60A99-ProofOfChebyshevsInequality1.tex

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\begin{document}
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 $$\Expect[Y] = \operatorname{Var}[X] = \sigma^2.$$

Applying Markov's inequality to $Y$, we see that $$\Prob{}{\left|X - \mu \right| \ge t} = \Prob{}{Y \ge t^2} \le \frac{1}{t^2}\Expect[Y] = \frac{\sigma^2}{t^2}.$$
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