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Chebyshev's inequality
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==History== The theorem is named after Russian mathematician [[Pafnuty Chebyshev]], although it was first formulated by his friend and colleague [[Irénée-Jules Bienaymé]].<ref>{{cite book |title=The Art of Computer Programming: Fundamental Algorithms, Volume 1 |edition=3rd |last1=Knuth |first1=Donald |author-link1=Donald Knuth |year=1997 |publisher=Addison–Wesley |location=Reading, Massachusetts |isbn=978-0-201-89683-1 |url=http://www-cs-faculty.stanford.edu/~uno/taocp.html |access-date=1 October 2012 |ref=KnuthTAOCP1 |archive-date=26 February 2009 |archive-url=https://web.archive.org/web/20090226183954/http://www-cs-faculty.stanford.edu/~uno/taocp.html |url-status=dead }}</ref>{{rp|98}} The theorem was first proved by Bienaymé in 1853<ref name="Bienaymé1853">{{cite journal | last1 = Bienaymé | first1 = I.-J. | year = 1853 | title = Considérations àl'appui de la découverte de Laplace | journal = Comptes Rendus de l'Académie des Sciences | volume = 37 | pages = 309–324 }}</ref> and more generally proved by Chebyshev in 1867.<ref name=Chebyshev1867>{{cite journal|last=Tchebichef|first=P.|title=Des valeurs moyennes|journal=Journal de Mathématiques Pures et Appliquées|year=1867|volume=12|series=2|pages=177–184}}</ref><ref>{{Cite book| last = Routledge| first = Richard| title = Chebyshev's inequality| url = https://www.britannica.com/science/Chebyshevs-inequality|publisher = Encyclopedia Britannica}}</ref> His student [[Andrey Markov]] provided another proof in his 1884 Ph.D. thesis.<ref name=Markov1884>Markov A. (1884) On certain applications of algebraic continued fractions, Ph.D. thesis, St. Petersburg</ref>
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