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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2891Two-State Regime SwitchingA market regime is either Calm or Volatile. From Calm, the chain moves to Volatile with probability ; from Volatile, it moves to Calm with probability . Find the stationary distribution.概率简单derivation未尝试面试订阅2892A Three-State Birth-Death Regime ChainConsider states Bull, Neutral, and Bear with transition matrix \[ P= \begin pmatrix 0.8 & 0.2 & 0\\ 0.3 & 0.4 & 0.3\\ 0 & 0.2 & 0.8 \end pmatrix . \] Find the stationary distribution.概率中等derivation未尝试面试订阅2893Uniform Stationarity for a Doubly Stochastic ChainSuppose a finite Markov chain has transition matrix P whose rows and columns both sum to 1. Show that the uniform distribution is stationary.概率简单derivation未尝试面试订阅2894Random Walk on an Undirected GraphA simple random walk moves on a connected undirected graph G=(V,E) by choosing a uniformly random neighbor at each step. Show that the stationary distribution is proportional to degree.概率简单derivation未尝试面试订阅2895Star Graph Long-Run OccupancyA simple random walk runs on a star graph with one hub and m leaves. What fraction of time is spent at the hub in stationarity, and what fraction at each leaf?概率简单derivation未尝试面试订阅2896Complete Bipartite Graph K_{2,3}A simple random walk runs on the complete bipartite graph K 2,3 . What is the stationary probability of each vertex, and what total stationary mass sits on each side of the bipartition?概率中等derivation未尝试面试订阅