Chebyshev's Inequality

thm. Chebyshev’s Inequality. let $X$ s.t. $E[X]=\mu$, $\text{Var}[X]={\sigma }^2$ where $\sigma$ is not infinite. Then:

$$\begin{gathered} \forall ϵ>0, \\ P(|X-\mu |>ϵ)\le \frac{{\sigma }^2}{ϵ} \end{gathered}$$

• Intuitively, it means that “only few” data points are “far away” from the mean for any “reasonable” distribution:
  • “only few”: “$\le \frac{{\sigma }^2}{ϵ}$”
  • “far away”: “$|X-\mu |>ϵ$”
  • “reasonable”: $\sigma$ is not infinite.