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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.
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.
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 2 n . Chebyshev's inequality , on the range of standard deviations around the mean, in statistics
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 statistics, the 68–95–99.7 rule, also known as the empirical rule, and sometimes abbreviated 3sr, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: approximately 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively.
In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if ...
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, 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.