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3483Expected Questions With a Rare-State TailA binary tree uses code lengths [2, 2, 2, 3, 3] for states with priors [1/4, 1/4, 1/4, 1/8, 1/8]. What is the expected number of questions?数学中等derivation未尝试面试订阅3484Expected Questions Under a Heavy First StateA binary tree uses code lengths [1, 3, 3, 3, 3] for priors [1/2, 1/8, 1/8, 1/8, 1/8]. What is the expected number of questions?数学中等derivation未尝试面试订阅3485Expected Questions in a Ternary-Then-Binary StrategyA strategy first asks a ternary question that isolates state A on one branch, state B on another branch, and groups states C and D on the last branch. If the prior over [A, B, C, D] is [0.4, 0.3, 0.2, 0.1], what is the expected number of questions if the grouped branch uses one extra yes/no question?数学中等derivation未尝试面试订阅3486Why Adaptive Testing Beats Static MenusWhy can an adaptive questioning strategy distinguish more states than a non-adaptive test battery with the same primitive tests?数学中等essay未尝试面试订阅3487Why Entropy Is a Lower Bound, Not Always an Exact CountWhy does the Shannon entropy of the prior give only a lower bound on expected yes/no questions rather than always the exact optimum?数学中等essay未尝试面试订阅3488Why a None State Changes the Coding ProblemIn one-defective search problems, why does adding the possibility of no defect change the counting argument in a substantive way?数学中等essay未尝试面试订阅3489Why Prior Skew Should Change the Query TreeWhy should the optimal query tree usually isolate high-probability states earlier than low-probability states?数学中等essay未尝试面试订阅3490Why Branch Constraints Can Matter More Than Alphabet SizeWhy can two testing systems with the same maximum branch count per question have very different identification power once branch-specific follow-up rules are imposed?数学中等essay未尝试面试订阅3539Why Reflection Helps With Barriers but Not Every FunctionalWhy is reflection so powerful for barrier events yet much less useful for path functionals like time averages?随机过程中等essay未尝试面试订阅3540Reflection Versus Optional StoppingIn one paragraph, contrast what reflection principle and optional stopping are each best at in Brownian barrier problems.随机过程中等essay未尝试面试订阅3556Interval Width When the Start Is One-Third of the Way UpBrownian motion starts at x = b/3 inside [0, b]. What b makes the expected exit time equal to 8?随机过程中等derivation未尝试面试订阅3557Expected Exit Time When Upper-Hit Probability Is 2/3 on [0, 9]Brownian motion starts somewhere in [0, 9] such that the probability of hitting 9 before 0 is 2/3. What is the expected exit time?随机过程中等derivation未尝试面试订阅3558Symmetric Interval and Start Matching Two TargetsBrownian motion starts at x inside [-L, L]. If the probability of hitting +L before -L is 0.75 and the expected exit time is 12, what is x?随机过程中等derivation未尝试面试订阅3559Lower Barrier and Hit Probability From a Target Exit TimeBrownian motion starts at x = 1 with upper barrier 5. The lower barrier a is chosen so that the expected exit time is 8. What is the resulting probability of hitting 5 before a?随机过程中等derivation未尝试面试订阅3560Start Needed to Match Both a Probability and an Exit Time on [0, 10]Brownian motion starts at x inside [0, 10]. If the probability of hitting 10 before 0 is 0.6, what is the expected exit time?随机过程中等derivation未尝试面试订阅3561Why the Hitting Probability Is HarmonicWhy does the probability of hitting the upper barrier before the lower barrier end up being a linear function of the start x for Brownian motion?随机过程中等essay未尝试面试订阅3562Why Expected Exit Time Is Quadratic in the StartWhy does expected exit time from a bounded interval depend quadratically on the starting point rather than linearly?随机过程中等essay未尝试面试订阅3563Finite-Horizon Touching vs Eventual HittingWhy should finite-horizon barrier-touch probabilities not be confused with the interval-hitting probabilities used in bounded first-passage problems?随机过程中等essay未尝试面试订阅3564Why Optional Stopping Needs a Bounded Setup HereWhy are bounded intervals the natural place where optional-stopping arguments for Brownian hitting questions are most reliable?随机过程中等essay未尝试面试订阅3601OU Long-Run Mean Implied by a Conditional Mean Target 1An OU process satisfies dX t = 0.8(theta - X t)dt + sigma dW t with X 0 = 7. If E[X 1] should equal 5.2, what theta is implied?随机过程中等derivation未尝试面试订阅