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: | ||
Revision as of 13:45, 18 June 2006
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.