Consider the following dataset with features X, Y and Z
[table]
X | Y | Z
Jack | 0 | A Jack | 1 | B Amy | 0 | A Amy | 1 | B Sam | 0 | B Sam | 1 | B
[/table] Let [math: H] denote the entropy function. Which of the following is true?

Question

BlackTom AI · Accepted Answer

To evaluate the statement, I will compute the entropy for each feature X, Y, and Z from the given dataset.
Option a: H(X) = 1.213, H(Y) = 1, H(Z) = 0.1544. Here, H(Y) = 1 is correct for a balanced binary variable (three 0s and three 1s), but H(X) = 1.213 is not the same as the actual H(X). Since X takes three categories (Jack, Amy, Sam) each appearing twice, the distribution is uniform (1/3, 1/3, 1/3), giving H(X) = log2(3) ≈ 1.585, not 1.213. The stated H(Z) = 0.1544 is also inconsistent with Z having two outcomes A and B with probabilities 1/3 and 2/3, which yields H(Z) ≈ 0.9183, not 0.1544. Overall, this option misreports both H(X) and H(Z).
Option b: H(X) = 1.785, H(Y) = 1, H(Z) = 0.656. While H(Y) = 1 is correct, H(X) = 1.785 would imply a non-uniform distribution over three X-values, which is not the case since each X occurs exactly twice (uniform over three categories). The actual H(X) is log2(3) ≈ 1.585. The H(Z) value 0.656 is closer to the real H(Z) but still not exact; the correct H(Z) is about 0.9183. So this option misstates H(X) and H(Z).
Option c: H(X) = 1.585, H(Y) = 0, H(Z) = 0.9183. This option has H(X) correct at 1.585, reflecting the uniform distribution over three X-values. However, H(Y) = 0 would mean Y is constant, which is false since Y takes values 0 and 1 equally often, giving H(Y) = 1. The Z value 0.9183 is correct, but the Y part is incorrect.
Option d: H(X) = 1.585, H(Y) = 1, H(Z) = 0.9183. This aligns with the dataset analyses: X is uniformly distributed over three categories, yielding H(X) ≈ 1.585; Y is balanced between 0 and 1, yielding H(Y) = 1; Z has probabilities 1/3 for A and 2/3 for B, yielding H(Z) ≈ 0.9183. All three components match the calculated entropies from the data.

Consider the following dataset with features X, Y and Z [table] X | Y | Z Jack | 0 | A Jack | 1 | B Amy | 0 | A Amy | 1 | B Sam | 0 | B Sam | 1 | B [/table] Let [math: H] denote the entropy function. Which of the following is true?

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Consider the following dataset with features X, Y and Z [table] X | Y | Z Jack | 0 | A Jack | 1 | B Amy | 0 | A Amy | 1 | B Sam | 0 | B Sam | 1 | B [/table] Let H denote the entropy function. Which of the following is true?

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