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* Markov's inequality and Chebyshev's inequality
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Markov's and inequality
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.
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.
inequality and Chebyshev's
* Chebyshev's inequality
* Chebyshev's inequality in probability and statistics
* Chebyshev's sum inequality
Chebyshev's inequality is usually stated for random variables, but can be generalized to a statement about measure spaces.
Several extensions of Chebyshev's inequality have been developed.
( This measure theoretic definition may sometimes be referred to as Chebyshev's inequality
Chebyshev's inequality uses the variance to bound the probability that a random variable deviates far from the mean.
* Chebyshev's inequality
* Chebyshev's sum inequality
In this respect, scenario analysis tries to defer statistical laws ( e. g., Chebyshev's inequality Law ), because the decision rules occur outside a constrained setting.
* The coarse result of Chebyshev's inequality that, for any probability distribution, the probability of an outcome greater than k standard deviations from the mean is at most 1 / k < sup > 2 </ sup >.
The theorem refines Chebyshev's inequality by including the factor of 4 / 9, made possible by the condition that the distribution be unimodal.
Without unimodality Chebyshev's inequality would give a looser bound of 1 / 9 = 0. 11111 ….
* Where the probability distribution is unknown, relationships like Chebyshev's or the Vysochanskiï – Petunin inequality can be used to calculate a conservative confidence interval
# redirect Chebyshev's inequality
In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if
There is also a continuous version of Chebyshev's sum inequality:
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