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The additional fraction of / present in these tail bounds lead to better confidence intervals than Chebyshev's inequality. For example, for any symmetrical unimodal distribution, the Vysochanskij–Petunin inequality states that 4/(9 x 3^2) = 4/81 ≈ 4.9% of the distribution lies outside 3 standard deviations of the mode.
Chebyshev's inequality guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. It was first formulated by his friend and colleague Irénée-Jules Bienaymé in 1853 and proved by Chebyshev in 1867.
In probability theory, the multidimensional Chebyshev's inequality [1] is a generalization of Chebyshev's inequality, which puts a bound on the probability of the event that a random variable differs from its expected value by more than a specified amount.
In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if ...
In fact, Chebyshev's proof works so long as the variance of the average of the first n values goes to zero as n goes to infinity. [15] As an example, assume that each random variable in the series follows a Gaussian distribution (normal distribution) with mean zero, but with variance equal to 2 n / log ( n + 1 ) {\displaystyle 2n/\log(n+1 ...
A weaker three-sigma rule can be derived from Chebyshev's inequality, stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. For unimodal distributions, the probability of being within the interval is at least 95% by the Vysochanskij–Petunin inequality ...
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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. Markov's inequality can also be used to upper bound the expectation of a non-negative random variable in terms of its distribution function.