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In probability theory, the law of large numbers (LLN) is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. [1] More formally, the LLN states that given a sample of independent and identically distributed values, the sample mean ...
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.
The Chebyshev inequality is used to prove the weak law of large numbers. [ citation needed ] The Bertrand–Chebyshev theorem (1845, 1852) states that for any n > 3 {\displaystyle n>3} , there exists a prime number p {\displaystyle p} such that n < p < 2 n {\displaystyle n<p<2n} .
Concentration inequality. In probability theory, concentration inequalities provide mathematical bounds on the probability of a random variable deviating from some value (typically, its expected value). The deviation or other function of the random variable can be thought of as a secondary random variable.
The law of the iterated logarithm specifies what is happening "in between" the law of large numbers and the central limit theorem. Specifically it says that the normalizing function √ n log log n, intermediate in size between n of the law of large numbers and √ n of the central limit theorem, provides a non-trivial limiting behavior.
The phenomenon that π 4,3 (x) is ahead most of the time is called Chebyshev's bias. The prime number race generalizes to other moduli and is the subject of much research; Pál Turán asked whether it is always the case that π(x;a,c) and π(x;b,c) change places when a and b are coprime to c. [33] Granville and Martin give a thorough exposition ...
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 ...
Continuous version. There is also a continuous version of Chebyshev's sum inequality: If f and g are real -valued, integrable functions over [a, b], both non-increasing or both non-decreasing, then. with the inequality reversed if one is non-increasing and the other is non-decreasing.