# Pafnuty Chebyshev and the Chebyshev Inequality

Pafnuty Chebyshev (1821 – 1894)

On May 16, 1821Russian mathematician Pafnuty Lvovich Chebyshev was born. Chebyshev is remembered primarily for his work on the theory of prime numbers, including the determination of the number of primes not exceeding a given number. Moreover, he is noted for his work in the fields of probability, statistics, mechanics, and number theory.

Pafnuty Chebyshev studied mathematical science at the University of Moscow starting from 1937. He later became Chebyshev became assistant professor of mathematics at the University of St. Petersburg. In 1874, Chebyshev became a foreign associate of the Institut de France.

Chebyshev is probably best known for developing an inequality of probability theory which was named Chebyshev’s inequality. It guarantees that, for a wide class of probability distributions, “nearly all” values are close to the mean. More exactly, no more than 1/k2 of the distribution’s values can be more than k standard deviations away from the mean. The inequality 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. Even though the theorem is named after Chebyshev, it was first formulated by his colleague Irénée-Jules Bienaymé who first stated the theorem without proof in 1853. Pafnuty Chebyshev later provided the proof and his student Andrey Markov provided another proof in his 1884 Ph.D. thesis. Further, the Bertrand–Chebyshev theorem states that for any n > 1, , there exists a prime number p such that n < p < 2 n. This is a consequence of the Chebyshev inequalities for the number π(n) of prime numbers less than n, which state that π(n) is of the order of n/log(n). A more precise form is given by the celebrated prime number theorem: the quotient of the two expressions approaches 1 as n tends to infinity.

Chebyshev is considered as one of the founding fathers of Russian mathematics. Among his well-known students were the mathematicians Dmitry Grave, Aleksandr Korkin, Aleksandr Lyapunov, and Andrei Markov. According to the Mathematics Genealogy Project, Chebyshev has 10,629 mathematical “descendants” as of 2015

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