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2066Second Reciprocal-Balance Newton Step 21The reciprocal equation has an interior positive solution, and the desk wants one explicit Newton move. Apply one Newton step to solve x + 1/x = 3 starting from x 0 = 2.数学简单数值题未尝试免费2067Second Two-Cashflow Yield Newton Step 22Another bond calibration step is computed with the same Newton machinery. Solve 6/(1+y) + 106/(1+y) 2 = 101 by one Newton step starting from y 0 = 0.04.数学中等数值题未尝试免费2068What a Good Stopping Rule Uses 23What two diagnostics do desks commonly monitor together before stopping a Newton solve?数学中等essay未尝试免费2069Newton Needs a Nonzero Derivative at the Root 24Why is a simple root f'(r) != 0 the standard clean setting for fast Newton convergence?数学简单derivation未尝试免费2371Infer Covariance From an Optimal Control Weight 1A control variate Y has Var(Y)=16. The desk estimates the optimal coefficient b*=0.75 in X-b(Y-E[Y]). What Cov(X,Y) is implied?数学简单数值题未尝试免费2372Infer the Known Control Mean From the Adjusted Estimate 2A raw Monte Carlo estimator has sample mean Xbar=9. The control sample mean is Ybar=41, the desk uses b*=0.5, and the adjusted estimate Xbar-b(Ybar-mu Y) equals 8. What known control mean mu Y is implied?数学简单数值题未尝试免费2373Infer Antithetic Correlation From the Variance Ratio 3A payoff is averaged with its antithetic partner. The variance of the antithetic average is 30% of the crude single-path variance, and the two legs have equal variance. What correlation rho is implied between the paired payoffs?数学中等数值题未尝试免费2374Why the Control-Variate Estimator Is Unbiased 4Why is X-b(Y-E[Y]) unbiased for E[X] for any fixed constant b?数学中等derivation未尝试免费2375Why the Optimal Control Weight Equals Cov/Var 5Why does the variance-minimizing control weight satisfy b*=Cov(X,Y)/Var(Y)?数学困难derivation未尝试免费2376Paired-Difference Standard Error 6Two pricing engines are compared with common random numbers. The paired differences have sample standard deviation s D=2.8 across n=49 shared paths. What is the standard error of the estimated mean difference?数学简单数值题未尝试免费2377Required Shared Paths for a Half-Width Target 7A common-random-numbers comparison has paired-difference standard deviation s D=3.5. Using a 95% normal half-width target of 0.49, about how many shared paths are needed?数学中等数值题未尝试免费2378Why Common Random Numbers Help Comparisons 8Why can common random numbers reduce the variance of the difference between two estimators even if each estimator on its own is unchanged?数学中等derivation未尝试免费2379Variance Formula for an Antithetic Average 9If X and X' have the same variance sigma 2 and correlation rho, what is Var((X+X')/2)?数学中等derivation未尝试面试订阅2380Why a Stratified Estimator Is Unbiased 10Why is the weighted sum of unbiased within-stratum sample means unbiased for the overall expectation?数学困难derivation未尝试面试订阅2382Remaining Variance Fraction Under Pairing 12Without common random numbers, two independent estimators have standard deviations 4 and 5, so the variance of their difference is 4 2+5 2. With pairing, the observed paired-difference standard deviation is 3. What fraction of the unpaired difference variance remains?数学简单数值题未尝试免费2383Law of Total Variance in a Regime Simulator 13What two pieces make up Var(X) under the law of total variance when a simulator first samples a regime Z and then samples X conditional on Z?数学中等derivation未尝试免费2384Why Regime Conditioning Can Reduce Variance 14Why can conditioning on a regime variable before sampling payoffs reduce Monte Carlo variance?数学困难derivation未尝试面试订阅2385When a Control Variate Can Make Things Worse 15Why can a badly chosen control coefficient increase variance instead of reducing it?数学困难derivation未尝试面试订阅2386Infer Regime Probability From Conditional Means 16A payoff has conditional mean 5 in a calm regime and -2 in a stress regime. The unconditional mean is 2.2. What calm-regime probability is implied?数学简单数值题未尝试免费2387Why Importance Sampling Stays Unbiased 17Why does reweighting by the likelihood ratio keep an importance-sampling estimator unbiased for the original expectation?数学中等derivation未尝试免费