmental protection.

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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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Vysochanskij–Petunin inequality

In probability theory, the Vysochanskij–Petunin inequality gives a lower bound for the probability that a random variable with finite variance lies within a certain number of standard deviations of the variable's mean, or equivalently an upper bound for the probability that it lies further away.

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68–95–99.7 rule

In statistics, the 68–95–99.7 rule is a shorthand used to remember the percentage of values that lie within a band around the mean in a normal distribution with a width of two, four and six standard deviations, respectively; more accurately, 68.27%, 95.45% and 99.73% of the values lie within one, two and three standard deviations of the mean, respectively.

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References

[1] https://en.wikipedia.org/wiki/Chebyshev's_inequality

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