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5930Why Full Information Beats RanksTwo candidates have qualities drawn iid from a KNOWN Uniform(0,1) distribution, revealed as actual numerical values one at a time; you must accept one (reject-the-first forces the second). You win only if you accept the candidate with the higher quality. Compare the best win probability achievable by a full-information policy that uses the numerical value of candidate 1 against the best win probability of a rank-only policy that sees only relative order. Which is larger and by how much?概率简单derivation未尝试免费5931Discounted Offer StoppingEach period an offer arrives, iid Uniform(0,1). If you accept an offer of value x at period t, you receive beta t * x, where beta = 0.9 is a per-period discount factor (so waiting shrinks the value of any future acceptance). No recall, infinite horizon. Find the optimal stationary acceptance threshold and the expected discounted payoff from the start.概率中等数值题未尝试免费5936Sell Before the Deadline in a Falling MarketYou must sell an asset within 3 periods. In period t one offer arrives, drawn uniformly on [0, 1 - 0.2*(t-1)] (a declining market: the range is [0,1] in period 1, [0,0.8] in period 2, [0,0.6] in period 3). You accept and stop, or reject forever (no recall). If you reach period 3 you must accept that offer. Knowing these distributions, find the optimal acceptance thresholds and the expected sale price under the optimal policy.概率中等数值题未尝试免费5937Reservation Wage With Exponential OffersJob offers arrive sequentially and independently, each an Exponential random variable with rate 1 (mean 1). After each offer you accept it (and stop) or reject it forever and pay a search cost c = 0.2 to see the next; there is no deadline. Find the optimal stationary reservation level a above which you accept, and the expected wage you end up accepting under that policy.概率中等数值题未尝试免费5938Bird in the Hand vs Discounted WaitYou face two periods. In period 1 a reward X1 ~ Uniform(0,1) is offered; accept it now to receive X1, or wait. If you wait, in period 2 you must accept X2 ~ Uniform(0,1), but a reward received in period 2 is worth only a fraction beta = 0.8 of its face value (discounting). No recall. Find the optimal period-1 acceptance threshold and the expected payoff of the optimal policy.概率简单数值题未尝试免费5941The 1/e Law of Best ChoiceIn the classic secretary problem with n candidates (relative ranks only, irrevocable choices), the look-then-leap rule observes the first r candidates without choosing and then accepts the first later candidate who beats all seen so far. For large n, write r = t*n and derive the limiting win probability as a function of the skip fraction t in (0,1). Then find the t that maximizes it and the resulting optimal asymptotic probability of selecting the single best candidate.概率中等derivation未尝试免费5989Variance of a Count WindowTrades hit a tape as a Poisson process with rate 6 per hour. Let N be the number of trades in a fixed 20-minute window. What is Var (N)?概率简单数值题未尝试免费5991Quiet Window on a Combined FeedTwo independent exchanges send quotes to your gateway. Exchange A is a Poisson process with rate 3 per minute and exchange B is an independent Poisson process with rate 5 per minute. Treating the combined stream as one process, what is the probability that no quote arrives during a 30-second window? Give a decimal to three places.概率中等数值题未尝试免费5993Waiting After a Quiet StretchCustomer arrivals at a help desk form a Poisson process with rate 12 per hour. You have already waited 2 minutes since the last arrival with no one appearing. What is the probability you must wait at least 5 more minutes for the next arrival? Give a decimal to three places.概率中等数值题未尝试免费5994At Least Two ArrivalsDefaults in a small credit book occur as a Poisson process with rate 8 per year. What is the probability that at least 2 defaults occur in the next 3 months? Give a decimal to three places.概率中等数值题未尝试免费5995Rate from a Mean GapTrades on an illiquid name arrive as a Poisson process. The average time between consecutive trades is observed to be 4 minutes. What is the implied arrival rate, expressed as trades per hour?概率简单数值题未尝试免费5998First Arrival Lands in a Target WindowSignals arrive as a Poisson process with rate 1 per second. What is the probability that the very first signal arrives strictly between 2 and 3 seconds after the start? Give a decimal to three places.概率困难数值题未尝试免费5999Forward Wait from a Spontaneous GlanceBuses arrive at a stop as a Poisson process with rate 10 per hour. You walk up at an arbitrary moment, unsynchronized with the buses. What is the expected time, in minutes, until the next bus arrives?概率困难数值题未尝试免费6000Window for a 90% Chance of an ArrivalQuote updates arrive as a Poisson process with rate 5 per hour. How long a window, in minutes, must you watch so that the probability of seeing at least one update is exactly 0.9? Give a decimal to one place.概率中等数值题未尝试免费6001Equally Likely Adjacent CountsArrivals follow a Poisson process. Over a fixed observation window the count N satisfies P(N=2)=P(N=3). Given this, compute P(N=3) as a decimal to three places.概率困难数值题未尝试免费6002Exactly Four in an Eighteen-Minute WindowCancellations hit an order book as a Poisson process with rate 10 per hour. What is the probability that exactly 4 cancellations occur in an 18-minute window? Give a decimal to three places.概率简单数值题未尝试免费6003Ratio of Adjacent Count ProbabilitiesTrades print as a Poisson process with rate 12 per hour. Over a 20-minute window with count N, what is the ratio P(N=5)/P(N=4)? Give a decimal.概率简单数值题未尝试免费6004Short-Window Linear Approximation ErrorSpikes arrive at a detector as a Poisson process with rate 3 per minute. For a very short 0.5-second window the probability of at least one spike is often approximated by t. What is the absolute error of that approximation, i.e. t - P(N\ge 1)? Give a decimal to four places.概率中等数值题未尝试免费6005Empty Interval Between Two Clock TimesAlarms trip as a Poisson process with rate 6 per hour. What is the probability of zero alarms during the interval from minute 10 to minute 25 (a 15-minute span starting partway through)? Give a decimal to three places.概率中等数值题未尝试免费6006Waiting Time Exceeds Three Quarters of an HourService requests arrive as a Poisson process with rate 4 per hour. What is the probability that the waiting time to the first request exceeds 45 minutes? Give a decimal to three places.概率简单数值题未尝试免费