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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