The Chebyshev inequality follows almost directly from Markov Inequality and is a key tool for proving the Weak and Strong Laws of Large Numbers.
Assume that for some random variable \(X\) we have that \(\mathbb{E}X<\infty\). Then \[\mathbb{P}(|X-\mathbb{E}X|\geq \epsilon )\leq {\frac{\operatorname{Var}(X)}{\epsilon ^2}}.\]
Proof: The proof of the full result requires martingale methods and is skipped for now. Instead we prove the result with the additional assumption that \(\operatorname{Var}(X)<\infty\). We apply Markov Inequality to \((X-\mathbb{E}X)^2\) and find that \[ \mathbb{P}(|X-\mathbb{E}X|\geq \epsilon )=\mathbb{P}((X-\mathbb{E}X)^2\geq \epsilon ^2)\leq \frac{\mathbb{E}[(X-\mathbb{E}X)^2]}{\epsilon ^2}=\frac{\operatorname{Var}(X)}{\epsilon }. \]