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3421Entropy Gain From Splitting a Quarter-Mass SymbolA source contains a symbol of probability 1/4. If you split that symbol into four equally likely sublabels while leaving everything else unchanged, by how many bits does the entropy increase?数学简单derivation未尝试面试订阅3422Entropy Gain From Splitting a 30% State Into ThreeA source has a single state of probability 0.3. If that state is refined into three equally likely sublabels, how many bits of entropy are added?数学简单derivation未尝试面试订阅3423Entropy Loss From Merging Two Equal 1/8 StatesIf two symbols each of probability 1/8 are merged into a single symbol while everything else stays unchanged, by how many bits does the entropy decrease?数学简单derivation未尝试面试订阅3424Entropy Loss From Merging a 20% State and a 30% StateTwo states of probabilities 0.2 and 0.3 are merged into a single 0.5-mass state. By how many bits does the entropy fall?数学简单derivation未尝试面试订阅3425Entropy Gain From Splitting a 12% State Into EightA symbol of probability 0.12 is split into eight equally likely sublabels. How many bits of entropy are added?数学简单derivation未尝试面试订阅3426Remaining Entropy After Revealing a 3-4-5 Bucket LabelA source is uniform over 12 states. You reveal which bucket the state lies in, where the bucket sizes are 3, 4, and 5. What is the remaining entropy H(X|bucket)?数学简单derivation未尝试面试订阅3427Remaining Entropy After Revealing a 1-3-4-8 PartitionA source is uniform over 16 states. A side signal reveals whether the realized state lies in a block of size 1, 3, 4, or 8. What is H(X|signal)?数学简单derivation未尝试面试订阅3428Remaining Entropy After a 4-vs-6 Split SignalA source is uniform over 10 states. You reveal whether the outcome lies in the first 4 states or in the last 6 states. What is the remaining entropy?数学简单derivation未尝试面试订阅3429Entropy of a Three-Regime Disjoint Alphabet SourceA regime label takes values A, B, C with probabilities 0.2, 0.5, 0.3. Conditional on A the source is deterministic, conditional on B it is uniform over 4 symbols, and conditional on C it is uniform over 2 symbols. Assuming the symbol sets are disjoint across regimes, what is the total source entropy?数学简单derivation未尝试面试订阅3430Entropy of a Fair 3-vs-5 Regime MixtureA fair regime bit first chooses source A or B. Source A is uniform over 3 symbols and source B is uniform over 5 symbols, with disjoint alphabets. What is the total entropy of the emitted symbol?数学简单derivation未尝试面试订阅3431Fixed-Length Gap for a Uniform 10-State SourceA source is uniform over 10 symbols. If you insist on a fixed-length binary code, what is the gap between the required fixed length and the entropy?数学中等derivation未尝试面试订阅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