Chebyshev's Inequality
(Redirected from Tchebyshev's Inequality)
Jump to navigation
Jump to search
Disambiguation
This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.
Chebyshev's Inequality may refer to:
Bienaymé-Chebyshev Inequality
Let $X$ be a random variable.
Let $\expect X = \mu$ for some $\mu \in \R$.
Let $\var X = \sigma^2$ for some $\sigma^2 \in \R_{> 0}$.
Then, for all $k > 0$:
- $\map \Pr {\size {X - \mu} \ge k \sigma} \le \dfrac 1 {k^2}$
Chebyshev's Sum Inequality: Discrete Version
Let $a_1, a_2, \ldots, a_n$ be real numbers such that:
- $a_1 \ge a_2 \ge \cdots \ge a_n$
Let $b_1, b_2, \ldots, b_n$ be real numbers such that:
- $b_1 \ge b_2 \ge \cdots \ge b_n$
Then:
- $\ds \dfrac 1 n \sum_{k \mathop = 1}^n a_k b_k \ge \paren {\dfrac 1 n \sum_{k \mathop = 1}^n a_k} \paren {\dfrac 1 n \sum_{k \mathop = 1}^n b_k}$
Chebyshev's Sum Inequality: Continuous Version
Let $u, v: \closedint 0 1 \to \R$ be integrable functions.
This article, or a section of it, needs explaining. In particular: Is it important in what sense integrable: Darboux, Lebesgue, etc.? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Let $u$ and $v$ both be either increasing or decreasing.
Then:
- $\ds \paren {\int_0^1 u \rd x} \cdot \paren {\int_0^1 v \rd x} \le \int_0^1 u v\rd x$
Source of Name
This entry was named for Pafnuty Lvovich Chebyshev.