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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6014Quadrupling the SampleYour current backtest detects a minimum edge of 6 bp at the desired power. You extend the sample so the number of observations is multiplied by 4, keeping sigma, alpha, and target power fixed. Because the minimum detectable effect scales as 1/sqrt(n), what is the new minimum detectable edge, in basis points?统计简单数值题未尝试免费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6016Posterior Mean Hit Rate After a Cold StreakA signal's hit probability p has prior Beta (4,4). Over the next batch you record 3 hits and 9 misses. By integrating the posterior density, what is the posterior mean of p?统计简单derivation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6018Posterior Mean Rate for a Rare FaultFaults arrive as a Poisson process with rate per day, prior Gamma (2,0.5) in shape-rate form. Over 3.5 days you observe 7 faults. What is the posterior mean of ?统计中等derivation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6020Predictive Chance the Next Trade FailsA fill probability p has prior Beta (2,1). After observing 3 fills and 4 misses, what is the posterior predictive probability that the very next attempt is a miss (a failure)?统计中等derivation未尝试免费6022MAP Estimate Versus Posterior MeanAfter updating, the posterior for a conversion rate p is Beta (8,4). Report both the MAP (posterior mode) estimate and the posterior mean, and state which is larger.统计中等derivation未尝试免费6023Long-Run Volatility (Not Variance) from GARCH ParametersA GARCH(1,1) model h t=\omega+ r t-1 2+ h t-1 has \omega=0.04, =0.12, =0.80. Here h t is the conditional variance of daily returns. Report the long-run (unconditional) daily volatility h as a decimal.统计中等derivation未尝试面试订阅6024Persistence and the Covariance-Stationarity VerdictA GARCH(1,1) has =0.20, =0.75. Compute the persistence + and state whether the process is covariance-stationary (i.e. has a finite, time-invariant unconditional variance). Give the persistence as a decimal.统计简单derivation未尝试面试订阅