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3450Second Sensor Noise Required for 0.4 Bits TotalA latent X ~ N(0,1) is already observed by one sensor with noise variance 3. A second independent sensor with noise variance v will be added. What v makes the total mutual information 0.4 bits?数学中等derivation未尝试面试订阅3451Mutual Information of an Asymmetric Binary AlertA latent regime R is 1 with probability 0.3. An alert Y has hit rate 0.8 and false-alarm rate 0.2. How many bits of mutual information does Y carry about R?数学简单derivation未尝试面试订阅3452Posterior-vs-Prior KL After a High-Quality Alert FiresA regime prior is 0.1. An alert fires with hit rate 0.9 and false-alarm rate 0.05. If the alert fires, what is the KL divergence in bits between the posterior on the regime and the prior?数学简单derivation未尝试面试订阅3454KL Between Signal Distributions Under the Two RegimesA binary signal Y has P(Y=1|R=1) = 0.8 and P(Y=1|R=0) = 0.3. What is KL(P(Y|R=1) || P(Y|R=0)) in bits?数学中等derivation未尝试面试订阅3455Posterior-vs-Prior KL After No Alert ArrivesA regime prior is 0.25. An alert has hit rate 0.85 and false-alarm rate 0.15. If no alert arrives, what is the KL divergence in bits between the posterior on the regime and the prior?数学困难derivation未尝试面试订阅3471One Ternary Plus Three Binary Assays for One Fault or NoneYou may run one ternary assay and three binary assays non-adaptively. There is either no faulty component or exactly one faulty component. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3472Two Ternary and One Binary Assays for One Fault or NoneYou may run two ternary assays and one binary assay non-adaptively. There is either no faulty component or exactly one faulty component. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3473One Four-Way and Two Binary Assays for One Fault or NoneYou may run one four-way assay and two binary assays non-adaptively. There is either no faulty component or exactly one faulty component. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3474Binary Assays When Each Faulty Unit Can Fail in Two ModesYou have three yes/no assays run non-adaptively. Either no component is faulty, or exactly one component is faulty and it can be in one of two distinct failure modes. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3475One Ternary and Two Binary Assays With Three Bad Modes Per ComponentYou may run one ternary assay and two binary assays non-adaptively. Either nothing is wrong, or exactly one component is wrong and it can be in one of three bad modes. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3476Entropy Lower Bound for a 1/2, 1/4, 1/8, 1/8 PriorA hidden state has prior probabilities [0.5, 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3477Entropy Lower Bound for a 1/2, 1/4, 1/4 PriorA hidden state has prior probabilities [0.5, 0.25, 0.25]. 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