Optimize interval-scoring strategy

You are given a five-round interval-estimation game used in market-making interviews. In each round, the interviewer asks a quantitative question with an unknown true numeric answer A. You must report a closed interval [L, U]. Scoring per round: if A ∉ [L, U], score = 0; if A ∈ [L, U], score = L/U. The goal after five rounds is a total score ≥ 2.0. Assume for each round you can form a subjective probability distribution for A. a) Formulate the optimization to choose L and U that maximizes expected per-round score E[(L/U)·1{A∈[L,U]}] given a known continuous pdf for A. b) Derive first-order optimality conditions and discuss how the optimal coverage probability compares to conventional confidence levels under light- vs heavy-tailed beliefs. c) Describe a strategy to allocate risk across five rounds (e.g., dynamic programming or heuristic thresholds) to target total ≥ 2.0, including how to adjust interval tightness after early wins/losses. d) Provide a quick mental method for approximately optimal [L, U] under log-normal beliefs; explain how you’d adapt for heavy tails. e) Outline a simple simulation to validate your strategy and estimate its probability of meeting the ≥ 2.0 target.