Chebyshev's inequalityChebyshevsInequality2001-10-17 01:28:422006-12-11 23:48:06TheoremInequality\usepackage{amssymb}
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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):
\begin{itemize}
\item if $x_1<x_2<\cdots<x_n$ and $y_1<y_2<\cdots<y_n$ then$$\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)\left(\frac{y_1+y_2+\cdots+y_n}{n}\right)
\le\frac{x_1y_1+x_2y_2+\cdots+x_ny_n}{n}.$$
\item if $x_1<x_2<\cdots<x_n$ and $y_1>y_2>\cdots>y_n$ then$$\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)\left(\frac{y_1+y_2+\cdots+y_n}{n}\right)
\ge\frac{x_1y_1+x_2y_2+\cdots+x_ny_n}{n}.$$\end{itemize}