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  2. Chebyshev's inequality - Wikipedia

    en.wikipedia.org/wiki/Chebyshev's_inequality

    By comparison, Chebyshev's inequality states that all but a 1/N fraction of the sample will lie within √ N standard deviations of the mean. Since there are N samples, this means that no samples will lie outside √ N standard deviations of the mean, which is worse than Samuelson's inequality.

  3. Law of large numbers - Wikipedia

    en.wikipedia.org/wiki/Law_of_large_numbers

    5.1 Proof using Chebyshev's inequality assuming finite variance. 5.2 Proof using convergence of characteristic functions. 6 Proof of the strong law. 7 Consequences.

  4. 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 ...

  5. 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.

  6. Markov's inequality - Wikipedia

    en.wikipedia.org/wiki/Markov's_inequality

    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 (sometimes, calling it the first Chebyshev inequality, while referring to Chebyshev's inequality as the second Chebyshev ...

  7. Chebyshev's theorem - Wikipedia

    en.wikipedia.org/wiki/Chebyshev's_theorem

    Chebyshev's sum inequality, about sums and products of decreasing sequences Chebyshev's equioscillation theorem , on the approximation of continuous functions with polynomials The statement that if the function π ( x ) ln ⁡ x / x {\textstyle \pi (x)\ln x/x} has a limit at infinity, then the limit is 1 (where π is the prime-counting function).

  8. Concentration inequality - Wikipedia

    en.wikipedia.org/wiki/Concentration_inequality

    Such inequalities are of importance in several fields, including communication complexity (e.g., in proofs of the gap Hamming problem [13]) and graph theory. [14] An interesting anti-concentration inequality for weighted sums of independent Rademacher random variables can be obtained using the Paley–Zygmund and the Khintchine inequalities. [15]

  9. Cantelli's inequality - Wikipedia

    en.wikipedia.org/wiki/Cantelli's_inequality

    In probability theory, Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for one-sided tail bounds. [1] [2] [3] The inequality states that, for >,