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In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev (Markov's teacher), and many sources, especially in analysis, refer to it as Chebyshev's inequality (sometimes, calling it the first Chebyshev inequality, while referring to the Chebyshev's inequality as the second Chebyshev's inequality) or Bienaymé's inequality. Markov's inequality (and other similar inequalities) relate probabilities to expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable. An example of an application of Markov's inequality is the fact that (assuming incomes are non-negative) no more than 1/5 of the population can have more than 5 times the average income. ==Statement== If is any nonnegative integrable random variable and , then : In the language of measure theory, Markov's inequality states that if is a measure space, is a measurable extended real-valued function, and , then : (This measure theoretic definition may sometimes be referred to as Chebyshev's inequality .〔E.M. Stein, R. Shakarchi, "Real Analysis, Measure Theory, Integration, & Hilbert Spaces", vol. 3, 1st ed., 2005, p.91〕) 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「Markov's inequality」の詳細全文を読む スポンサード リンク
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