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2388Why Rare Events Cause Huge Monte Carlo Noise 18Why can a rare-event payoff have an unstable Monte Carlo estimate even when most simulated paths look harmless?数学中等derivation未尝试免费2389Why Regime Averaging Can Beat Crude Mixing 19Why can separately estimating conditional means by regime and then averaging them outperform one crude pooled Monte Carlo run?数学困难derivation未尝试面试订阅2390Why Antithetics May Fail on a Highly Nonmonotone Payoff 20Why can antithetic pairing fail to reduce variance for a payoff that oscillates strongly across the state space?数学困难derivation未尝试面试订阅239195% Half-Width From Sample Volatility and Path Count 21A Monte Carlo estimator has sample standard deviation 5 across n=400 paths. Using the normal approximation, what is the 95% half-width?数学简单数值题未尝试免费2392Required Paths for a Target Standard Error 22A crude estimator has standard deviation 6. How many paths are needed to bring the standard error down to 0.3?数学简单数值题未尝试免费2393RMSE of an Unbiased Monte Carlo Estimate 23An unbiased estimator has standard deviation 4 over one path and uses n=64 independent paths. What is the RMSE of the sample mean?数学中等数值题未尝试免费2394Confidence Interval From Mean and Standard Error 24A Monte Carlo estimate is 12.0 with standard error 0.4. Using the normal approximation, what 95% confidence interval do you report?数学中等数值题未尝试免费2395Why Bias Still Matters Alongside Variance 25Why is a low-variance Monte Carlo estimator still problematic if it is biased?数学困难derivation未尝试面试订阅3368Midpoint Rule Exact on a LineUsing a single-panel midpoint rule, approximate \int 0 4 (1+x)\,dx.数学简单derivation未尝试面试订阅3369Trapezoid Rule on a Symmetric Convex FunctionUsing a single-panel trapezoid rule, approximate \int -1 1 x 2\,dx.数学简单derivation未尝试面试订阅3370Midpoint Rule Misses a Symmetric BowlUsing a single-panel midpoint rule, approximate \int -1 1 x 2\,dx.数学简单derivation未尝试面试订阅3371Simpson on a CubicUsing one Simpson panel, approximate \int 0 2 x 3\,dx.数学中等derivation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3381Degree of Exactness of Two-Point GaussFor two-point Gauss-Legendre quadrature on [-1,1], what is the highest polynomial degree integrated exactly?数学中等derivation未尝试面试订阅3382Two-Point Gauss on x^2+1Using two-point Gauss-Legendre on [-1,1], approximate \int -1 1 (x 2+1)\,dx.数学中等derivation未尝试面试订阅3383Two-Point Gauss on x^4Using two-point Gauss-Legendre on [-1,1], what approximation do you get for \int -1 1 x 4\,dx?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