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# 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

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

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## Mathematics Subject Classification

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