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1765Fixed Effects Still Miss a Moving Stress ChannelPanel fixed effects remove each desk's persistent skill, but an omitted intraday stress variable still changes day by day. Suppose higher stress raises both inventory pressure X and slippage Y within the same desk. After adding desk fixed effects, what confounding channel remains, and in what direction does it bias the within-desk slope on X?统计中等essay未尝试面试订阅1766What Remains After Controlling the Hedge ChannelA signal X changes the hedge ratio H immediately, and both X and H affect desk PnL Y. The direct effect of X on Y is +1.2 bps, while the channel through H contributes another +0.8 bps. If H is measured perfectly and you regress Y on X and H, what effect does the coefficient on X identify, and what number should it equal?统计简单derivation未尝试免费1770Why Selection on Implemented Trades Distorts Treatment EffectsWhy can studying only executed trades bias the estimated effect of an execution rule, even if the rule assignment itself was randomized upstream?统计简单essay未尝试免费1800Signal Stationarity Classification 5A candidate signal is defined by X t = t ε t. Is it weakly stationary?统计困难derivation未尝试面试订阅1802Measurement Noise Effect 2A latent stationary signal Y t has gamma(0) = 8 and gamma(1) = 3. You observe X t = Y t + eta t, where eta t is iid noise with variance 2 independent of Y t. What are gamma X(0) and gamma X(1)?统计中等derivation未尝试免费1820Measurement Noise Flattens ACFWhy can adding independent observation noise make an otherwise persistent signal look less autocorrelated?统计困难derivation未尝试面试订阅1823AR(1) Multi-Step Forecast 3A signal follows X t = -1 + 0.8 X (t-1) + e t with Var(e t) = 1 and current value X t = 3. What is the h = 4 step forecast E[X (t+4) | X t]?统计中等derivation未尝试免费1861Long-Run Residual Variance 1A mean-reverting residual follows X (t+1) = 1/2 X t + epsilon (t+1) with Var(epsilon (t+1)) = 4. What is the stationary variance of X t?统计简单essay未尝试免费1868RW Versus MR Diagnosis 3A five-day variance ratio comes in well below 1. What does that suggest about serial dependence in returns?统计中等essay未尝试面试订阅1869RW Versus MR Diagnosis 4If a spread's conditional mean always equals today's level, regardless of horizon, which model is the closer description?统计中等derivation未尝试面试订阅1871Hidden Launch Mix from Survivor Panel 1A fund database currently shows 54 live statistical-arbitrage funds and 18 live macro funds. Historical one-year survival rates for these styles were 90% for statistical-arbitrage and 60% for macro. Assuming the current live panel comes from one launch cohort, what fraction of the original launches were macro funds?统计简单essay未尝试免费1872Hidden Launch Mix from Survivor Panel 2A live manager database now contains 32 market-neutral funds and 24 credit relative-value funds. Historical survival rates for those styles were 80% and 50%, respectively. What fraction of the original launches were credit relative-value funds?统计简单essay未尝试面试订阅1873Displayed High-Turnover Share BiasA live fund panel shows 45 low-turnover funds and 15 high-turnover funds. Historical survival rates were 90% for low-turnover funds and 30% for high-turnover funds. By how many percentage points does the displayed live panel understate the original share of high-turnover launches?统计简单essay未尝试免费1880Pre-2020 Share Hidden by AttritionA database of currently live funds contains 12 funds launched before 2020 and 36 funds launched in or after 2020. The survival rates for those two groups were 25% and 75%, respectively. What share of the original launches occurred before 2020?统计简单essay未尝试面试订阅2211Forward Default Slice Between Two Dates 16A reduced-form model reports survival probabilities S(T1)=0.96 and S(T2)=0.9 with T2>T1. What conditional default probability over the interval (T1,T2], given survival to T1, is implied?数理金融中等数值题未尝试面试订阅2246State-Mixture Copula Calibration 1A desk uses a two-state one-factor copula toy model: conditional on a calm systemic state a name defaults with probability 0.40%, and conditional on a stress state it defaults with probability 6.40%. The unconditional one-year default probability is 1.60%. What stress-state probability is implied?数理金融中等数值题未尝试面试订阅2247State-Mixture Copula Calibration 2In a two-state conditional-independence model, a name has calm-state default probability 0.60% and stress-state weight 20.00%. If the unconditional one-year default probability is 2.40%, what stress-state default probability is implied?数理金融中等数值题未尝试面试订阅2248State-Mixture Copula Calibration 3A name's one-year default probability is modeled as a mixture of a calm state and a stress state. The stress-state probability is 25.00%, the stress-state default probability is 5.50%, and the unconditional default probability is 1.93%. What calm-state default probability is implied?数理金融中等数值题未尝试面试订阅2256Copula Desk Intuition 1A junior trader says 'conditional independence means no dependence.' Why is that wrong in a one-factor copula?数理金融困难essay未尝试面试订阅2257Copula Desk Intuition 2Why can a senior tranche be more sensitive to stronger dependence even when the equity tranche barely moves?数理金融困难essay未尝试面试订阅