Editing Chebyshev's inequality (section)

===Bounds for specific distributions===

DasGupta has shown that if the distribution is known to be normal<ref name=DasGupta2000>{{cite journal | last1 = DasGupta | first1 = A | year = 2000 | title = Best constants in Chebychev inequalities with various applications | journal = Metrika | volume = 5 | issue = 1| pages = 185–200 | doi = 10.1007/s184-000-8316-9 | s2cid = 121436601 }}</ref>

: <math> \Pr( | X - \mu | \ge k \sigma ) \le \frac{ 1 }{ 3 k^2 } .</math>

From DasGupta's inequality it follows that for a normal distribution at least 95% lies within approximately 2.582 standard deviations of the mean. This is less sharp than the true figure (approximately 1.96 standard deviations of the mean).

*DasGupta has determined a set of best possible bounds for a [[normal distribution]] for this inequality.<ref name=DasGupta2000 />
*Steliga and Szynal have extended these bounds to the [[Pareto distribution]].<ref name=Steliga2010>{{cite journal |last1=Steliga |first1=Katarzyna |last2=Szynal |first2=Dominik |title=On Markov-Type Inequalities |journal=International Journal of Pure and Applied Mathematics |year=2010 |volume=58 |issue=2 |pages=137–152 |url=http://ijpam.eu/contents/2010-58-2/2/2.pdf |access-date=10 October 2012 |issn=1311-8080}}</ref>
*Grechuk et al. developed a general method for deriving the best possible bounds in Chebyshev's inequality for any family of distributions, and any [[deviation risk measure]] in place of standard deviation. In particular, they derived Chebyshev inequality for distributions with [[Logarithmically concave function|log-concave]] densities.<ref name="cheb">Grechuk, B., Molyboha, A., Zabarankin, M. (2010).
[https://www.researchgate.net/publication/231939730_Chebyshev_inequalities_with_law-invariant_deviation_measures Chebyshev Inequalities with Law Invariant Deviation Measures], Probability in the Engineering and Informational Sciences, 24(1), 145-170.</ref>