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.

Markov's inequality states that for any real-valued random variable Y and any positive number a, we have Pr (| Y | > a ) ≤ E (| Y |)/ a.

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.

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.

In the probabilistic setting, the inequality can be further generalized to its full strength.

The inequality can be used to show that the error of the approximation by the quasi-Monte Carlo method is, whereas the Monte Carlo method has a probabilistic error of.

Note that the inequality is not really applicable either to electrons or photons, since it builds in no probabilistic properties in the measurement process.

The name triangular norm refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize triangle inequality of ordinary metric spaces.

In probabilistic metric spaces, t-norms are used to generalize triangle inequality of ordinary metric spaces.

The right-hand side of the first equation would be the upper bound of the work, which would now be converted into an inequality

Specifically, he demonstrated an upper limit, known as Bell's inequality, regarding the strength of correlations that can be produced in any theory obeying local realism, and he showed that quantum theory predicts violations of this limit for certain entangled systems.

In the TSP with triangle inequality case it is possible to prove upper bounds in terms of the minimum spanning tree and design an algorithm that has a provable upper bound on the length of the route.

For a 90 % probability, covering the range from the 5 % to the 95 % range on the probability curve, the upper and lower limits can be found using the inequality:

There are also converses of the Jensen's inequality, which estimate the upper bound of the integral of the convex function.

A member of the upper classes, he still has an innate dislike of hereditary wealth and a horror of social inequality.

Boole's inequality may be generalised to find upper and lower bounds on the probability of finite unions of events.

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.

When the number of observations is relatively small, Chebychev's inequality can be used for an upper bound on probabilities, regardless of any assumptions about the distribution of experimental errors: the maximum probabilities that a parameter will be more than 1, 2 or 3 standard deviations away from its expectation value are 100 %, 25 % and 11 % respectively.

where is an objective, is a vector of design variables, is a vector of inequality constraints, is a vector of equality constraints, and and are vectors of lower and upper bounds on the design variables.

One can consider sharpening the upper bound by minimizing the right hand side of the inequality.

had the conscious aim of perpetuating unfreedom and inequality "; because the true goal was to end history upon becoming the perpetual High ruling class – composed not of aristocrats or plutocrats, but of " bureaucrats, scientists, technicians, trade-union organizers, publicity experts, sociologists, teachers, journalists and professional politicians " originally from " the salaried middle class and the upper grades of the working class ".

Senate, the upper house of the Parliament, has equal representation from the federating units balancing the provincial inequality in the National Assembly, where the number of members is based on population of the provinces.

Tsirelson's bound, also known as Tsirelson's inequality, or in another transliteration, Cirel ' son's inequality, is an inequality that imposed an upper limit to quantum mechanical correlations between distant events.

* Using the Griffiths inequality in the formulation of Ginibre, Aizenman and Simon proved that the two point spin correlation of the ferromagnetics XY model in dimension, coupling and inverse temperature is dominated by ( i. e. has upper bound given by ) the two point correlation of the ferromagnetic Ising model in dimension, coupling and inverse temperature

The inequality states that on the complex plane, within the disk of radius 1, the degree of a polynomial times the maximum value of a polynomial is an upper bound for the similar maximum of its derivative.

In many societies, attempts have been made, through property redistribution, taxation, or regulation, to redistribute wealth, sometimes in support of the upper class, and sometimes to diminish extreme inequality.

We resume our derivation by expressing the upper bound on our in light of the geometric inequality above:

In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of random variables deviates from its expected value.