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Chebyshev's inequality
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(Theorem)
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If $x_1,x_2,\ldots,x_n$ and $y_1,y_2,\ldots,y_n$ are two sequences (at least one of them consisting of positive numbers):
- if $x_1<x_2<\cdots<x_n$ and $y_1<y_2<\cdots<y_n$ then$$\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)\left(\frac{y_1+y_2+\cdots+y_n}{n}\right) \le\frac{x_1y_1+x_2y_2+\cdots+x_ny_n}{n}.$$
- if $x_1<x_2<\cdots<x_n$ and $y_1>y_2>\cdots>y_n$ then$$\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)\left(\frac{y_1+y_2+\cdots+y_n}{n}\right) \ge\frac{x_1y_1+x_2y_2+\cdots+x_ny_n}{n}.$$
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"Chebyshev's inequality" is owned by drini. [ full author list (2) | owner history (1) ]
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Cross-references: numbers, positive, sequences
There are 3 references to this entry.
This is version 2 of Chebyshev's inequality, born on 2001-10-17, modified 2006-12-11.
Object id is 277, canonical name is ChebyshevsInequality.
Accessed 17460 times total.
Classification:
| AMS MSC: | 26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals) |
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Pending Errata and Addenda
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