Chebyshev’s inequality

If $x_1,x_2,\mathrm{\dots },x_n$ and $y_1,y_2,\mathrm{\dots },y_n$ are two sequences (at least one of them consisting of positive numbers):

• if and then

  $$\left(\frac{x_1+x_2+\mathrm{\cdots }+x_n}{n}\right)\left(\frac{y_1+y_2+\mathrm{\cdots }+y_n}{n}\right)\le \frac{x_1{y}_1+x_2{y}_2+\mathrm{\cdots }+x_n{y}_n}{n}.$$

• if and $y_1>y_2>\mathrm{\cdots }>y_n$ then

  $$\left(\frac{x_1+x_2+\mathrm{\cdots }+x_n}{n}\right)\left(\frac{y_1+y_2+\mathrm{\cdots }+y_n}{n}\right)\ge \frac{x_1{y}_1+x_2{y}_2+\mathrm{\cdots }+x_n{y}_n}{n}.$$

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