| Chebyshev's Inequality |
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In Probability Theory , Chebyshev's inequality (also known as '''Tchebysheff's inequality''', '''Chebyshev's theorem''', or the '''Bienaymé-Chebyshev inequality'''), named after Pafnuty Chebyshev , who first proved it, states that in any data sample or Probability Distribution , nearly all the values are close to the Mean Value , and provides a quantitative description of "nearly all" and "close to". For example, no more than 1/4 of the values are more than 2 Standard Deviation s away from the mean, no more than 1/9 are more than 3 standard deviations away, no more than 1/25 are more than 5 standard deviations away, and so on. GENERAL STATEMENT The inequality can be stated quite generally using Measure Theory ; the statement in the language of probability theory then follows as a particular case, for a space of measure 1. Measure-theoretic statement Let (''X'',Σ,μ) be a Measure Space , and let ''f'' be an Extended Real -valued Measurable Function defined on ''X''. Then for any real number ''t'' > 0, | ||
| Let ''A''<sub>''t''</sub> Be Defined As ''A''<sub>''t''</sub> | {''x'' ∈ ''X'' ''f''(''x'') ≥ ''t''}, and let |
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| "http://wwwinformationdelightinfo/encyclopedia/entry/Markov's_inequality" class="copylinks">Markov's Inequality states that for any real-valued random variable ''Y'' and any positive number ''a'', we have Pr(''Y'' > ''a'') ≤ E(''Y'')/''a'' One way to prove Chebyshev's inequality is to apply Markov's inequality to the random variable ''Y'' = (''X'' − μ)<sup>2</sup> with ''a'' = (σ''k'')<sup>2</sup> |
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| ::<math>\Pr(X-\mu \geq K\sigma) | \operatorname{E}(I_{X-\mu \geq k\sigma}) |
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