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3432Fixed-Length Gap for a Uniform 5-State SourceA source is uniform over 5 symbols. What is the fixed-length binary coding gap in bits?数学中等derivation未尝试面试订阅3433Entropy of a Dyadic Four-State SourceWhat is the entropy of the distribution (1/2, 1/4, 1/8, 1/8)?数学中等derivation未尝试面试订阅3434Remaining Entropy After Revealing 'Special Pair or Not'A source is uniform over 8 states. A side signal reveals whether the state lies in a special pair of states or in the other 6 states. What is the remaining entropy?数学中等derivation未尝试面试订阅3435Entropy Reduction From Revealing One Prefix BitA source is uniform over 16 equiprobable states. If a side signal reveals the first binary prefix bit of the state label, by how many bits does entropy drop?数学中等derivation未尝试面试订阅3436Why Uniform Distributions Maximize Entropy Under a Support ConstraintWhy does spreading probability mass more evenly over a fixed finite support raise entropy?数学中等essay未尝试面试订阅3437Why Coarse-Graining Lowers EntropyWhy does merging labels generally reduce entropy rather than increase it?数学中等essay未尝试面试订阅3438Why Side Information Cannot Increase Remaining UncertaintyWhy is it impossible for an informative side signal to make the conditional entropy of the original source larger on average?数学中等essay未尝试面试订阅3439Why Fixed-Length Codes Waste Bits on Non-Power-of-Two AlphabetsWhy does a fixed-length binary code necessarily waste some average code length when the number of equally likely symbols is not a power of two?数学中等essay未尝试面试订阅3440Why Entropy Is Interpreted as Average SurpriseWhy is entropy often described as the average surprise of a source?数学中等essay未尝试面试订阅3441Base-Rate Forecast Penalty on a 70% EventA binary event truly occurs with probability 0.7. A forecast uses probability 0.55 instead of the true probability. What is the expected extra log-loss in bits relative to a calibrated forecast?数学困难derivation未尝试面试订阅3446Mutual Information of a Signal With Noise Variance FourA latent Gaussian alpha X ~ N(0,1) is observed through Y = X + eps with eps ~ N(0,4). How many bits of mutual information does Y carry about X?数学中等derivation未尝试面试订阅3447Mutual Information When Signal Variance Is Two and Noise Is SixA latent Gaussian factor has variance 2 and observation noise has variance 6. How many bits of mutual information does one observation carry about the factor?数学困难derivation未尝试面试订阅3448Incremental Information From a Second Independent SensorA latent X ~ N(0,1) is observed by two conditionally independent sensors with noise variances 4 and 9. How many extra bits do you gain by adding the second sensor after already seeing the first?数学中等derivation未尝试面试订阅3449Noise Variance Needed for 0.25 Bits of InformationA latent X ~ N(0,1) is observed as Y = X + eps with eps ~ N(0,sigma 2). What sigma 2 gives mutual information exactly 0.25 bits?数学中等derivation未尝试面试订阅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3456Why KL Is the Right Tax for MiscalibrationIn probability forecasting, why does KL divergence naturally show up as the penalty for using the wrong predictive distribution?数学中等essay未尝试面试订阅