Given $n$ (real-valued) random variables $X_1, X_2, ..., X_n \in [0, B]$, it can be derived from Hoeffding's Inequality that: $$\mathbb{P}^n\left[ \bar{X} - \mathbb{E}_n[ \bar{X} ] \geq t \right] \leq \exp \left( - \frac{2 n t^2}{B^2} \right) $$ where $\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i$, and $\mathbb{P}^n$ is the product measure $\mathbb{P} \times \mathbb{P} \times \ldots \mathbb{P}$ ($n$ times).
Is there a tighter bound if we have (integer-valued) random variables $X_1, X_2, ..., X_n \in \{0, 1, ..., B-1, B\}$? What similar inequalities are available for discrete, bounded, random variables?