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1969Which Side of Zero Is Optimal in the Linear-Softplus Problem 24For K(x)=m x - n ln(1+e x) with 0<m<n/2, is the unique optimizer positive, zero, or negative?数学中等derivation未尝试免费1978Why the Three-Book Solution Is Inverse-Coefficient Weighted 8In min a x 2 + b y 2 + c z 2 subject to x+y+z=N, why do larger quadratic coefficients receive smaller allocations?数学中等derivation未尝试免费1994Spread-Constrained Allocation With Uneven Penalties 24The spread target fixes how far apart the outer books must sit, while the middle book absorbs the rest. Minimize 1x 2 + 2y 2 + 3z 2 subject to x+y+z=11 and x-z=1.数学中等derivation未尝试免费1995Alpha Maximization With Uneven Risk Units 25The second sleeve has both larger alpha and a different risk unit, so the optimal point must balance both effects. Maximize 4x + 6y subject to 4x 2 + 9y 2 = 225.数学困难derivation未尝试面试订阅2010Smoothed Worst of Two Affine Stress Terms 15The worst-case proxy is no longer a hard max, but a smooth convex substitute. Show that g(x) = ln(exp(2x) + exp(-1x + 3)) is convex on R.数学困难derivation未尝试免费2025Reciprocal Funding Buffer Is Convex 5Let phi(L)=1/(1+L) on L>-1. State the Jensen inequality relation between E[phi(L)] and phi(E[L]).数学困难derivation未尝试面试订阅2041Probability-Implied Reciprocal Buffer Score 21Leverage L takes values 1 and 4. If E[L] = 2.2, what probability p is on L=1, and what is E[1/(1+L)]?数学简单derivation未尝试免费2045Comparing Two Equal-Mean Schedules Under an Alternative Surcharge 25A utilization surcharge is c(q)=1/(2-q) on q<2. Schedule A is deterministic with Q=1. Schedule B uses Q=1/2 or 3/2 with probability 1/2 each. Compute E[c(Q)] for Schedule B and c(E[Q]) for the shared mean.数学困难数值题未尝试面试订阅2396Variance of an Equal-Weight Correlated EnsembleFive base models each have prediction variance 4, and every pair of model predictions has correlation 0.25. If you average the five predictions equally, what is the ensemble variance?机器学习简单derivation未尝试免费2399Optimal Weight on a Noisy Unbiased ModelModel A is unbiased with variance 9. Model B has variance 1.44 and fixed bias 0.6. If you blend them as P w = wA + (1-w)B and treat their errors as independent, what weight w minimizes MSE?机器学习困难derivation未尝试面试订阅2413Why Bagging Mainly Targets VarianceWhy is bagging usually described as a variance-reduction tool rather than a bias-reduction tool?机器学习中等essay未尝试面试订阅2415Why a Stable but Biased Model May Be Preferred OperationallyWhy might a desk prefer a slightly biased model that behaves predictably over a lower-bias model whose outputs swing wildly retrain to retrain?机器学习困难essay未尝试面试订阅2418Why Small Validation Sets Overreact to Complex ModelsWhy can small validation sets make model-comparison results look much noisier for complex models?机器学习中等essay未尝试面试订阅2427Decision Threshold Under Asymmetric Classification CostA false negative costs 5 and a false positive costs 1. If p is the predicted probability of the positive class, above what threshold should you classify as positive?机器学习中等derivation未尝试面试订阅2428Weighted-Absolute-Loss Constant ForecastDemand Y takes values 0, 2, and 5 with probabilities 0.5, 0.3, and 0.2. Under a loss of 3(Y-a) + + 1(a-Y) +, what constant forecast a minimizes expected loss?机器学习中等derivation未尝试面试订阅2446Hidden Validation Positives Implied by a Leaky Target EncoderA category appears 40 times in train with 18 positives and 10 times in validation. A target encoder is incorrectly fit on train plus validation and outputs 0.56 for that category. How many validation positives did the encoder implicitly use?机器学习简单数值题未尝试免费2448Held-Out Base Rate Implied by a Full-Sample Class WeightA training set has 100 labels with 30 positives. A class-weighting routine is mistakenly fit on all 125 labels and reports an overall positive rate of 0.36. What is the positive rate in the 25 held-out labels?机器学习中等数值题未尝试面试订阅2451Rare Category Survival Caused by Held-Out RowsA categorical preprocessor keeps a level only if it appears at least 5 times. In train alone, level Z appears 4 times. After the preprocessor is wrongly fit on the full sample, level Z appears to have frequency 7 and is kept. How many held-out Z rows caused the leak?机器学习简单数值题未尝试免费2455Repeated Validation Peeking During ResearchA researcher keeps trying new transformations and only retains the ones that improve the same validation score. Why does the validation set stop being a clean model-selection tool?机器学习困难essay未尝试面试订阅2459Using Revised Index Membership in Historical FilteringA backtest filters the universe using current index membership and then evaluates historical predictions on that restricted universe. Why is this also a train/test discipline problem?机器学习困难essay未尝试面试订阅