Editing Chebyshev's inequality (section)

==== Known correlation ====

If the variables are independent this inequality can be sharpened.<ref name=Kotz2000>{{cite book |last1=Kotz |first1=Samuel  |author-link1=Samuel Kotz |last2=Balakrishnan |first2=N. |last3= Johnson |first3=Norman L. |author-link3=Norman Lloyd Johnson |title=Continuous Multivariate Distributions, Volume 1, Models and Applications |year=2000 |publisher=Houghton Mifflin |location=Boston [u.a.] |isbn=978-0-471-18387-7 |url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471183873.html |edition=2nd |access-date=7 October 2012}}</ref>

:<math>\Pr\left (\bigcap_{i = 1}^n \frac{|X_i - \mu_i|}{\sigma_i} \le k_i \right ) \ge \prod_{i=1}^n \left (1 - \frac{1}{k_i^2} \right)</math>

Berge derived an inequality for two correlated variables {{math|''X''<sub>1</sub>, ''X''<sub>2</sub>}}.<ref name=Berge1938>{{cite journal | last1 = Berge | first1 = P. O. | year = 1938 | title = A note on a form of Tchebycheff's theorem for two variables | journal = Biometrika | volume = 29 | issue = 3/4| pages = 405–406 | doi=10.2307/2332015| jstor = 2332015 }}</ref> Let {{mvar|ρ}} be the correlation coefficient between ''X''<sub>1</sub> and ''X''<sub>2</sub> and let ''σ''<sub>''i''</sub><sup>2</sup> be the variance of {{mvar|X<sub>i</sub>}}. Then

: <math> \Pr\left( \bigcap_{ i = 1}^2 \left[ \frac{ | X_i - \mu_i | } { \sigma_i }  < k \right] \right) \ge 1 - \frac{ 1 + \sqrt{ 1 - \rho^2 } } { k^2 }.</math>

This result can be sharpened to having different bounds for the two random variables<ref name=Lal1955>Lal D. N. (1955) A note on a form of Tchebycheff's inequality for two or more variables. [[Sankhya (journal)|Sankhya]] 15(3):317–320</ref> and having asymmetric bounds, as in Selberg's inequality.
<ref name=Isii1959>Isii K. (1959) On a method for generalizations of Tchebycheff's inequality. Ann Inst Stat Math 10: 65–88</ref>

Olkin and Pratt derived an inequality for {{mvar|n}} correlated variables.<ref name=Olkin1958>{{cite journal|last1=Olkin|first1=Ingram |author-link1=Ingram Olkin | last2=Pratt |first2=John W. |author-link2=John W. Pratt |title=A Multivariate Tchebycheff Inequality| journal=The Annals of Mathematical Statistics|year=1958|volume=29|issue=1|pages=226–234|doi=10.1214/aoms/1177706720|zbl=0085.35204 |mr=93865 |doi-access=free}}</ref>

: <math> \Pr\left(\bigcap_{i = 1 }^n \frac{|X_i - \mu_i|}{\sigma_i} < k_i \right) \ge 1 - \frac{1}{n^2} \left(\sqrt{u} + \sqrt{n-1} \sqrt{n \sum_i \frac 1 { k_i^2} - u} \right)^2 </math>

where the sum is taken over the ''n'' variables and

: <math> u = \sum_{i=1}^n \frac{1}{ k_i^2} + 2\sum_{i=1}^n \sum_{j<i} \frac{\rho_{ij}}{k_i k_j} </math>

where {{mvar|ρ<sub>ij</sub>}} is the correlation between {{mvar|X<sub>i</sub>}} and {{mvar|X<sub>j</sub>}}.

Olkin and Pratt's inequality was subsequently generalised by Godwin.<ref name=Godwin1964>Godwin H. J. (1964) Inequalities on distribution functions. New York, Hafner Pub. Co.</ref>