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5984Expected Inspection Cost Until the First DefectA quality line inspects items one at a time; each item is defective independently with probability 0.05. Inspection stops at the first defective item. Each inspection (defective or not) costs an i.i.d. amount C i with E[C i]=\8, independent of the defect outcomes. Let N be the number of items inspected. Find the expected total inspection cost E\! [\sum i=1 N C i ].概率简单数值题未尝试免费5985Expected Total Slippage With Negative DriftA market-making desk incurs i.i.d. per-trade adverse-selection costs X 1,X 2,\dots with E[X i]=-0.4 (a net loss per trade). The number of trades in a session, N, is independent of the costs and is Poisson with mean 15. Compute the expected cumulative cost E\! [\sum i=1 N X i ].概率简单derivation未尝试免费5986Expected Winnings Over a Random Number of BetsA gambler places bets until a random stopping rule halts play; the number of bets N is a stopping time for the i.i.d. bet outcomes with E[N]=8. Each bet has an i.i.d. net result X i with E[X i]=-0.05 (a 5\% house edge per unit staked, with unit stakes), and the decision to stop after bet n depends only on outcomes up to bet n. Compute the gambler's expected total winnings E\! [\sum i=1 N X i ], and state whether any stopping rule with E[N]=8 can make this positive.概率中等derivation未尝试免费5987When the Stopping Rule Looks at the Last DrawDraw i.i.d. values X 1,X 2,\dots uniform on \ 1,2,3\ (so E[X i]=2). Define N as follows: keep drawing and stop the first time you draw a 3; let N be the number of draws. Let S N=\sum i=1 N X i. A candidate computes E[N]E[X 1]=3 2=6 and claims E[S N]=6. Compute the correct value of E[S N] and explain in one sentence why E[N]E[X 1] is the wrong formula here.概率困难essay未尝试面试订阅5988Expected Sample Size of a Sequential Drift TestA sequential test accumulates i.i.d. log-likelihood increments X 1,X 2,\dots with E[X i]=0.25. The test stops at N=\min\ n: |S n|\ge 3\ where S n=\sum i=1 n X i, and it is given that E[N]< and that the expected stopped statistic is E[S N]=2.0 (reflecting that the upper boundary is hit far more often under this positive drift). Each observation costs \6 to collect. Using a Wald-style identity, find the expected total data-collection cost.概率中等数值题未尝试免费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5990Expected Time to the Third ArrivalOrders arrive at a matching engine as a Poisson process with rate 4 per minute. What is the expected time, in seconds, until the 3rd order arrives (measured from time 0)?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5992Most Likely Number of FillsFills on a passive order arrive as a Poisson process with rate 7 per hour. Over a fixed 30-minute window, what is the single most likely number of fills (the mode of the count distribution)?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5996Variance of the Fourth Arrival TimePackets arrive at a sensor as a Poisson process with rate 2 per minute. Let T 4 be the time of the 4th packet. What is Var (T 4), in minutes squared?概率中等数值题未尝试免费5997Expected Count Under an Unknown RegimeOn any given day the market is in a 'calm' regime with probability 0.5, where news events arrive as a Poisson process with rate 6 per hour, or a 'busy' regime with probability 0.5, with rate 14 per hour. Before observing the day you do not know the regime. What is the expected number of news events in a 1-hour window?概率中等数值题未尝试免费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