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2876概率中等derivationmedium

Sub-Gaussian Tail from an MGF Assumption

题目

Suppose a centered random variable XX satisfies \[ E[e^{tX}]\le e^{\sigma^2 t^2/2}\qquad\text{for all }t\in\mathbb R. \] Use exponential Markov to prove that \[ P(X\ge x)\le e^{-x^2/(2\sigma^2)}. \]

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