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3236Can a p-Value Be Combined with Prior Conviction?A trader says, "I had a strong prior view, and the p-value came out 0.03, so now I can blend the two and call the trade 97% likely." Why is that not a coherent frequentist calculation?统计简单essay未尝试面试订阅3237Why a Posterior Mean Can Move While the MLE Does NotA desk observes only 4 new defaults for a rare event. Using a strong historical Beta prior, the Bayesian posterior mean default rate is much lower than the sample proportion, while the frequentist MLE equals the sample proportion exactly. Explain why these two answers can legitimately differ and what each one is conditioning on.统计简单essay未尝试面试订阅3238Empirical Bayes Sits Between the CampsWhy does empirical Bayes often feel like it sits between Bayesian and frequentist workflows?统计简单essay未尝试面试订阅3239Large p-Value Does Not Mean High Posterior Null ProbabilityWhy is it dangerous to read a large p-value as strong evidence that the null is true?统计简单essay未尝试面试订阅3240Prior Sensitivity Is a Feature, Not Always a BugWhy can prior sensitivity analysis be a feature rather than a bug in a high-stakes decision problem?统计简单essay未尝试面试订阅3376Halving h in the Trapezoid RuleA smooth integrand is approximated with the trapezoid rule. If the current error at step size h is about 0.08, what is the rough error after halving the step size?数学中等derivation未尝试面试订阅3378Halving h in Simpson's RuleA smooth integrand is approximated with the Simpson rule. If the current error at step size h is about 0.016, what is the rough error after halving the step size?数学中等derivation未尝试面试订阅3386Why Trapezoid Overestimates a Convex FunctionWhy does the trapezoid rule typically overestimate the integral of a convex function on one panel?数学中等essay未尝试面试订阅3387Why Midpoint Often Beats Trapezoid on Smooth ProblemsWhy can the midpoint rule outperform the trapezoid rule on smooth integrands even though both are second-order?数学中等essay未尝试面试订阅3388When Simpson's Rule Is Especially AttractiveWhen is Simpson's rule especially attractive compared with midpoint or trapezoid?数学中等essay未尝试面试订阅3389Why Oscillatory Integrals Need CareWhy can standard coarse-grid quadrature be unreliable on highly oscillatory integrals?数学中等essay未尝试面试订阅3390Why Adaptive Refinement Helps Near KinksWhy is adaptive refinement often better than a uniform fine grid when the integrand has a kink or localized sharp feature?数学中等essay未尝试面试订阅3396Max Mesh Under a Central-Difference Error BudgetA central-difference first-derivative formula has truncation error bounded by M h 2 / 6, and a third-derivative bound gives M=12. How large can h be if you want the truncation error below 0.002?数学中等derivation未尝试面试订阅3397Forward-Difference Mesh Under a Linear Error BoundA forward-difference derivative formula has truncation error bounded by M h / 2 with M=8. How large can h be if the truncation budget is 0.01?数学中等derivation未尝试面试订阅3398Second-Derivative Mesh From a Fourth-Derivative BoundA centered second-derivative formula has error bounded by M h 2 / 12 with M=24. What h keeps the truncation error at or below 0.005?数学中等derivation未尝试面试订阅3399Balancing Truncation and Rounding ErrorSuppose a numerical derivative has total error model E(h)=3h 2 + 10 -4 /h. What h minimizes this approximation?数学中等derivation未尝试面试订阅3400Minimum Error at the Balanced MeshUsing E(h)=3h 2 + 10 -4 /h and the optimal h from first-order balancing, what is the minimum modeled error?数学中等derivation未尝试面试订阅3401Explicit Heat Stability Cap 1For an explicit heat-equation update, diffusion coefficient kappa = 0.5, and grid spacing Delta x = 0.1, what is the largest stable time step under the standard Delta t <= Delta x 2 / (2 kappa) rule?数学中等derivation未尝试面试订阅3402Explicit Heat Stability Cap 2For an explicit heat update with kappa = 1.2 and Delta x = 0.05, what is the largest stable Delta t under the same rule?数学中等derivation未尝试面试订阅3403Upwind Advection CFL LimitAn explicit upwind advection scheme has speed u = 2 and grid spacing Delta x = 0.03. What is the largest stable Delta t under the CFL rule Delta t <= Delta x / |u|?数学中等derivation未尝试面试订阅