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2036Quadratic Jensen Gap From Mean and Variance 16Let phi(x)=x 2. If a random variable X has E[X]=2 and Var(X)=5, what is E[phi(X)] - phi(E[X])?数学简单数值题未尝试免费2037Why Jensen Matters for Nonlinear Risk Transforms 17Why is it dangerous to plug an average state into a nonlinear convex risk transform and treat that as the average transformed risk?数学中等essay未尝试面试订阅2038Universal Lower Bound for a Convex Stress Multiplier 18A convex stress multiplier is phi(x)=e x. If a signal X has mean 0.2, what lower bound does Jensen's inequality give for E[e X]?数学中等derivation未尝试免费2039Convex Penalty for Mixed Schedules 19Two execution schedules have penalties phi(q 1) and phi(q 2) under a convex phi. What does Jensen say about a random 50-50 mix versus the penalty at the average size?数学困难derivation未尝试面试订阅2040Three-Scenario Square-Root Impact Gap 20Suppose V takes values 0, 3, and 8 with equal probability. Compute E[sqrt(1+V)] and sqrt(1+E[V]).数学困难数值题未尝试面试订阅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2042Jensen Upper Bound for an Expected Log Score 22If X > -1 almost surely and E[X]=0.2, what upper bound does Jensen give for E[ln(1+X)]?数学简单derivation未尝试免费2044Why Equality Holds Only Without Dispersion 24For a strictly convex phi, when can Jensen's inequality E[phi(X)] >= phi(E[X]) hold with equality?数学中等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2052Derive the Newton Update for exp(x) + cx = d 7Derive the Newton iteration for solving exp(x) + cx = d.数学中等derivation未尝试免费2060Newton Versus Fixed-Point Iteration 15Why does Newton usually converge in fewer iterations than a naive fixed-point map when both behave well near the same root?数学困难derivation未尝试免费2424Convexity of the Log-Cosh Loss 4Show that ell(r)=ln cosh(r) is convex in the residual r.机器学习中等derivation未尝试免费2425Why Asymmetric Loss Changes the Optimal Prediction 21Why does changing the relative penalty on overprediction versus underprediction generally move the Bayes act away from the conditional mean?机器学习困难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2429Total Huber Loss on a Residual SetUsing Huber loss with delta = 1, compute the total loss on residuals 0.5, -1.2, and 3.0.机器学习中等数值题未尝试面试订阅2430Why the Weighted-Brier Bayes Act Is Still a Weighted Mean 7For binary Y and weighted squared loss alpha Y (1-q) 2 + beta (1-Y) q 2, derive the Bayes probability q as a function of p=P(Y=1|X).机器学习困难derivation未尝试面试订阅2433Pinball Loss Subgradient at the Kink 9For pinball loss rho tau(r)=tau r if r>=0 and (tau-1)r if r<0, what is the subgradient set at r=0?机器学习中等derivation未尝试面试订阅2434Why a Strictly Proper Loss Must Prefer Truthful Probabilities 10Why is it desirable for a probabilistic classification loss to be strictly proper rather than merely classification-accurate?机器学习困难derivation未尝试面试订阅2435Why Log-Loss Punishes Overconfidence More Than Brier LossWhy does log-loss react much more harshly than Brier loss when a model assigns near-certainty to the wrong class?机器学习困难essay未尝试面试订阅2437Why Huber Sits Between Squared and Absolute LossWhy is Huber loss often described as sitting 'between' squared loss and absolute loss?机器学习中等essay未尝试面试订阅