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2863A Batch-Size Compound Poisson Desk FlowTrades arrive according to N\sim Poisson ( ). Each trade contributes a batch size B taking values 0,1,2 with probabilities 1/2,1/3,1/6, independently across trades and from N. Let \[ S=\sum k=1 N B k. \] Find the MGF of S, and compute E[S] and Var (S).概率中等derivation未尝试面试订阅2864Exponential Random Intensity Gives Geometric CountsA latent intensity \Lambda is Exponential ( ) with rate . Conditional on \Lambda, the count N is Poisson (\Lambda). Use MGFs to identify the unconditional law of N, and compute E[N].概率中等derivation未尝试面试订阅2871Hoeffding for a Bernoulli MeanLet X 1,\dots,X n be i.i.d. Bernoulli(p) and let X n be the sample mean. Use Hoeffding's inequality to bound \[ P( X n-p\ge \varepsilon). \]概率简单derivation未尝试面试订阅2874Hoeffding for Bounded Daily PnLSuppose daily centered PnL increments X 1,\dots,X 50 are independent and each lies almost surely in [-2,3]. Bound \[ P\! ( 1 50 \sum i=1 50 X i\ge 0.5 ) \] using Hoeffding's inequality.概率中等derivation未尝试面试订阅2875Monte Carlo Pricing Error with Bounded PayoffsA Monte Carlo pricer averages 500 i.i.d. discounted payoff samples, each in [0,1]. Use Hoeffding's inequality to bound the probability that the estimated price differs from the true price by at least 0.05.概率简单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