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3478Entropy Lower Bound for a 1/4, 1/4, 1/4, 1/8, 1/8 PriorA hidden state has prior probabilities [0.25, 0.25, 0.25, 0.125, 0.125]. What is the Shannon lower bound, in yes/no questions on average, for any adaptive identification strategy?数学中等derivation未尝试面试订阅3480Entropy Lower Bound for a Six-State Dyadic PriorA hidden state has prior probabilities [0.25, 0.25, 0.125, 0.125, 0.125, 0.125]. What is the Shannon lower bound, in yes/no questions on average, for any adaptive identification strategy?数学中等derivation未尝试面试订阅3481Expected Questions in a Hot-State-First Binary TreeA binary decision tree asks first whether the state is S1. If not, it asks whether it is S2. If not, it asks whether it is S3, and otherwise concludes S4. If prior probabilities are [0.5, 0.25, 0.15, 0.10], what is the expected number of questions?数学中等derivation未尝试面试订阅3482Expected Questions for a Balanced Three-State TriageA binary tree first asks whether the state is in S1, S2 . If yes, one more question separates S1 from S2. If no, the answer is S3. If prior probabilities are [0.5, 0.25, 0.25], what is the expected number of questions?数学中等derivation未尝试面试订阅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3491Coefficient Making W_5 - aW_2 Independent of W_2Choose a so that W 5 - aW 2 is independent of W 2.随机过程简单derivation未尝试面试订阅3493Coefficient Making W_6 - aW_2 Independent of W_4 - W_1Choose a so that W 6 - aW 2 is independent of W 4 - W 1.随机过程简单derivation未尝试面试订阅3494Coefficient Making a Sample Average Residual Independent of W_3Choose a so that (W 1 + W 2 + W 3)/3 - aW 3 is independent of W 3.随机过程简单derivation未尝试面试订阅3495Coefficient Making W_4 - a(W_2 + W_1) Independent of W_1Choose a so that W 4 - a(W 2 + W 1) is independent of W 1.随机过程简单derivation未尝试面试订阅3496Variance of 2W_1 - W_4What is Var(2W 1 - W 4)?随机过程简单derivation未尝试面试订阅3497Covariance of Two Overlapping IncrementsWhat is Cov(W 4 - W 1, W 6 - W 3)?随机过程简单derivation未尝试面试订阅3498Variance of the Average of W_1, W_2, W_3What is Var((W 1 + W 2 + W 3)/3)?随机过程简单derivation未尝试面试订阅3499Variance of W_5 - 0.5W_2What is Var(W 5 - 0.5W 2)?随机过程简单derivation未尝试面试订阅