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

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