Chebyshev Inequality
1 Statement
The Chebyshev inequality bounds the probability that a random variable deviates from its mean by a given amount, in terms of its variance. It follows directly from the Markov Inequality and is the key tool in the proof of the weak and strong Law of Large Numbers.
Let \(X\) be a random variable with finite mean \(\mathbb{E}X<\infty\) and finite variance \(\operatorname{Var}(X)<\infty\). Then for every \(\epsilon>0\) \[ \mathbb{P}(|X-\mathbb{E}X|\geq \epsilon )\leq {\frac{\operatorname{Var}(X)}{\epsilon ^2}}. \]
Proof. Apply the Markov Inequality to the non-negative random variable \((X-\mathbb{E}X)^2\) and the level \(\epsilon^2\): \[ \mathbb{P}(|X-\mathbb{E}X|\geq \epsilon )=\mathbb{P}((X-\mathbb{E}X)^2\geq \epsilon ^2)\leq \frac{\mathbb{E}[(X-\mathbb{E}X)^2]}{\epsilon ^2}=\frac{\operatorname{Var}(X)}{\epsilon ^2}. \]
If \(\operatorname{Var}(X)=\infty\) the bound is trivially true, so only the finite mean is really needed.