proof of Chebyshev’s inequality

The proof of Chebyshev’s inequality follows from the application of Markov’s inequality.

Define $Y={(X-\mu )}^2$. Then $Y\ge 0$ is a random variable, and

$$𝔼[Y]=\mathrm{Var}[X]={\sigma }^2.$$

Applying Markov’s inequality to $Y$, we see that

$$\mathbb{P}\left\{\right|X-\mu |\ge t\}=\mathbb{P}\{Y\ge t^2\}\le \frac{1}{t^2}𝔼[Y]=\frac{{\sigma }^2}{t^2}.$$

Title	proof of Chebyshev’s inequality
Canonical name	ProofOfChebyshevsInequality
Date of creation	2013-03-22 12:47:58
Last modified on	2013-03-22 12:47:58
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	6
Author	PrimeFan (13766)
Entry type	Proof
Classification	msc 60A99