Editing Chebyshev's inequality (section)

===Finite-dimensional vector===
{{main|Multidimensional Chebyshev's inequality}}

Chebyshev's inequality naturally extends to the multivariate setting, where one has ''n'' random variables {{mvar|X<sub>i</sub>}} with mean {{mvar|μ<sub>i</sub>}} and variance ''σ''<sub>i</sub><sup>2</sup>. Then the following inequality holds.

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

This is known as the Birnbaum–Raymond–Zuckerman inequality after the authors who proved it for two dimensions.<ref name=Birnbaum1947>{{cite journal |last1=Birnbaum |first1=Z. W. |last2=Raymond |first2=J. |last3=Zuckerman |first3=H. S. |title=A Generalization of Tshebyshev's Inequality to Two Dimensions |journal=The Annals of Mathematical Statistics |issn=0003-4851 |year=1947 |volume=18 |issue=1 |pages=70–79 |doi=10.1214/aoms/1177730493 |mr=19849 |zbl=0032.03402 |url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177730493 |access-date=7 October 2012|doi-access=free }}</ref> This result can be rewritten in terms of [[Multivariate random variable|vectors]] {{math|''X'' {{=}} (''X''<sub>1</sub>, ''X''<sub>2</sub>, ...)}} with mean {{math|''μ'' {{=}} (''μ''<sub>1</sub>, ''μ''<sub>2</sub>, ...)}}, standard deviation ''σ'' = (''σ''<sub>1</sub>, ''σ''<sub>2</sub>, ...), in the Euclidean norm {{math|{{!!}} ⋅ {{!!}}}}.<ref name=Ferentinos1982>{{cite journal | last1 = Ferentinos | first1 = K | year = 1982 | title = On Tchebycheff type inequalities | journal = Trabajos Estadıst Investigacion Oper | volume = 33 | pages = 125–132 | doi = 10.1007/BF02888707 | s2cid = 123762564 }}</ref>

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

One can also get a similar [[Multidimensional Chebyshev's inequality#Infinite dimensions|infinite-dimensional Chebyshev's inequality]]. A second related inequality has also been derived by Chen.<ref name=Chen2007>{{cite arXiv |author=Xinjia Chen |eprint=0707.0805v2 |title=A New Generalization of Chebyshev Inequality for Random Vectors |year=2007 |class=math.ST }}</ref> Let {{mvar|n}} be the [[dimension]] of the stochastic vector {{mvar|X}} and let {{math|E(''X'')}} be the mean of {{mvar|X}}. Let {{mvar|S}} be the [[covariance matrix]] and {{math|''k'' > 0}}. Then

: <math> \Pr \left( ( X - \operatorname{E}(X) )^T S^{-1} (X - \operatorname{E}(X)) < k  \right) \ge 1 - \frac{n}{k} </math>

where ''Y''<sup>T</sup> is the [[transpose]] of {{mvar|Y}}.
The inequality can be written in terms of the [[Mahalanobis distance]] as

: <math> \Pr \left( d^2_S(X,\operatorname{E}(X)) < k  \right) \ge 1 - \frac{n}{k} </math>

where the Mahalanobis distance based on S is defined by

: <math> d_S(x,y) =\sqrt{ (x -y)^T S^{-1} (x -y) } </math>

Navarro<ref name=Navarro2014>{{cite journal |author=Jorge Navarro |volume=91 |pages=1–5 |title=Can the bounds in the multivariate Chebyshev inequality be attained?   |journal=Statistics and Probability Letters  |year=2014 |doi=10.1016/j.spl.2014.03.028}}</ref> proved that these bounds are sharp, that is, they are the best possible bounds for that regions when we just know the mean and the covariance matrix of X.

Stellato et al.<ref name=":0">{{Cite journal|last1=Stellato|first1=Bartolomeo|last2=Parys|first2=Bart P. G. Van|last3=Goulart|first3=Paul J.|date=2016-05-31|title=Multivariate Chebyshev Inequality with Estimated Mean and Variance|journal=The American Statistician|volume=71|issue=2|pages=123–127|doi=10.1080/00031305.2016.1186559|issn=0003-1305|arxiv=1509.08398|s2cid=53407286}}</ref> showed that this multivariate version of the Chebyshev inequality can be easily derived analytically as a special case of Vandenberghe et al.<ref>{{Cite journal|last1=Vandenberghe|first1=L.|last2=Boyd|first2=S.|last3=Comanor|first3=K.|date=2007-01-01|title=Generalized Chebyshev Bounds via Semidefinite Programming|journal=SIAM Review|volume=49|issue=1|pages=52–64|doi=10.1137/S0036144504440543|issn=0036-1445|bibcode=2007SIAMR..49...52V|citeseerx=10.1.1.126.9105}}</ref> where the bound is computed by solving a [[Semidefinite programming|semidefinite program (SDP).]]

==== 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>