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arrow_back State the Hoeffding inequality for bounded independent random variables

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State the Hoeffding inequality for bounded independent random variables

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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 .

by Platinum
(106,844 points)

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