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2886Gaussian Tail via Exponential MarkovLet Z\sim N(0, 2). Use the Gaussian MGF to derive the Chernoff bound \[ P(Z\ge a)\le e -a 2/(2 2) . \]概率简单derivation未尝试面试订阅2887A/B Gap ConcentrationYou run an A/B test with n Bernoulli observations in treatment and n in control, all independent. Let X and Y be the sample means. Use Hoeffding's inequality to bound \[ P\bigl(( X- Y)-E[ X- Y]\ge \varepsilon\bigr). \]概率中等derivation未尝试面试订阅2888Heterogeneous Range Hoeffding BoundIndependent centered shocks satisfy \[ X 1\in[-1,1],\quad X 2\in[-2,2],\quad X 3\in[-3,3],\quad X 4\in[-4,4] \] almost surely. Use Hoeffding's inequality to bound P(X 1+X 2+X 3+X 4\ge 6).概率中等derivation未尝试面试订阅2889How Large Must the Mean Be for a 2x Poisson Spike to Be Rare?For N\sim Poisson ( ), use the upper-tail Chernoff bound to find a sufficient condition on guaranteeing \[ P(N\ge 2 )\le 0.01. \]概率中等derivation未尝试面试订阅2890Best Available Bound for a Bounded MeanYou average n=200 independent observations in [0,1]. Compare the Chebyshev and Hoeffding upper bounds on \[ P ( X-E[ X]\ge 0.1 ). \] Use the worst-case variance for the Chebyshev side.概率中等derivation未尝试面试订阅