Chebyshev's inequality

Upper bound on deviation probability using variance.

Chebyshev’s inequality

Chebyshev’s inequality: Let $X$ be a random variable with expectation $\mu=\mathbb{E}[X]$ and finite variance $\sigma^2=\mathrm{Var}(X)$ . Then for every $t>0$ ,

\mathbb{P}\bigl(|X-\mu|\ge t\bigr)\le \frac{\sigma^2}{t^2}.

Equivalently, for every $k>0$ ,

\mathbb{P}\bigl(|X-\mu|\ge k\sigma\bigr)\le \frac{1}{k^2}.

Here $\mathbb{P}$ denotes the probability measure on the underlying probability space . Chebyshev’s inequality is a direct consequence of Markov's inequality applied to $(X-\mu)^2$ , and it is a standard tool for proving the weak law of large numbers .