# Chebyshev’s inequality

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} and $y_{1} then

 $\left(\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)\left(\frac{y_{1}+y_{2}+\cdots+% y_{n}}{n}\right)\leq\frac{x_{1}y_{1}+x_{2}y_{2}+\cdots+x_{n}y_{n}}{n}.$
• if $x_{1} and $y_{1}>y_{2}>\cdots>y_{n}$ then

 $\left(\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)\left(\frac{y_{1}+y_{2}+\cdots+% y_{n}}{n}\right)\geq\frac{x_{1}y_{1}+x_{2}y_{2}+\cdots+x_{n}y_{n}}{n}.$
 Title Chebyshev’s inequality  Canonical name ChebyshevsInequality Date of creation 2013-03-22 11:47:36 Last modified on 2013-03-22 11:47:36 Owner drini (3) Last modified by drini (3) Numerical id 7 Author drini (3) Entry type Theorem Classification msc 26D15 Classification msc 18F99 Classification msc 58Z05 Related topic RearrangementInequality Related topic ProofOfRearrangementInequality Related topic KolmogorovsInequality Related topic ChebyshevsInequality2