# 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 $Y\geq 0$ is a random variable, and

 $\mathbb{E}[Y]=\operatorname{Var}[X]=\sigma^{2}.$

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

 $\mathbb{P}\left\{\left|X-\mu\right|\geq t\right\}=\mathbb{P}\left\{Y\geq t^{2}% \right\}\leq\frac{1}{t^{2}}\mathbb{E}[Y]=\frac{\sigma^{2}}{t^{2}}.$
Title proof of Chebyshev’s inequality ProofOfChebyshevsInequality 2013-03-22 12:47:58 2013-03-22 12:47:58 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Proof msc 60A99