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The bounds these inequalities give on a finite sample are less tight than those the Chebyshev inequality gives for a distribution. To illustrate this let the sample size N = 100 and let k = 3. Chebyshev's inequality states that at most approximately 11.11% of the distribution will lie at least three standard deviations away from the mean.
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 ...
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.
In mathematical analysis, the Chebyshev–Markov–Stieltjes inequalities are inequalities related to the problem of moments that were formulated in the 1880s by Pafnuty Chebyshev and proved independently by Andrey Markov and (somewhat later) by Thomas Jan Stieltjes. [1]
Chebyshev is known for his work in the fields of probability, statistics, mechanics, and number theory. The Chebyshev inequality states that if is a random variable with standard deviation σ > 0, then the probability that the outcome of is = or more away from its mean is at most / = /:
Chebyshev's theorem is any of several theorems proven by Russian mathematician Pafnuty Chebyshev. Bertrand's postulate, that for every n there is a prime between n and 2n. Chebyshev's inequality, on the range of standard deviations around the mean, in statistics; Chebyshev's sum inequality, about sums and products of decreasing sequences
Change detection. Change detection (GIS) Chapman–Kolmogorov equation; Chapman–Robbins bound; Characteristic function (probability theory) Chauvenet's criterion; Chebyshev center; Chebyshev's inequality; Checking if a coin is biased – redirects to Checking whether a coin is fair; Checking whether a coin is fair; Cheeger bound; Chemometrics