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4974Reduction in Expected Hitting Time After Lowering the Reset RateIn the same 0→1→2 with reset model, let a=1 and b=2. If the return rate c drops from 3 to 1, by how much does the expected hitting time from 0 to 2 fall?随机过程中等数值题未尝试面试订阅4975Stationary Distribution of a Three-State Birth-Death ChainA three-state birth-death CTMC has rates 0->1 = 1, 1->0 = 2, 1->2 = 3, and 2->1 = 1.5. What is its stationary distribution (pi 0, pi 1, pi 2)?随机过程中等数值题未尝试面试订阅4988Stationary Does Not Mean SlowWhy can a state have a small stationary probability even if it has a large exit rate?随机过程困难essay未尝试面试订阅4990Why First-Step WorksWhy is first-step analysis so effective for expected hitting times in jump processes?随机过程困难essay未尝试面试订阅5001Infer Volatility Strike From Matched Vega Notional 11Around the volatility strike K vol, a variance swap with variance notional N var is often locally matched to a vol swap with vega notional N vega via N vega = 2*K vol*N var. If N var=100000 and N vega=40000, what K vol is implied?金融与交易中等数值题未尝试面试订阅5002Infer Volatility Strike From Matched Vega Notional 12Using N vega = 2*K vol*N var, if N var=75000 and N vega=27000, what volatility strike K vol is implied?金融与交易中等数值题未尝试面试订阅5003Infer Volatility Strike From Matched Vega Notional 13Using N vega = 2*K vol*N var, if N var=120000 and N vega=60000, what K vol is implied?金融与交易中等数值题未尝试面试订阅5006Infer Realized Vol Where Variance And Vol Swaps Match 16A variance swap has notional 100000 and variance strike 0.04, while a vol swap has vega notional 45000 and volatility strike 0.2. Other than the trivial case sigma real = 0.2 where both are at strike, what realized volatility makes the two payouts equal?金融与交易困难数值题未尝试面试订阅5007Infer Realized Vol Where Variance And Vol Swaps Match 17A variance swap has notional 80000 and variance strike 0.0324, while a vol swap has vega notional 26400 and volatility strike 0.18. Other than sigma real = 0.18, what realized volatility makes the payouts equal?金融与交易困难数值题未尝试面试订阅5008Infer Realized Vol Where Variance And Vol Swaps Match 18A variance swap has notional 120000 and variance strike 0.0625, while a vol swap has vega notional 66000 and volatility strike 0.25. Other than sigma real = 0.25, what realized volatility makes the two payouts equal?金融与交易困难数值题未尝试面试订阅5031BIC Improvement Threshold for an Extra Component 16A mixture model with one extra component would add Delta k = 3 free parameters on a sample of size n = 100. Under BIC = -2 log L + k log n, what minimum log-likelihood improvement Delta log L is needed before the larger model is preferred?机器学习中等数值题未尝试面试订阅5036EM Mixture Diagnostic 21Why does one EM component sometimes collapse onto a single point with near-zero variance during mixture training?机器学习困难essay未尝试面试订阅5037EM Mixture Diagnostic 22Why can two EM runs on the same data land on noticeably different mixture parameters even when they reach similar likelihoods?机器学习困难essay未尝试面试订阅5038EM Mixture Diagnostic 23Why is label switching not a bug in a mixture model but still a headache when you compare runs?机器学习困难essay未尝试面试订阅5039EM Mixture Diagnostic 24Why can k-means and a Gaussian mixture with shared spherical covariances give similar clusters, yet still disagree on borderline points?机器学习困难essay未尝试面试订阅5040EM Mixture Diagnostic 25Why should a desk be careful before interpreting mixture components as economic regimes rather than as a flexible density fit?机器学习困难essay未尝试面试订阅5081Recover Epsilon From a Logged Action Probability 16An epsilon-greedy policy has 5 available actions and exactly one greedy action. A log file says the greedy action was chosen with probability 0.84. What epsilon does that imply?机器学习简单数值题未尝试面试订阅5111Why Earlier Cashflow MattersTwo projects pay the same total cash over three years, but one front-loads more of it. Why is its present value typically higher when discount rates are positive?金融与交易困难essay未尝试面试订阅5112Why Discount Factors Are UsefulWhy do rates desks often think in discount factors rather than only in interest rates?金融与交易困难essay未尝试面试订阅5113Compounding Convention RiskWhy can using the wrong compounding convention create pricing discrepancies even if the quoted number looks similar?金融与交易困难essay未尝试面试订阅