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

BlackTom AI · Accepted Answer

Question restatement and options:
- Given a dataset with features X, Y, Z as shown:
  Rows: (0,0,0), (0,1,1), (1,0,0), (1,1,1), (2,0,1), (2,1,1).
- We are asked to compute the entropy of the variable described by XY/(Z+1).
- Answer options: a) 2.11328, b) Undefined, c) 1.2516, d) 1.

Step-by-step analysis of each option:
- Option a: 2.11328
  To evaluate entropy, we first determine the distribution of the random variable XY/(Z+1) across the six samples.
  Compute XY and Z+1 per row:
  - (0,0,0): XY=0, Z+1=1 → value 0/1 = 0
  - (0,1,1): XY=0, Z+1=2 → value 0/2 = 0
  - (1,0,0): XY=0, Z+1=1 → value 0/1 = 0
  - (1,1,1): XY=1, Z+1=2 → value 1/2 = 0.5
  - (2,0,1): XY=0, Z+1=2 → value 0/2 = 0
  - (2,1,1): XY=2, Z+1=2 → value 2/2 = 1
  Thus the observed values are: 0 (occurs 4 times), 0.5 (1 time), and 1 (1 time).
  The probability masses are p(0)=4/6, p(0.5)=1/6, p(1)=1/6. The entropy H = -[ (4/6) log2(4/6) + (1/6) log2(1/6) + (1/6) log2(1/6) ].
  Evaluating: log2(4/6) ≈ -0.585, so (4/6)·(-0.585) ≈ -0.390; each (1/6)·log2(1/6) ≈ (1/6)·(-2.585) ≈ -0.431, and there are two of them, totaling ≈ -0.862. Summing gives H ≈ 0.390 + 0.862 ≈ 1.252 bits. This is not 2.11328, so option a is incorrect.
- Option b: Undefined
  Entropy is defined for a discrete distribution over the observed values. Since XY/(Z+1) yields a finite set of observed values with nonzero probabilities, the entropy is defined. There is no mathematical reason for it to be undefined here, so option b is incorrect.
- Option c: 1.2516
  This matches the computed entropy using the distribution p(0)=4/6, p(0.5)=1/6, p(1)=1/6 and H = -[ (4/6) log2(4/6) + 2·(1/6) log2(1/6) ]. The numerical value is approximately 1.2516 bits, aligning with the calculation above. This option is consistent with the derived distribution and formula.
- Option d: 1
  An entropy of exactly 1 bit would require a distribution with two outcomes each having probability 0.5 (or an equivalent distribution yielding the same entropy). Our observed distribution has three outcomes with probabilities 0.6667, 0.1667, 0.1667, which yields a different entropy (~1.2516). Therefore option d is incorrect.

Summary of reasoning:
- The key step is building the distribution of XY/(Z+1) across all rows, confirming values {0, 0.5, 1} with frequencies 4, 1, and 1 respectively.
- Using the entropy formula with base 2 (bits), we compute H ≈ -[ (4/6) log2(4/6) + (1/6) log2(1/6) + (1/6) log2(1/6) ] ≈ 1.2516.
- The value 2.11328 (option a) does not fit the observed distribution, and undefined (option b) is incorrect since the distribution exists. The exact entropy matches option c, and 1 (option d) does not reflect the actual probabilities of the three distinct outcomes.

Therefore, the option that aligns with the computed entropy is the one that corresponds to approximately 1.2516.

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}

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