Chebyshev Inequality

1. Statement

Theorem (Chebyshev inequality). For every $ϵ>0$,

$$\mathbb{P}(|X|\ge ϵ)\le \frac{\mathbb{E}[X^2]}{{ϵ}^2}.$$

2. Proof

Proof

$$\mathbb{E}|X{|}^2={\int }_{\{|X|\ge ϵ\}\cup \{|X|<ϵ\}}|x{|}^2{f}_X(x)dx\ge {\int }_{|X|\ge ϵ}{x}^2{f}_X(x)dx\ge {\int }_{|X|\ge ϵ}{ϵ}^2{f}_X(x)dx={ϵ}^2\mathbb{P}(|X|\ge ϵ).$$

Dividing by ${ϵ}^2$ gives the result. $\square$

3. Corollary

Corollary (Centered Chebyshev inequality).

$$\mathbb{P}(|X-\mathbb{E}[X]|\ge ϵ)\le \frac{\mathrm{Var}[X]}{{ϵ}^2}.$$

Related Notes

• Convergence of Random Variables
• Central Limit Theorem