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Bound on probability of a random variable being far from its mean
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) provides an upper bound on the probability of deviation
Chebyshev's_inequality
probability theory, the multidimensional Chebyshev's inequality is a generalization of Chebyshev's inequality, which puts a bound on the probability of
Multidimensional Chebyshev's inequality
Multidimensional_Chebyshev's_inequality
Topics referred to by the same term
between n and 2n. Chebyshev's inequality, on the range of standard deviations around the mean, in statistics Chebyshev's sum inequality, about sums and
Chebyshev's_theorem
Concept in probability theory
first Chebyshev inequality, while referring to Chebyshev's inequality as the second Chebyshev inequality) or Bienaymé's inequality. Markov's inequality (and
Markov's_inequality
Mathematical inequality
In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if a 1 ≥ a 2 ≥ ⋯ ≥ a n {\displaystyle a_{1}\geq a_{2}\geq \cdots
Chebyshev's_sum_inequality
Inequality in probability theorem
Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for
Cantelli's_inequality
Averages of repeated trials converge to the expected value
¯ n ) = μ . {\displaystyle E({\overline {X}}_{n})=\mu .} Using Chebyshev's inequality on X ¯ n {\displaystyle {\overline {X}}_{n}} results in P ( |
Law_of_large_numbers
Exponentially decreasing bounds on tail distributions of random variables
Markov's inequality or Chebyshev's inequality. The Chernoff bound is related to the Bernstein inequalities. It is also used to prove Hoeffding's inequality, Bennett's
Chernoff_bound
Inequality in probability theory
inequality follows by Chebyshev's inequality. This inequality was generalized by Hájek and Rényi in 1955. Chebyshev's inequality Etemadi's inequality
Kolmogorov's_inequality
Russian mathematician (1821–1894)
the Chebyshev inequality (which can be used to prove the weak law of large numbers), the Bertrand–Chebyshev theorem, Chebyshev polynomials, Chebyshev linkage
Pafnuty_Chebyshev
Probabilistic inequality
Vysochanskiï–Petunin inequality, a similar result for the distance from the mean rather than the mode Chebyshev's inequality, concerns distance from
Gauss's_inequality
Concept in statistics
{n-1}{\sqrt {n}}}.} Chebyshev's inequality locates a certain fraction of the data within certain bounds, while Samuelson's inequality locates all the data
Samuelson's_inequality
Measure of variation in statistics
Accuracy and precision Algorithms for calculating variance Chebyshev's inequality An inequality on location and scale parameters Coefficient of variation
Standard_deviation
Pair of polynomial sequences
Mathematics portal Chebyshev rational functions Function approximation Discrete Chebyshev transform Markov brothers' inequality Rivlin, Theodore J. (1974)
Chebyshev_polynomials
Average value of a random variable
finite expectation, then Markov's inequality may be applied to the random variable |X−E[X]|2 to obtain Chebyshev's inequality P ( | X − E [ X ] | ≥ a ) ≤
Expected_value
Topics referred to by the same term
polynomials Chebyshev filter Chebyshev's inequality Chebyshev distance Chebyshev (crater): A lunar crater 2010 Chebyshev: An asteroid from the asteroid
Chebyshev_(disambiguation)
Rule in statistics
95% confidence)[citation needed]. Cantelli's inequality is the one-tailed version of Chebyshev's inequality. Binomial proportion confidence interval Rule
Rule_of_three_(statistics)
Mathematical relation making a non-equal comparison
Azuma's inequality Bernoulli's inequality Bell's inequality Boole's inequality Cauchy–Schwarz inequality Chebyshev's inequality Chernoff's inequality Cramér–Rao
Inequality_(mathematics)
Mathematical theorem
the Chebyshev–Markov–Stieltjes inequalities are inequalities related to the problem of moments that were formulated in the 1880s by Pafnuty Chebyshev and
Chebyshev–Markov–Stieltjes inequalities
Chebyshev–Markov–Stieltjes_inequalities
Theorem in probability theory
In probability theory, Azuma's inequality or the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result
Azuma's_inequality
Inequalities in probability theory
Bernstein inequalities are also known as the Chernoff bound, Hoeffding's inequality and Azuma's inequality. The martingale case of the Bernstein inequality is
Bernstein inequalities (probability theory)
Bernstein_inequalities_(probability_theory)
Property of having a unique mode or maximum value
second is the Vysochanskij–Petunin inequality, a refinement of the Chebyshev inequality. The Chebyshev inequality guarantees that in any probability distribution
Unimodality
{\displaystyle P\left(\left|X_{n}\right|\leq K(\eta )\right)\geq 1-\eta } . Chebyshev's inequality states: P ( | X − μ | ≤ h σ ) ≥ 1 − h − 2 {\displaystyle P\left(\left|X-\mu
Big_O_in_probability_notation
inequality Chebyshev–Markov–Stieltjes inequalities Chebyshev's sum inequality Clarkson's inequalities Eilenberg's inequality Fekete–Szegő inequality Fenchel's
List_of_inequalities
Chebyshev's inequality Chebyshev pseudospectral method Chebyshev space Chebyshev's sum inequality Chebyshev's theorem (disambiguation) Chebyshev linkage,
List of things named after Pafnuty Chebyshev
List_of_things_named_after_Pafnuty_Chebyshev
Mathematical inequality explaining concentration of random variables
deviation of X {\displaystyle X} . Chebyshev's inequality can be seen as a special case of the generalized Markov's inequality applied to the random variable
Concentration_inequality
Without unimodality Chebyshev's inequality would give a looser bound of 1/9 = 0.11111.... A version of the Vysochanskij-Petunin inequality for one-sided tail
Vysochanskij–Petunin inequality
Vysochanskij–Petunin_inequality
Shorthand used in statistics
qualify as a discovery. A weaker three-sigma rule can be derived from Chebyshev's inequality, stating that even for non-normally distributed variables, at least
68–95–99.7_rule
French mathematician
demography and social sciences. In particular, he formulated the Bienaymé–Chebyshev inequality concerning the law of large numbers and the Bienaymé formula for
Irénée-Jules_Bienaymé
Stochastic process
the central limit theorem, law of large numbers, Chernoff's inequality, Chebyshev's inequality or similar tools. When analyzing similar objects where the
Doob_martingale
Concept in mathematics
|f(x)|^{p}} . This representation can be used to prove Markov's inequality and Chebyshev's inequality. Symmetric decreasing rearrangement Willem, Michel (2013)
Layer_cake_representation
In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez (Remez 1936), gives a bound on the sup norms of certain
Remez_inequality
Mathematical function
by x {\displaystyle x} to obtain the inequality in the theorem. The following bounds are known for the Chebyshev functions:[1][2] (in these formulas pk
Chebyshev_function
Mathematical theory by discovered by Józef Marcinkiewicz
inequality ‖ f ‖ 1 , w ≤ ‖ f ‖ 1 . {\displaystyle \|f\|_{1,w}\leq \|f\|_{1}.} This is nothing but Markov's inequality (a.k.a. Chebyshev's Inequality)
Marcinkiewicz interpolation theorem
Marcinkiewicz_interpolation_theorem
Number theory related to prime numbers
π(x; 4, 3) > π(x; 4, 1) occurs much more frequently. For example, this inequality holds for all primes x < 26833 except 5, 17, 41 and 461, for which π(x; 4
Chebyshev's_bias
Middle quantile of a data set or probability distribution
one-sided Chebyshev inequality; it appears in an inequality on location and scale parameters. This formula also follows directly from Cantelli's inequality. For
Median
Characteristic function (probability theory) Chauvenet's criterion Chebyshev center Chebyshev's inequality Checking if a coin is biased – redirects to Checking whether
List_of_statistics_articles
Markov's inequality Chebyshev's inequality = Chernoff bound Chernoff's inequality Bernstein inequalities (probability theory) Hoeffding's inequality Kolmogorov's
List_of_probability_topics
Theorem in mathematics
geometric mean inequality, the Cauchy–Schwarz inequality, and Chebyshev's sum inequality. As a simple example, consider real numbers x 1 ≤ ⋯ ≤ x n {\displaystyle
Rearrangement_inequality
Inequality type in mathematical analysis
general functions. This finishes the proof. Rearrangement inequality Chebyshev's sum inequality Lorentz space Lieb, Elliott; Loss, Michael (2001). Analysis
Hardy–Littlewood_inequality
Type of polynomial used in Numerical Analysis
relation holds uniformly in x, which can be seen from its proof via Chebyshev's inequality, taking into account that the variance of 1⁄n K, equal to 1⁄n x(1−x)
Bernstein_polynomial
statistics and number theory, author of the Chebyshev's inequality, Chebyshev distance, Chebyshev function, Chebyshev equation etc. Sergei Chernikov, significant
List of Russian mathematicians
List_of_Russian_mathematicians
Inequality proved by Andrey Markov and Vladimir Markov
x\leq 1}|P(x)|.} This inequality is tight, as equality is attained for Chebyshev polynomials of the first kind. Bernstein's inequality (mathematical analysis)
Markov_brothers'_inequality
Overview of and topical guide to probability
the monotone and dominated convergence theorems Markov's inequality and Chebyshev's inequality Independent random variables Discrete: constant (see also
Outline_of_probability
Correlation inequality
any measure μ. In case the measure μ is uniform, the FKG inequality is Chebyshev's sum inequality: if the two increasing functions take on values a 1 ≤ a
FKG_inequality
Inequality in probability theory
zero. Apply Chebyshev's inequality to the right-hand side of Etemadi's inequality and replace α by α / 3. The result is Kolmogorov's inequality with an extra
Etemadi's_inequality
Machine learning technique
Chebyshev's inequality. Since cos , sin {\displaystyle \cos ,\sin } are bounded, there is a stronger convergence guarantee by Hoeffding's inequality.
Random_feature
Wald test Bernstein inequalities (probability theory) Binomial regression Binomial proportion confidence interval Chebyshev's inequality Chernoff bound Gauss's
List of analyses of categorical data
List_of_analyses_of_categorical_data
Theorems about certain polynomial families
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Pál Turán (and first published by Szegö (1948)). There are
Turán's_inequalities
Marcinkiewicz–Zygmund inequality / mnt Markov's inequality / (1:R) McDiarmid's inequality Multidimensional Chebyshev's inequality Paley–Zygmund inequality / (1:R) Pinsker's
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Theorems concerning stochastic processes
integrability of the random variables N t {\displaystyle N_{t}} . By Chebyshev's inequality, convergence in L1 implies convergence in probability and convergence
Doob's martingale convergence theorems
Doob's_martingale_convergence_theorems
Problem in probability theory
{1}{n^{2}}}+\cdots } (see Basel problem). Bound the desired probability using the Chebyshev inequality: P ( | T − n H n | ≥ c n ) ≤ π 2 6 c 2 . {\displaystyle \operatorname
Coupon_collector's_problem
Charles Chebyshev distance, equation, filter, linkage, polynomials – Pafnuty Chebyshev Chebyshev's inequality (a.k.a. Bienaymé–Chebyshev inequality) – Pafnuty
Scientific phenomena named after people
Scientific_phenomena_named_after_people
Statistical estimator
(in which case it is known as Markov inequality), or the quadratic function (respectively Chebyshev's inequality). Another useful result is the continuous
Consistent_estimator
Theorem on operator interpolation
Th_{n}\to Th} in measure: For any ϵ > 0 {\textstyle \epsilon >0} , Chebyshev’s inequality yields μ 2 ( y ∈ Ω 2 : | T g − T g n | > ϵ ) ≤ ‖ T g − T g n ‖ q
Riesz–Thorin_theorem
Correlation inequality in statistical mechanics
inequality, sometimes also called Griffiths–Kelly–Sherman inequality or GKS inequality, named after Robert B. Griffiths, is a correlation inequality for
Griffiths_inequality
American mathematician
MR 0024099. Birnbaum, Z. W.; Marshall, A.W. (1961). "Some multivariate Chebyshev inequalities with extensions to continuous parameter processes". Annals of Mathematical
Z._W._Birnbaum
Theorem in probability theory
used to prove the Strong Law of Large Numbers. In many proofs, Chebyshev's inequality is applied to bound the probability that a sum of random variables
Borel–Cantelli_lemma
Parts of a whole which carry only relative information
of such noise, any attempt to use the central limit theorem and Chebyshev's inequality to define a strict boundary between signal and noise fails, as the
Compositional_data
Hölder's inequality Minkowski inequality Jensen's inequality Chebyshev's inequality Inequality of arithmetic and geometric means Generalized mean Pythagorean
List_of_real_analysis_topics
Statistical measure of how far values spread from their average
information that a variance does not. For inequalities associated with the semivariance, see Chebyshev's inequality § Semivariances. The term variance was
Variance
Statistical measure that quantifies the robustness of a manufacturing process
conform to particular distributions, but the Chebyshev's inequality and the Vysochanskij–Petunin inequality allow the inference that for any unimodal distribution
Process_window_index
Feature selection algorithm used in binary classification
determined by inspection. However, it can also be determined by Chebyshev's inequality for a given confidence level (α) that a τ of 1/sqrt(α*m) is good
Relief_(feature_selection)
E. C. Stoner, and later improved by Subrahmanyan Chandrasekhar. Chebyshev's inequality guarantees that, for a wide class of probability distributions,
List of examples of Stigler's law
List_of_examples_of_Stigler's_law
Statistic used in statistical hypothesis testing
population that falls within k standard deviations for any k (see: Chebyshev's inequality). Two-sample z-test z = ( x ¯ 1 − x ¯ 2 ) − d 0 σ 1 2 n 1 + σ 2
Test_statistic
Tool to assess control of a manufacturing process
deviation) limits on the following basis. The coarse result of Chebyshev's inequality that, for any probability distribution, the probability of an outcome
Control_chart
French mathematician (born 1948)
Gilles Courtois, a Chebyshev inequality for the minimal entropy of locally symmetrical spaces of negative curvature; the inequality gives a new and simpler
Sylvestre_Gallot
statistics and number theory, author of the Chebyshev's inequality, Chebyshev distance, Chebyshev function, Chebyshev equation Boris Delaunay, inventor of Delaunay
List_of_Russian_scientists
almost surely if d ≥ 2 {\displaystyle \scriptstyle d\geq 2} proof By Chebyshev's inequality and the Borel–Cantelli lemma, there is the equation below: P ( ρ
Voter_model
Length of a line segment
spaces, this inequality may not be true. Euclidean distance geometry studies properties of Euclidean distance such as Ptolemy's inequality, and their application
Euclidean_distance
Model in statistical mechanics
_{2}{\frac {s}{1-p}}+(1-s)\log _{2}{\frac {1-s}{p}}\end{aligned}}} By the Chebyshev inequality, if Σ > 0 {\displaystyle \Sigma >0} , then n ( s ) {\displaystyle
Random_subcube_model
χ[1−ε,1](hN) tends to 0 in L1(Y,ν). But this follows easily by Chebyshev's inequality: indeed (1−ε) χ[1−ε,1](hN) ≤ hN, so that ν(χ[1−ε,1](hN)) ≤ (1−ε)−1
Ergodic_flow
Notion of convergence of random variables
− Q P ( h ) ) {\displaystyle m\cdot Q_{P}(h)(1-Q_{P}(h))} . By Chebyshev's inequality we get P m { | Q P ( h ) − Q s ( h ) ^ | > ε 2 } ≤ m ⋅ Q P ( h )
Uniform convergence in probability
Uniform_convergence_in_probability
Theory of getting acceptably close inexact mathematical calculations
Bordeaux. 20: 281–7. doi:10.5802/jtnb.627. Erdélyi, T. (2009). "The Remez inequality for linear combinations of shifted Gaussians". Mathematical Proceedings
Approximation_theory
Characterization of how many integers are prime
= 1 and ζ(x + 2iy) stays analytic, the left hand side in the previous inequality tends to 0, a contradiction. Finally, we can conclude that the PNT is
Prime_number_theorem
statistics and number theory, author of the Chebyshev's inequality, Chebyshev distance, Chebyshev function, Chebyshev equation Boris Delaunay, inventor of Delaunay
List_of_Russian_people
after Andrey Markov, an influential Russian mathematician. Chebyshev–Markov–Stieltjes inequalities Dynamics of Markovian particles Dynamic Markov compression
List of things named after Andrey Markov
List_of_things_named_after_Andrey_Markov
Russian mathematician (1856–1922)
death in 1922. List of things named after Andrey Markov Chebyshev–Markov–Stieltjes inequalities Gauss–Markov theorem Gauss–Markov process Hidden Markov
Andrey_Markov
Country in Eastern Europe and North Asia
fifth-highest number of billionaires in the world. However, its income inequality remains high compared to other developed countries. The variance of natural
Russia
Soviet mathematician
function theory, in particular, for the Remez algorithm and the Remez inequality. His doctoral students include Boris Korenblum. Cheney, Elliott Ward (1998)
Evgeny_Yakovlevich_Remez
Mathematical space with a notion of distance
weaker form of the triangle inequality, such as: The ρ-inframetric inequality implies the ρ-relaxed triangle inequality (assuming the first axiom), and
Metric_space
Constants related to interpolation errors
{\displaystyle \|f-X(f)\|\leq \|f-p^{*}\|+\|p^{*}-X(f)\|} by the triangle inequality. But X {\displaystyle X} is a projection on Πn, so p∗ − X( f ) = X(p∗)
Lebesgue_constant
Function on an integer n which is log(p) if n equals p^k and zero otherwise
with diverging oscillations: there exists a value K > 0 such that both inequalities F ( y ) < − K y , and F ( z ) > K z {\displaystyle F(y)<-{\frac {K}{\sqrt
Von_Mangoldt_function
Operation in calculus
dx\right)^{1/q}.} For p = q = 2, Hölder's inequality becomes the Cauchy–Schwarz inequality. Minkowski inequality. Suppose that p ≥ 1 is a real number and
Integral
Vector distance function
The Minkowski distance is a metric as a result of the Minkowski inequality, ‖ X + Y ‖ p ≤ ‖ X ‖ p + ‖ Y ‖ p . {\displaystyle \|X+Y\|_{p}\leq \|X\|_{p}+\|Y\|_{p}
Minkowski_distance
Solved prime-number problem
First conjectured in 1845 by Joseph Bertrand, it was first proven by Chebyshev, and a shorter but also advanced proof was given by Ramanujan. The following
Proof_of_Bertrand's_postulate
Polynomial sequence
scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles
Hermite_polynomials
Function representing the number of primes less than or equal to a given number
{x}{\log x}}\quad {\text{for }}x\geq 17.} The left inequality holds for x ≥ 17 and the right inequality holds for x > 1. The constant 1.25506 is 30log 113/113
Prime-counting_function
Baltic German mathematician
will be recompensed but I am still grateful to Hitler". Chebyshev function Isoperimetric inequality Low-rank approximation List of Baltic German scientists
Erhard_Schmidt
Conjecture on zeros of the zeta function
and a rank-trace inequality derived from von Neumann's trace inequality. The logical core of the proof, including the matrix inequalities and trace asymptotics
Riemann_hypothesis
Absolute value of (x - y), a metric
− y | + | y − z | {\displaystyle |x-z|\leq |x-y|+|y-z|} (the triangle inequality); equality holds if and only if x ≤ y ≤ z {\displaystyle x\leq y\leq z}
Absolute_difference
words, the radius of the largest inscribed disk with center z. Both inequalities are best possible: The upper bound is clearly attained by taking D =
Conformal_radius
Russian mathematician and academic (1877–1943)
by her instructor, David Hilbert, and was one of the first to include inequalities in the calculus of variations. Gernet was born on April 18, 1877 in Simbirsk
Nadeschda_Gernet
secretary problem and for the Samuels Conjecture involving a Chebyshev-type inequality for sums of independent, non-negative random variables. After
Stephen_Mitchell_Samuels
Polynomial sequence
The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. The Jacobi polynomials
Jacobi_polynomials
Polynomial equation of degree 3
definition of a principal part is not purely algebraic, since it involves inequalities for comparing real parts. Also, the use of principal cube root may give
Cubic_equation
Extension of the factorial function
consists of only positive terms. Logarithmic convexity and Jensen's inequality together imply, for any positive real numbers x 1 , … , x n {\displaystyle
Gamma_function
Type of metric geometry
the geometry of numbers, Hermann Minkowski established his Minkowski inequality, stating that these spaces define normed vector spaces. The name taxicab
Taxicab_geometry
corresponding term in the sum above is just (sin x)n.) Aristarchus's inequality Derivatives of trigonometric functions Exact trigonometric values (values
List of trigonometric identities
List_of_trigonometric_identities
Polynomial sequence
weight function (1 − x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named
Gegenbauer_polynomials
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CHEBYSHEVS INEQUALITY
CHEBYSHEVS INEQUALITY
CHEBYSHEVS INEQUALITY
CHEBYSHEVS INEQUALITY
CHEBYSHEVS INEQUALITY
CHEBYSHEVS INEQUALITY
CHEBYSHEVS INEQUALITY
CHEBYSHEVS INEQUALITY
CHEBYSHEVS INEQUALITY
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