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4974Reduction in Expected Hitting Time After Lowering the Reset RateIn the same 0→1→2 with reset model, let a=1 and b=2. If the return rate c drops from 3 to 1, by how much does the expected hitting time from 0 to 2 fall?随机过程中等数值题未尝试面试订阅4976Infer Missing Birth-Death Rate From Stationarity 11A three-state birth-death CTMC has stationary distribution (0.5, 0.3, 0.2). The rates 0->1 and 1->2 are 0.6 and 0.4, and the rate 1->0 is 1. What rate 2->1 is needed for stationarity?随机过程困难数值题未尝试面试订阅4979Infer Missing Birth-Death Rate From Stationarity 14A three-state birth-death CTMC has stationary distribution (0.2, 0.5, 0.3). The rates 0->1 and 1->2 are 1.5 and 0.9, and the rate 1->0 is 0.6. What rate 2->1 is implied?随机过程困难数值题未尝试面试订阅4981Expected Hitting Time 1A CTMC has states 0,1,2 with state 2 absorbing. From 0 it jumps to 1 at rate 0.8. From 1 it jumps to 2 at rate 1.2 and back to 0 at rate 0.4. What is the expected time to hit state 2 starting from state 0?随机过程困难数值题未尝试面试订阅4983Infer Return Rate From Expected Hitting Time 17A CTMC has states 0,1,2 with state 2 absorbing. From 0 it jumps to 1 at rate a=0.5. From 1 it jumps to 2 at rate b=1 and back to 0 at rate c. If the expected time to hit 2 from 0 is 4, what is c?随机过程困难数值题未尝试面试订阅4985Infer Return Rate From Expected Hitting Time 19A CTMC has states 0,1,2 with state 2 absorbing. From 0 to 1 the rate is a=1.5, and from 1 to 2 the rate is b=0.5. If the expected hitting time of state 2 from 0 is 3, what is the return rate c from 1 back to 0?随机过程困难数值题未尝试面试订阅4986Why Fixed Waiting Times Break the CTMC PropertyA simulator gets the jump-chain routing probabilities right but replaces exponential waits by deterministic one-minute waits in every state. Why is the resulting calendar-time process generally not a CTMC anymore?随机过程困难essay未尝试面试订阅4987Why Uniformization Can Use One Poisson ClockWhy can a CTMC with different exit rates across states still be simulated using one common Poisson clock plus occasional virtual self-jumps?随机过程困难essay未尝试面试订阅4989Same Jump Chain, Different Calendar-Time BehaviorWhy can two jump processes share exactly the same jump chain but still look very different when observed in real time?随机过程困难essay未尝试面试订阅5654Asian Path Payoff 4A simulated path for an arithmetic-average Asian call is [95, 92, 90, 97] with strike 94. What payoff does this single path contribute to the Monte Carlo estimator?数理金融中等数值题未尝试面试订阅5901Expected Rounds to Double a Kelly BankrollA gambler bets the Kelly fraction on an even-money coin with win probability p=0.6 every round, so log-wealth is a random walk with positive drift. Let G be the per-round expected log-growth (the maximal Kelly growth rate). Using an optional-stopping argument on a suitable martingale, estimate the expected number of rounds until wealth first doubles. You may ignore overshoot past the doubling level.概率困难数值题未尝试面试订阅5902Kelly Sizing with an Unknown Win ProbabilityA coin's win probability is unknown, with prior \sim Beta (2,2). You observe 7 wins and 3 losses in calibration trials, then must place one even-money bet on the next flip, choosing a fraction f of wealth to maximize the expected log-wealth after that bet. What fraction should you bet, and why is the posterior mean (rather than, say, the posterior mode) the right quantity to plug into the Kelly formula?概率中等数值题未尝试面试订阅5910The Martingale Doubling System on RouletteOn a roulette red bet you win (even money) with probability 18/38 and lose with probability 20/38. You run the doubling (martingale) system aiming to win \1: bet \1; if it loses, bet \2; if that loses, bet \4. You stop after a win or after three straight losses (your bankroll of \7 is then gone). Find (a) the probability the campaign ends in ruin and (b) the expected net profit of the campaign.概率中等数值题未尝试面试订阅5918Defective-Batch Inspection With an Imperfect DetectorA batch is defective with prior probability \frac14. Accepting a good batch pays +20; accepting a defective batch pays -40; rejecting pays 0. Before deciding you may run a detector that flags 'defective.' It flags a truly defective batch with probability 9 10 and a good batch with probability \frac15 (false positive). What is the value of running the detector (the increase in expected payoff from using it optimally)?概率困难derivation未尝试面试订阅5919One Free Draw Before Betting on the Majority ColorAn urn is type-R with probability \frac35 (then it is 80\% red balls) or type-B with probability \frac25 (then it is 80\% blue balls). You will bet on the urn's majority color: a correct bet pays 1, a wrong bet pays 0. You may first draw one ball (with replacement) and observe its color for free. By how much does observing this single draw raise your expected payoff over betting with no draw?概率中等derivation未尝试免费5920Clairvoyance Across Three StatesThe state is High, Mid, or Low with probabilities \frac12,\frac13,\frac16. You pick action Long or Flat once. Long pays 12,\ -3,\ -9 in High, Mid, Low respectively; Flat pays 0 in every state. A clairvoyant will tell you the exact state before you choose. What is the difference between your expected payoff acting on the clairvoyant's report and your expected payoff using the single best action chosen in advance?概率中等derivation未尝试免费5921Is the Analyst's Report Worth Its FeeAn investment pays +14 if a deal closes and -10 if it falls through; closing has prior probability \frac12. You may invest or pass (pass pays 0). For a fee of 2 you may buy an analyst report that correctly predicts the outcome with probability 7 10 , after which you decide. Should you buy the report, and what is its value net of the no-report optimum?概率中等derivation未尝试免费5922Three-Candidate Best-ChoiceThree candidates of distinct, unknown qualities arrive in uniformly random order. After each interview you learn only the candidate's rank relative to those seen so far, and you must immediately and irrevocably hire or reject. You want to maximize the probability of hiring the single best candidate. What is the optimal policy and the resulting probability of success?概率简单数值题未尝试免费5924House-Selling Stationary ThresholdOffers arrive sequentially and independently, each Uniform(0,1). After each offer you either accept it (and stop) or reject it forever (no recall) and wait for the next, paying a fixed search cost c per rejected offer. There is no deadline. For c = 0.02, find the optimal stationary acceptance threshold and the expected net payoff under it.概率中等数值题未尝试免费5926Settling For Top TwoFour items arrive in uniformly random order; you observe only relative ranks and must accept one irrevocably (if you reach the last item you take it). Unlike the classic problem, you win if the item you accept is either the best OR the second-best overall. Find the policy that maximizes your winning probability and that maximum probability.概率困难数值题未尝试面试订阅