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Difference between revisions of "Chebyshev inequality"

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''for finite monotone sequences
 
''for finite monotone sequences
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021880/c0218801.png" /></td> </tr></table>
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$$a_1\leq\dotsb\leq a_n,\quad b_1\leq\dotsb\leq b_n$$
  
 
''
 
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The inequality
 
The inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021880/c0218802.png" /></td> </tr></table>
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$$\sum_{k=1}^na_k\sum_{k=1}^nb_k\leq n\sum_{k=1}^na_kb_k.$$
  
Chebyshev's inequality for monotone functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021880/c0218803.png" /> is the inequality
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Chebyshev's inequality for monotone functions $f,g\geq0$ is the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021880/c0218804.png" /></td> </tr></table>
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$$\int\limits_a^bf(x)dx\int\limits_a^bg(x)dx\leq(b-a)\int\limits_a^bf(x)g(x)dx,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021880/c0218805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021880/c0218806.png" /> are either both increasing or both decreasing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021880/c0218807.png" />.
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where $f$ and $g$ are either both increasing or both decreasing on $[a,b]$.
  
 
The inequalities were established by P.L. Chebyshev in 1882.
 
The inequalities were established by P.L. Chebyshev in 1882.
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====Comments====
 
====Comments====
It is not important that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021880/c0218808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021880/c0218809.png" /> be non-negative. The proof consists of simply integrating the non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021880/c02188010.png" /> over the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021880/c02188011.png" />.
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It is not important that $f$ and $g$ be non-negative. The proof consists of simply integrating the non-negative function $\tau(x,y)=[f(x)-f(y)][g(x)-g(y)]$ over the square $[a,b]\times[a,b]$.

Latest revision as of 12:59, 14 February 2020

for finite monotone sequences

$$a_1\leq\dotsb\leq a_n,\quad b_1\leq\dotsb\leq b_n$$

The inequality

$$\sum_{k=1}^na_k\sum_{k=1}^nb_k\leq n\sum_{k=1}^na_kb_k.$$

Chebyshev's inequality for monotone functions $f,g\geq0$ is the inequality

$$\int\limits_a^bf(x)dx\int\limits_a^bg(x)dx\leq(b-a)\int\limits_a^bf(x)g(x)dx,$$

where $f$ and $g$ are either both increasing or both decreasing on $[a,b]$.

The inequalities were established by P.L. Chebyshev in 1882.


Comments

It is not important that $f$ and $g$ be non-negative. The proof consists of simply integrating the non-negative function $\tau(x,y)=[f(x)-f(y)][g(x)-g(y)]$ over the square $[a,b]\times[a,b]$.

How to Cite This Entry:
Chebyshev inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_inequality&oldid=15534
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article