Difference between revisions of "Chebyshev's Inequality"
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Now, by adding the inequalities: | Now, by adding the inequalities: | ||
| − | <math>\sum_{i=1}^{n}a_ib_i\geq a_1b_1+a_2b_2+...+a_n b_{n}</math> | + | <math>\sum_{i=1}^{n}a_ib_i\geq a_1b_1+a_2b_2+...+a_n b_{n}</math> |
| − | <math>\sum_{i=1}^{n}a_ib_i\geq a_1b_2+a_2b_3+...+a_nb_1</math> | + | <math>\sum_{i=1}^{n}a_ib_i\geq a_1b_2+a_2b_3+...+a_nb_1</math> |
| − | + | <math>\cdots</math> | |
<math>\sum_{i=1}^{n}a_ib_i\geq a_1b_n+a_2b_1+...+a_nb_{n-1}</math> | <math>\sum_{i=1}^{n}a_ib_i\geq a_1b_n+a_2b_1+...+a_nb_{n-1}</math> | ||
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we get the initial inequality. | we get the initial inequality. | ||
| − | [[Category: | + | [[Category:Algebra]] |
| − | [[Category: | + | [[Category:Inequalities]] |
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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.
