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In probability theory, Chebyshev's inequality (also spelled as Tchebysheff's inequality, (ロシア語:Нера́венство Чебышёва)) guarantees that in any probability distribution, "nearly all" values are close to the mean — the precise statement being that no more than 1/''k''2 of the distribution's values can be more than ''k'' standard deviations away from the mean (or equivalently, at least 1−1/''k''2 of the distribution's values are within ''k'' standard deviations of the 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 completely arbitrary distributions (unknown except for mean and variance). For example, it can be used to prove the weak law of large numbers. In practical usage, in contrast to the 68-95-99.7% rule, which applies to normal distributions, under Chebyshev's inequality a minimum of just 75% of values must lie within two standard deviations of the mean and 89% within three standard deviations. The term ''Chebyshev's inequality'' may also refer to Markov's inequality, especially in the context of analysis. ==History== The theorem is named after Russian mathematician Pafnuty Chebyshev, although it was first formulated by his friend and colleague Irénée-Jules Bienaymé.〔 〕 The theorem was first stated without proof by Bienaymé in 1853〔Bienaymé I.-J. (1853) Considérations àl'appui de la découverte de Laplace. Comptes Rendus de l'Académie des Sciences 37: 309–324 〕 and later proved by Chebyshev in 1867. His student Andrey Markov provided another proof in his 1884 Ph.D. thesis.〔Markov A. (1884) On certain applications of algebraic continued fractions, Ph.D. thesis, St. Petersburg〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Chebyshev's inequality」の詳細全文を読む スポンサード リンク
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