Hoeffding Inequality
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Let \(X_1, X_2, \ldots, X_n\) be independent bounded random variables with bound \(X_i \in [a, b]\) for all \(i\), where \(-\infty < a \leq b < \infty\). Then \[ \mathbb{P}\left(\frac{1}{n} \sum_{i=1}^{n}\left(X_{i}-\expected\left[X_{i}\right]\right) \geq t\right) \leq \exp \left(-\frac{2 n t^{2}}{(b-a)^{2}}\right) \] and \[ \mathbb{P}\left(\frac{1}{n} \sum_{i=1}^{n}\left(X_{i}-\mathbb{E}\left[X_{i}\right]\right) \leq-t\right) \leq \exp \left(-\frac{2 n t^{2}}{(b-a)^{2}}\right) \]