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Chebyshev's inequality follows from Markov's inequality by considering the random variable
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Chebyshev's and inequality
* 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.
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 >.
Several extensions of Chebyshev's inequality have been developed.
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.
( 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.
* Markov's inequality and Chebyshev's inequality
* 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:
Chebyshev's and by
His conjecture was completely proved by Chebyshev ( 1821 – 1894 ) in 1850 and so the postulate is also called the Bertrand-Chebyshev theorem or Chebyshev's theorem.
Chebyshev's theorem is a name given to several theorems proven by Russian mathematician Pafnuty Chebyshev
inequality and follows
This follows directly from the triangle inequality.
This follows from the triangle inequality and homogeneity of the norm.
from which readily follows the Cauchy – Schwarz inequality.
The triangle inequality for the inner product is often shown as a consequence of the Cauchy – Schwarz inequality, as follows: given vectors x and y:
The desired inequality then follows from the properties of positive 2 × 2 ( operator ) matrices.
The desired inequality follows from dividing the above inequality by g ( t ).
* The inequality at the heart of the uncertainty principle of quantum mechanics follows from the properties of Fourier integrals and from assuming time invariance.
( Note that this particular case of the Bernoulli distribution has the lowest possible excess kurtosis ; this can be proved by Jensen's inequality as follows.
This inequality can be formulated as follows.
For a classical model we can prove the Bogoliubov inequality as follows.
( This follows from the Kraft-McMillan inequality.
In measure-theoretic terms, Boole's inequality follows from the fact that a measure ( and certainly any probability measure ) is σ-sub-additive.
From Jensen's inequality it follows that and, for, we have
The best estimate, C < 0. 4784, follows from the inequality
More generally, by Hölder's inequality, it follows that if ƒ ∈ L < sup > p </ sup >( a, b ), then I < sup > α </ sup > ƒ ∈ L < sup > p </ sup >( a, b ) as well, and the analogous inequality holds
We claim that without loss of generality, the latter inequality is always strict ; once we do this the theorem can be proved as follows.
A simple proof of this follows from the crossing number inequality: if m cells have a total of x + n edges, one can form a graph with m nodes ( one per cell ) and x edges ( one per pair of consecutive cells on the same line ).
it follows immediately from the preceding inequality that the particle associated with the wave should possess an energy which is not perfectly defined ( since different frequencies are involved in the superposition ) and consequently there is indeterminacy in energy:
The latter formulation follows from the former through an application of Hölder's inequality and a duality argument.
Bessel's inequality follows from the identity:
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