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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).
The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.
His conjecture was completely proved by Chebyshev (1821–1894) in 1852 [3] and so the postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem. Chebyshev's theorem can also be stated as a relationship with π ( x ) {\displaystyle \pi (x)} , the prime-counting function (number of primes less than or equal to x ...
In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if ... This page was last edited on 11 December 2024, at 17:02 (UTC).
In mathematics, Bertrand's postulate (now a theorem) states that, for each , there is a prime such that < <. First conjectured in 1845 by Joseph Bertrand, [1] it was first proven by Chebyshev, and a shorter but also advanced proof was given by Ramanujan. [2]
This theorem makes rigorous the intuitive notion of probability as the expected long-run relative frequency of an event's occurrence. It is a special case of any of several more general laws of large numbers in probability theory. Chebyshev's inequality. Let X be a random variable with finite expected value μ and finite non-zero variance σ 2.
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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.