# arrow_back State the Hoeffding inequality for bounded independent random variables

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State the Hoeffding inequality for bounded independent random variables

Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent random variables, such that $\operatorname{Pr}\left(a_{k} \leq\right.$ $\left.X_{k} \leq b_{k}\right)=1$ for all $k$, where $a_{k}$ and $b_{k}$ are constant, $a_{k}<b_{k} .$ Let $S_{n}$ be the sum $X_{1}+\ldots+X_{n}$. Then
$\begin{gathered} \operatorname{Pr}\left(S_{n}-E\left[S_{n}\right]>\epsilon\right) \leq \exp \left(-\frac{2 \epsilon^{2}}{\sum_{k=1}^{n}\left(b_{k}-a_{k}\right)^{2}}\right) \\ \operatorname{Pr}\left(\left|S_{n}-E\left[S_{n}\right]\right|>\epsilon\right) \leq 2 \exp \left(-\frac{2 \epsilon^{2}}{\sum_{k=1}^{n}\left(b_{k}-a_{k}\right)^{2}}\right) \end{gathered}$
References
[1] W. Hoeffding, "Probability inequalities for sums of bounded random variables", J. Amer. Statist. Assoc., vol. 58, pp.13-30, 1963 .

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