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3815Why the Martingale Argument Sits Underneath Feynman-KacWhat is the martingale intuition that secretly sits underneath the Feynman-Kac theorem?随机过程中等essay未尝试面试订阅4141Generative Threshold from Equal-Variance Gaussians 1A discriminative model was trained at class prior P(Y=1)=0.5 and outputs posterior probability 0.7 for a case x. Overnight the base rate shifts to P(Y=1)=0.2, while the class-conditional evidence for x is assumed unchanged. What posterior probability should you use after this pure prior shift?机器学习中等数值题未尝试面试订阅4146Naive Bayes Posterior 1A generative regime model assigns posterior probability P(trend|x)=0.7 to the trend regime. If the next-day expected payoff is 12 bps in trend and -4 bps in mean reversion, what conditional expected payoff E[r|x] does the model imply?机器学习中等数值题未尝试面试订阅4148Naive Bayes Posterior 3A generative regime model assigns posterior probability P(trend|x)=0.6 to the trend regime. If the next-day expected payoff is 0.015 return units in trend and -0.01 return units in mean reversion, what conditional expected payoff E[r|x] does the model imply?机器学习中等数值题未尝试面试订阅4149Naive Bayes Posterior 4A generative regime model assigns posterior probability P(trend|x)=0.4 to the trend regime. If the next-day expected payoff is 3 return units in trend and 1 return units in mean reversion, what conditional expected payoff E[r|x] does the model imply?机器学习中等数值题未尝试面试订阅4150Naive Bayes Posterior 5A generative regime model assigns posterior probability P(trend|x)=0.8 to the trend regime. If the next-day expected payoff is -2 bps in trend and 5 bps in mean reversion, what conditional expected payoff E[r|x] does the model imply?机器学习中等数值题未尝试面试订阅4151Generative Classification with a Missing Feature 1A two-feature naive Bayes model was trained generatively, but at prediction time X2 is missing. Prior P(Y=1)=0.5, P(X1=1|Y=1)=0.8, P(X1=1|Y=0)=0.3, P(X2=1|Y=1)=0.75, P(X2=1|Y=0)=0.4. You only observe X1=1. What posterior P(Y=1|X1) should the generative model use?机器学习中等数值题未尝试面试订阅4591Martingale Carry Inference 1Spot is 100, maturity is 1 year, and the prepaid forward price is 97.0446. What continuous dividend yield q is implied?数理金融简单数值题未尝试面试订阅4592Martingale Carry Inference 2Spot is 80, maturity is 1 year, the forward price is 82.4364, and the risk-free rate is 5%. What continuous dividend yield q is implied?数理金融简单数值题未尝试面试订阅4593Martingale Carry Inference 3A stock has spot 90 and 1-year discounted risk-neutral expectation E[e -rT S T] = 88.218. What continuous dividend yield q is implied?数理金融简单数值题未尝试面试订阅4594Martingale Carry Inference 4Under risk-neutral pricing, spot is 120, risk-free rate is 4%, dividend yield is 1.5%, and maturity is 2 years. What is E[S T]?数理金融简单数值题未尝试面试订阅4595Martingale Carry Inference 5A 1-year prepaid forward on a stock is 96, and the corresponding forward price is 100.9221. What risk-free rate r is implied?数理金融简单数值题未尝试面试订阅4881Infer Regime Probability From Conditional Means 16A payoff has conditional mean 4 in a calm regime and -3 in a stress regime. The overall expected payoff is 1.9. What probability of the calm regime is implied?数理金融困难数值题未尝试面试订阅4882Infer Missing Stress Mean From Tower Equation 17A payoff has conditional mean 5 in a calm regime and unknown mean m in a stress regime. The calm regime probability is 0.6 and the overall expected payoff is 2.6. What is m?数理金融困难数值题未尝试面试订阅4883Infer Stress Probability From Conditional Means 18A payoff has conditional mean 2 in a normal regime and -4 in a stress regime. The overall expected payoff is 0.8. What probability of the stress regime is implied?数理金融困难数值题未尝试面试订阅4884Infer Missing Mid-Regime Mean In a Three-State Mixture 19A payoff has regime probabilities 0.5, 0.3, and 0.2. The conditional means are 1, m, and -2, and the overall expected payoff is 0.7. What is m?数理金融困难数值题未尝试面试订阅4885Infer Regime Exercise Probability From Overall Expectation 20In regime A, which occurs with probability 0.4, an option pays 10 if exercised. In regime B, which occurs with probability 0.6, it pays 4 if exercised, and the regime-B exercise probability is 0.25. The overall expected payoff is 2.6. What exercise probability in regime A is implied?数理金融困难数值题未尝试面试订阅4971Expected Time to Absorption With One Reset LinkA CTMC has states 0, 1, 2 with state 2 absorbing. From 0 it jumps to 1 at rate a=1, and from 1 it jumps to 2 at rate b=2 or back to 0 at rate c=1. What is the expected time to hit state 2 starting from 0?随机过程中等数值题未尝试面试订阅4972Reset Rate Implied by a Target Expected Hitting TimeA CTMC has states 0,1,2 with state 2 absorbing. From 0 it jumps to 1 at rate a=1, and from 1 it jumps to 2 at rate b=1.5 or back to 0 at rate c. If the expected time to hit state 2 from state 0 is 3, what is c?随机过程中等数值题未尝试面试订阅4973Forward Rate Implied by a Target Expected Hitting TimeA CTMC has states 0,1,2 with state 2 absorbing. The rates are a=2 from 0 to 1 and c=1 from 1 back to 0, while the rate from 1 to 2 is b. If the expected time to hit state 2 from state 0 is 1.5, what is b?随机过程中等数值题未尝试面试订阅