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3493Coefficient Making W_6 - aW_2 Independent of W_4 - W_1Choose a so that W 6 - aW 2 is independent of W 4 - W 1.随机过程简单derivation未尝试面试订阅3494Coefficient Making a Sample Average Residual Independent of W_3Choose a so that (W 1 + W 2 + W 3)/3 - aW 3 is independent of W 3.随机过程简单derivation未尝试面试订阅3495Coefficient Making W_4 - a(W_2 + W_1) Independent of W_1Choose a so that W 4 - a(W 2 + W 1) is independent of W 1.随机过程简单derivation未尝试面试订阅3501Conditional Mean of W_1 Given W_4 = 2Given W 4 = 2, what is E[W 1 | W 4 = 2]?随机过程简单derivation未尝试面试订阅3502Conditional Variance of W_1 Given W_4What is Var(W 1 | W 4)?随机过程简单derivation未尝试面试订阅3504Conditional Mean of W_2 + W_4 Given W_6 = 3Given W 6 = 3, what is E[W 2 + W 4 | W 6 = 3]?随机过程简单derivation未尝试面试订阅3505Conditional Mean of W_2 - W_1 Given W_5 = 1Given W 5 = 1, what is E[W 2 - W 1 | W 5 = 1]?随机过程简单derivation未尝试面试订阅3511Why Zero Covariance Gives Independence HereWhy does making two Brownian linear combinations uncorrelated automatically make them independent in these questions?随机过程中等essay未尝试面试订阅3512Why Brownian Scaling Is Operationally UsefulWhy is Brownian scaling more than a formal symmetry and actually useful in interview-style calculations?随机过程中等essay未尝试面试订阅3513Why Conditioning on an Endpoint Creates a BridgeWhy does conditioning Brownian motion on a future endpoint naturally produce bridge-style residuals?随机过程中等essay未尝试面试订阅3514Why Independent Increments Matter in Quant PracticeWhy is the independent-increment property operationally important when reasoning about multi-horizon risk or signal aggregation?随机过程中等essay未尝试面试订阅3515Why Brownian Paths Feel Smooth but Are NotWhy can Brownian paths be continuous everywhere and still fail to behave like ordinary smooth curves?随机过程中等essay未尝试面试订阅3591GBM Drift for a Median Target Over Two YearsA geometric Brownian motion satisfies dS t = mu S t dt + 0.3 S t dW t with S 0 = 100. If the median of S 2 should equal 128, what drift mu is required?随机过程中等derivation未尝试面试订阅3594GBM Drift Implied by a Mild Median UpliftA geometric Brownian motion satisfies dS t = mu S t dt + 0.15 S t dW t with S 0 = 60. If the median of S 1.5 should equal 66, what drift mu is required?随机过程中等derivation未尝试面试订阅3596Volatility from a 95th-to-Median Ratio in One YearFor a GBM, the ratio of the 95th percentile of S 1 to its median is observed to be 1.9. What volatility sigma does this imply?随机过程中等derivation未尝试面试订阅3598Volatility Matching a 97.5th-to-Median SpreadFor a GBM, the ratio of the 97.5th percentile of S 1.5 to its median is observed to be 2.2. What volatility sigma does this imply?随机过程中等derivation未尝试面试订阅3599Short-Horizon Volatility from an Upper Quantile RatioFor a GBM, the ratio of the 90th percentile of S 0.5 to its median is observed to be 1.3. What volatility sigma does this imply?随机过程中等derivation未尝试面试订阅3601OU Long-Run Mean Implied by a Conditional Mean Target 1An OU process satisfies dX t = 0.8(theta - X t)dt + sigma dW t with X 0 = 7. If E[X 1] should equal 5.2, what theta is implied?随机过程中等derivation未尝试面试订阅3603OU Long-Run Mean Implied by a Conditional Mean Target 3An OU process satisfies dX t = 0.6(theta - X t)dt + sigma dW t with X 0 = 10. If E[X 0.5] should equal 8.9, what theta is implied?随机过程中等derivation未尝试面试订阅3606OU Volatility Implied by a Conditional Variance Target 1An OU process satisfies dX t = 1(theta - X t)dt + sigma dW t. If Var(X 1 | X 0) should equal 0.45, what sigma is required?随机过程中等derivation未尝试面试订阅