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?

View Explanation

Log in for full answers

Similar Questions

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?

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?

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?

Consider the following dataset with features X, Y and Z [table] X | Y | Z 0 | 0 | 0 0 | 1 | 1 1 | 0 | 0 1 | 1 | 1 2 | 0 | 1 2 | 1 | 1 [/table] Compute the entropy of [math: XYZ+1]\frac{XY}{Z+1}

Consider the following dataset with features X, Y and Z [table] X | Y | Z 0 | 0 | 0 0 | 1 | 1 1 | 0 | 0 1 | 1 | 1 2 | 0 | 1 2 | 1 | 1 [/table] Compute the entropy of [math: XYZ+1]\frac{XY}{Z+1}

Question at position 33 True/False Question: The entropy of a homogeneous set (a set of instances with the same label) is not always zero. TrueFalse

Question at position 16 Multiple Choice Question: Entropy can be negativeis used to calculate information gainis a measure of information gainis a measure of correlation between numeric variables

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler