For any randomized trial, some variation from the mean is expected, of course, but the randomization ensures that the experimental groups have mean values that are close, due to the central limit theorem and Markov's inequality.

The term Chebyshev ’ s inequality may also refer to the Markov's inequality, especially in the context of analysis.

One way to prove Chebyshev's inequality is to apply Markov's inequality to the random variable Y = ( X − μ )< sup > 2 </ sup > with a = ( σk )< sup > 2 </ sup >.

Common tools used in the probabilistic method include Markov's inequality, the Chernoff bound, and the Lovász local lemma.

Markov's inequality gives an upper bound for the measure of the set ( indicated in red ) where exceeds a given level.

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.

It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev ( Markov's teacher ), and many sources, especially in analysis, refer to it as Chebyshev's inequality or Bienaymé's inequality.

Markov's inequality ( and other similar inequalities ) relate probabilities to expectations, and provide ( frequently ) loose but still useful bounds for the cumulative distribution function of a random variable.

An example of an application of Markov's inequality is the fact that ( assuming incomes are non-negative ) no more than 1 / 5 of the population can have more than 5 times the average income.

In the language of measure theory, Markov's inequality states that if ( X, Σ, μ ) is a measure space, ƒ is a measurable extended real-valued function, and, then

Chebyshev's inequality follows from Markov's inequality by considering the random variable

for which Markov's inequality reads

This identity is used in a simple proof of Markov's inequality.

If μ is less than 1, then the expected number of individuals goes rapidly to zero, which implies ultimate extinction with probability 1 by Markov's inequality.

* Markov's inequality and Chebyshev's inequality

Observe that any Las Vegas algorithm can be converted into a Monte Carlo algorithm ( via Markov's inequality ), by having it output an arbitrary, possibly incorrect answer if it fails to complete within a specified time.

* Markov's inequality, a probabilistic upper bound

By an application of Markov's inequality, a Las Vegas algorithm can be converted into a Monte Carlo algorithm via early termination ( assuming the algorithm structure provides for such a mechanism ).

It is a sharper bound than the known first or second moment based tail bounds such as Markov's inequality or Chebyshev inequality, which only yield power-law bounds on tail decay.

For finite games, and games where the appropriate instance of Markov's rule can be constructively established by means of bar induction, then the non-constructive proof of a winning strategy for the first player can be converted into a winning strategy.

He made his debut for Plamen Markov's Bulgaria in a friendly against Spain on 20 November 2002, when he was a CSKA Sofia player, coming on as a second half substitute during 0 – 1 defeat at Los Cármenes in Granada.

By Markov's Inequality, the chance that it will yield an answer before we stop it is 1 / 2.

From Markov's inequality and using independence we can derive the following useful inequality:

A single-move version of Markov's theorem, was published by.

This is nothing but Markov's inequality.

The inequality states that

" Also known as the kinship theory of genomic imprinting, this hypothesis states that the inequality between parental genomes due to imprinting is a result of the differing interests of each parent in terms of the evolutionary fitness of their genes.

Samuelson's inequality is a result that states, given that the sample mean and variance have been calculated from a particular sample, bounds on the values that individual values in the sample can take.

The Cauchy – Schwarz inequality states that for all vectors x and y of an inner product space it is true that

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ( and, if the setting is a Euclidean space, then the inequality is strict if the triangle is non-degenerate ).< ref name = Khamsi >

For instance, Daisy Myers has been hailed as " The Rosa Parks of the North ", who helped expose the northern states ' problems with racial inequality of that time.

The Chebyshev inequality states that if is a random variable with standard deviation σ, then the probability that the outcome of is no less than away from its mean is no more than:

In physics, the CHSH inequality can be used in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics cannot be reproduced by local hidden variable theories.

It would appear from both these later derivations that the only assumptions really needed for the inequality itself ( as opposed to the method of estimation of the test statistic ) are that the distribution of the possible states of the source remains constant and the detectors on the two sides act independently.

An equivalent formulation states that for any ε > 0, there exists a constant K such that, for all triples of coprime positive integers ( a, b, c ) satisfying a + b = c, the inequality

This provides an interesting twist on Wallerstein's neo-Marxist interpretation of the international order which faults differences in power relations between ' core ' and ' periphery ' states as the chief cause for economic and political inequality ( However, the Singer-Prebisch thesis also works with different bargaining positions of labour in developed and developing countries ).

# The Bishop – Gromov inequality states that if a complete m-dimensional Riemannian manifold has non-negative Ricci curvature, then the volume of a ball is smaller or equal to the volume of a ball of the same radius in Euclidean m-space.

In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean after convex transformation ; it is a simple corollary that the opposite is true of concave transformations.

Specifically, the isoperimetric inequality states, for the length L of a closed curve and the area A of the planar region that it encloses, that

The isoperimetric inequality states that

Denote by L the length of C and by A the area enclosed by C. The spherical isoperimetric inequality states that

In full generality, the isoperimetric inequality states that for any set S ⊂ R < sup > n </ sup > whose closure has finite Lebesgue measure

Each inequality states that your opponent's net gain is more than zero.

On the issue of markets in a socialist society, Webb states, " Admittedly, market mechanisms in a socialist society can generate inequality, disproportions and imbalances, destructive competition, downward pressure on wages, and monopoly cornering of commodity markets – even the danger of capitalist restoration.

Specifically, the Koksma-Hlawka inequality states that the error

The geodesic minimizes the entropy, due to the Cauchy – Schwarz inequality, which states that the action is bounded below by the length of the curve, squared.

Then Bernstein's inequality states that for M non-zero,