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  2. Chebyshev's inequality - Wikipedia

    en.wikipedia.org/wiki/Chebyshev's_inequality

    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.

  3. List of examples of Stigler's law - Wikipedia

    en.wikipedia.org/wiki/List_of_examples_of_Stigler...

    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.

  4. Multidimensional Chebyshev's inequality - Wikipedia

    en.wikipedia.org/wiki/Multidimensional_Chebyshev...

    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.

  5. Chebyshev's sum inequality - Wikipedia

    en.wikipedia.org/wiki/Chebyshev's_sum_inequality

    In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if ...

  6. Law of large numbers - Wikipedia

    en.wikipedia.org/wiki/Law_of_large_numbers

    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 ...

  7. 68–95–99.7 rule - Wikipedia

    en.wikipedia.org/wiki/68–95–99.7_rule

    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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  9. Markov's inequality - Wikipedia

    en.wikipedia.org/wiki/Markov'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. Markov's inequality can also be used to upper bound the expectation of a non-negative random variable in terms of its distribution function.