Hoeffding's Inequality

Let
$X_1, \ldots, X_n$ be iid with mean $\mu$ and suppose that $a_i \leq X_i \leq b_i$ for all i.
Then, for every $\epsilon > 0$,

(1)

\begin{align} P(|\overline{X} - \mu| > \epsilon) \leq 2 e^{-2c \, n \epsilon^2} \end{align}

where

(2)

\begin{align} \overline{X} = \frac{1}{n}\sum_{i=1}^n X_i \end{align}

and

(3)

\begin{align} \frac{1}{c} = \sum_{i=1}^n (b_i-a_i)^2. \end{align}