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2027Log Barrier Jensen Direction 7A utilization score explodes as it nears 1, so the convexity direction matters for stress design. Let u(x)=-ln(1-x) on x<1. If U is random and almost surely below 1, compare E[u(U)] and u(E[U]).数学中等derivation未尝试免费2028Log Carry Score Is Concave 8Let psi(x)=ln(1+x) on x>-1. Compare E[psi(X)] and psi(E[X]).数学中等derivation未尝试面试订阅2029Exact Utilization Penalty Gap With Unequal Weights 9A utilization surcharge uses phi(q)=q/(1-q) on 0<=q<1. Suppose Q equals 0 with probability 1/3 and 3/4 with probability 2/3. Compute E[phi(Q)] and phi(E[Q]).数学困难数值题未尝试面试订阅2030Recovering Scenario Weights From a Concave Impact Average 10Let V take values 0 and 3, with probabilities p and 1-p. If E[sqrt(1+V)] = 5/4, determine p and then compute sqrt(1+E[V]).数学中等derivation未尝试面试订阅2031Backing Out the Stress State From a Reciprocal Plug-In Score 11A funding-buffer score uses phi(L)=1/(1+L). Suppose leverage L equals 0 with probability 1/2 and H with probability 1/2. If phi(E[L]) = 1/3, what is H and what is E[phi(L)]?数学简单derivation未尝试免费2032Barrier Score Gap for a Two-State Utilization Model 12Let u(x)=-ln(1-x) on x<1. Suppose U equals 0 with probability 1/2 and 3/4 with probability 1/2. Compute E[u(U)] and u(E[U]).数学简单数值题未尝试免费2033Certainty-Equivalent Return From Expected Log Growth 13A strategy produces one-period returns of 0% or 60%, each with probability 1/2 on one dollar of wealth. Compute E[ln(1+R)] and the certainty-equivalent constant return r ce satisfying ln(1+r ce)=E[ln(1+R)].数学中等derivation未尝试面试订阅2034Conditional Jensen Lower Bound 14If phi is convex, what inequality holds between E[phi(X)|F] and phi(E[X|F]) almost surely?数学困难derivation未尝试面试订阅2035Funding Buffer Gap With Unequal Scenario Weights 15The high-leverage state is rarer, but still materially affects the convex average. A funding-buffer model uses phi(L)=1/(1+L). Suppose L takes values 1 and 4 with probabilities 1/4 and 3/4. Compute E[phi(L)] and phi(E[L]).数学困难数值题未尝试面试订阅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2043Log Carry Gap From Two Scenarios 23The desk wants to see the exact concave Jensen gap, not just the inequality direction. A desk scores carry through psi(x)=ln(1+x). Suppose X takes values 0 and 3 with probabilities 1/2 and 1/2. Compute E[psi(X)] and psi(E[X]).数学中等数值题未尝试面试订阅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2046One Newton Step for a Log-Balance Equation 1A smooth calibration map mixes a linear term and a logarithmic correction. Apply one Newton step to solve x + 2 ln x = 3 starting from x 0 = 1.数学简单数值题未尝试免费