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2872How Many Bernoulli Trials for 99% Accuracy?You estimate a Bernoulli success probability by the sample mean of n i.i.d. trials. How large must n be to guarantee, via Hoeffding's inequality, that \[ P(| X n-p|\ge 0.02)\le 0.01? \]概率中等derivation未尝试面试订阅2873Hoeffding Versus Chebyshev for 65 Heads in 100 TossesA fair coin is tossed 100 times. Compare the Hoeffding and Chebyshev upper bounds on the event that the fraction of heads is at least 0.65.概率中等derivation未尝试面试订阅2876Sub-Gaussian Tail from an MGF AssumptionSuppose a centered random variable X satisfies \[ E[e tX ]\le e 2 t 2/2 \qquad for all t\in R. \] Use exponential Markov to prove that \[ P(X\ge x)\le e -x 2/(2 2) . \]概率中等derivation未尝试面试订阅2877Rademacher-Sum Upper TailLet X 1,\dots,X 100 be i.i.d. with P(X i=1)=P(X i=-1)=1/2. Use a Chernoff-style bound to estimate \[ P (\sum i=1 100 X i\ge 20 ). \]概率中等derivation未尝试面试订阅2878Poisson Upper Tail in Multiplicative FormLet N\sim Poisson ( ). Show that for any >0, \[ P(N\ge (1+ ) )\le \exp\! (- \bigl((1+ )\ln(1+ )- \bigr) ). \]概率中等derivation未尝试面试订阅2879Numerical Poisson Overload BoundAn exchange gateway receives N\sim Poisson (100) messages in a fixed interval. Use the Poisson Chernoff upper-tail bound to estimate P(N\ge 130).概率中等derivation未尝试面试订阅2880Poisson Lower Tail BoundLet N\sim Poisson ( ). Show that for 0< <1, \[ P(N\le (1- ) )\le \exp\! (- \bigl( +(1- )\ln(1- )\bigr) ). \]概率中等derivation未尝试面试订阅2881A Numerical Poisson Shortfall BoundIf N\sim Poisson (100), use the lower-tail Chernoff bound to estimate P(N\le 80).概率中等derivation未尝试面试订阅2882Optimizing a Chernoff Bound for an Exponential VariableLet X\sim Exponential (1). Use its MGF to derive the best Chernoff-type upper bound you can on P(X\ge a) for a>1.概率中等derivation未尝试面试订阅2883A Chernoff Bound for a Sum of ExponentialsLet S=X 1+\cdots+X k where X i\overset i.i.d. \sim Exponential (1). Use the MGF to derive a Chernoff upper bound for P(S\ge a) when a>k.概率困难derivation未尝试面试订阅2884Multiplicative Chernoff for a Binomial CountLet X\sim Binomial (n,p) with mean =np. Show that for any >0, \[ P(X\ge (1+ ) )\le ( e (1+ ) 1+ ) . \]概率困难derivation未尝试面试订阅2885Sub-Gaussian Sum with a Volatility ProxySuppose X 1,\dots,X n are independent centered random variables and each satisfies \[ E[e tX i ]\le e 2 t 2/2 \qquad for all t\in R. \] Show that for S n=\sum i=1 n X i, \[ P(S n\ge x)\le \exp\! (- x 2 2n 2 ). \]概率中等derivation未尝试面试订阅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