Abstract
The mean μ of a random variable X is arguably the most common one number summary of the distribution of X. Although averaging is a primitive concept with some natural appeal, the mean μ is a useful summary only when the random variable X is concentrated around the mean μ, that is, probabilities of large deviations from the mean are small. The most basic large deviation inequality is Chebyshev’s inequality, which says that if X has a finite variance σ2, then \(P(\vert X - \mu \vert > k\sigma ) \leq \frac{1} {{k}^{2}}\). But, usually, this inequality is not strong enough in specific applications, in the sense that the assurance we seek is much stronger than what Chebyshev’s inequality will give us.
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DasGupta, A. (2011). Large Deviations. In: Probability for Statistics and Machine Learning. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9634-3_17
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