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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