Editing Chebyshev's inequality (section)

===Exponential moment===
A related inequality sometimes known as the exponential Chebyshev's inequality<ref name=RassoulAgha2010>[http://www.math.utah.edu/~firas/Papers/rassoul-seppalainen-ldp.pdf  Section 2.1] {{webarchive |url=https://web.archive.org/web/20150430075226/http://www.math.utah.edu/~firas/Papers/rassoul-seppalainen-ldp.pdf |date=April 30, 2015 }}</ref> is the inequality

:<math> \Pr(X \ge \varepsilon) \le e^{ -t \varepsilon }\operatorname{E}\left (e^{ t X } \right), \qquad t > 0.</math>

Let {{math|''K''(''t'')}} be the [[cumulant generating function]],

: <math> K( t ) = \log \left(\operatorname{E}\left( e^{ t x } \right) \right). </math>

Taking the [[Legendre–Fenchel transformation]]{{clarify|reason=articles should be reasonably self contained, more explanation needed|date=May 2012}} of {{math|''K''(''t'')}} and using the exponential Chebyshev's inequality we have

: <math>-\log( \Pr (X \ge \varepsilon )) \ge \sup_t( t \varepsilon - K( t ) ). </math>

This inequality may be used to obtain exponential inequalities for unbounded variables.<ref name=Baranoski2001>{{cite journal |last1=Baranoski |first1=Gladimir V. G. |last2=Rokne |first2=Jon G. |last3=Xu |first3=Guangwu |title=Applying the exponential Chebyshev inequality to the nondeterministic computation of form factors |journal=Journal of Quantitative Spectroscopy and Radiative Transfer |date=15 May 2001 |volume=69 |issue=4 |pages=199–200 |doi=10.1016/S0022-4073(00)00095-9 |bibcode=2001JQSRT..69..447B}} (the references for this article are corrected by {{cite journal|last1=Baranoski |first1=Gladimir V. G. |last2=Rokne |first2=Jon G. |author3=Guangwu Xu |title=Corrigendum to: 'Applying the exponential Chebyshev inequality to the nondeterministic computation of form factors' |journal=Journal of Quantitative Spectroscopy and Radiative Transfer |date=15 January 2002 |volume=72 |issue=2 |pages=199–200 |doi=10.1016/S0022-4073(01)00171-6 |bibcode=2002JQSRT..72..199B|doi-access=free }})</ref>