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5913Minimizing Ruin in a Favorable GameYou have 2 chips and play a favorable even-money game you win with probability p=0.6, intending to play forever (no cash-out target) and stake whole chips. To minimize the chance of ever going broke you bet the smallest stake, 1 chip per round. What is the probability you are eventually ruined under this minimum-stake (timid) play?概率中等数值题未尝试免费5917Free Peek Before Calling the Bigger BoxTwo boxes each independently contain an amount drawn uniformly from \ 1,2,3,4\ . You must guess which box holds the strictly larger amount; a correct guess pays 1 and a tie or wrong guess pays 0. Before guessing you may take a free peek at the contents of one box (your choice of which). By how much does this peek increase your probability of a correct guess compared with guessing blind?概率简单derivation未尝试免费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5982Variance of Fills Over a Binomial Number of QuotesOut of n=10 resting quotes, each fills independently with probability 0.3, so the number of fills N is Binomial(10,0.3). Each fill produces an i.i.d. PnL X i with E[X i]=2 and Var (X i)=9, independent of which quotes fill. For the stopped sum S N=\sum i=1 N X i, compute Var (S N).概率中等derivation未尝试免费5983Number of Trades to Cross a Profit TargetA strategy books i.i.d. positive profits X 1,X 2,\dots with E[X i]=2.5. Let N be the first time the running total S n=\sum i=1 n X i strictly exceeds 10; that is, N=\min\ n:S n>10\ . Assuming E[N]< , the expected overshoot is known to satisfy E[S N]=14. Use a Wald-style identity to compute E[N].概率中等derivation未尝试免费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6007Buys Before the First SellBuy fills and sell fills arrive as independent Poisson processes with rates \lambda 1=12 and \lambda 2=3 per hour. Counting from now, what is the expected number of buy fills that occur strictly before the first sell fill?概率中等derivation未尝试面试订阅6008Which of Three Feeds Ticks FirstThree independent market-data feeds emit ticks as Poisson processes with rates 5, 8, and 2 ticks per second. What is the probability that the very next tick across all feeds comes from the rate-8 feed?概率简单derivation未尝试面试订阅6009Expected Large Trades in Two HoursTrades print as a Poisson process at rate =30 per hour. Independently, each trade is a block (large) trade with probability 0.15. What is the expected number of block trades over the next 2 hours?概率简单derivation未尝试面试订阅6010No Orders Routed to Venue COrders arrive as a Poisson process at rate =20 per minute and are independently routed to venue A, B, or C with probabilities 0.5, 0.25, and 0.25. What is the probability that venue C receives no orders during the next 6 seconds?概率中等derivation未尝试面试订阅6011Joint Count of Two Split StreamsA Poisson process at rate =30 per hour is split by independent fair-coin-style labeling into a 'lit' stream (prob 0.2) and a 'dark' stream (prob 0.8). Over the next 30 minutes, what is the probability of observing exactly 4 lit prints and exactly 9 dark prints?概率中等derivation未尝试面试订阅6012A Three-Arrival Head StartAggressive and passive child orders fill as independent Poisson processes with rates \lambda A=10 and \lambda B=5 per minute. What is the probability that the first three fills in the merged stream are all aggressive (stream A)?概率简单derivation未尝试面试订阅6013Power of a One-Sided Z-TestYou run a one-sided z-test at level alpha = 0.05 (critical value 1.645) for a positive mean edge. The true edge is delta = 3 bp, the per-observation standard deviation is sigma = 8 bp, and you collect n = 64 observations. The power equals Phi(delta*sqrt(n)/sigma - 1.645). Compute the power, using Phi(1.355) approx 0.9123.统计中等数值题未尝试免费6015Track Record to Confirm a SharpeA strategy has a true annualized Sharpe ratio of 0.5. The t-statistic of the mean return over a track record of T years is approximately t = SR * sqrt(T). How many years of returns are needed before the t-statistic reaches 2 (the usual significance bar)?统计中等数值题未尝试免费6017Posterior Distribution After Ten Wins in TwelveA win probability p has the uniform prior Beta (1,1). You then see k=10 wins out of n=12 games. Name the exact posterior distribution of p and give its mode.统计简单数值题未尝试免费6019Blending a Prior View With Four Noisy QuotesA latent fair value \sim N(10,4). You collect n=4 independent quotes with known per-quote variance 2=8 and sample mean x=12. Compute the posterior mean of .统计中等derivation未尝试免费