Difference between revisions of "Chebyshev's Inequality"
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| − | Chebyshev's inequality, named after [[Pafnuty Chebyshev]], states that if | + | '''Chebyshev's inequality''', named after [[Pafnuty Chebyshev]], states that if |
<math> a_1\geq a_2\geq ... \geq a_n </math> and <math> b_1\geq b_2\geq ... \geq b_n </math> then the following inequality holds: | <math> a_1\geq a_2\geq ... \geq a_n </math> and <math> b_1\geq b_2\geq ... \geq b_n </math> then the following inequality holds: | ||
| − | <math> | + | <math>n \left(\sum_{i=1}^{n}a_ib_i\right)\geq\left(\sum_{i=1}^{n}a_i\right)\left(\sum_{i=1}^{n}b_i\right)</math>. |
On the other hand, if <math> a_1\geq a_2\geq ... \geq a_n </math> and | On the other hand, if <math> a_1\geq a_2\geq ... \geq a_n </math> and | ||
<math> b_n\geq b_{n-1}\geq ... \geq b_1 </math> then: | <math> b_n\geq b_{n-1}\geq ... \geq b_1 </math> then: | ||
| − | <math> | + | <math>n \left(\sum_{i=1}^{n}a_ib_i\right)\leq\left(\sum_{i=1}^{n}a_i\right)\left(\sum_{i=1}^{n}b_i\right)</math>. |
==Proof== | ==Proof== | ||
Chebyshev's inequality is a consequence of the [[Rearrangement inequality]], which gives us that the sum <math>S=a_1b_{i_1}+a_2b_{i_2}+...+a_nb_{i_n} </math> is maximal when <math>i_k=k</math>. | Chebyshev's inequality is a consequence of the [[Rearrangement inequality]], which gives us that the sum <math>S=a_1b_{i_1}+a_2b_{i_2}+...+a_nb_{i_n} </math> is maximal when <math>i_k=k</math>. | ||
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Now, by adding the inequalities: | Now, by adding the inequalities: | ||
| − | <math> | + | <math>\sum_{i=1}^{n}a_ib_i\geq a_1b_1+a_2b_2+...+a_n b_{n}</math> |
| − | <math> | + | <math>\sum_{i=1}^{n}a_ib_i\geq a_1b_2+a_2b_3+...+a_nb_1</math> |
| − | + | <math>\cdots</math> | |
| − | <math> | + | <math>\sum_{i=1}^{n}a_ib_i\geq a_1b_n+a_2b_1+...+a_nb_{n-1}</math> |
we get the initial inequality. | we get the initial inequality. | ||
| + | |||
| + | [[Category:Algebra]] | ||
| + | [[Category:Inequalities]] | ||
Latest revision as of 19:32, 13 March 2022
Chebyshev's inequality, named after Pafnuty Chebyshev, states that if
and 
then the following inequality holds:
.
On the other hand, if and
then:
.
Proof
Chebyshev's inequality is a consequence of the Rearrangement inequality, which gives us that the sum 
is maximal when 
.
Now, by adding the inequalities:
we get the initial inequality.
