GATE EC 2025 Set 1 — Question 54
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Communications → Information Theory → Information Measure & Entropy
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Question
The random variable takes values in with probabilities
and , where .
Let denote the entropy of (in bits), parameterized by .
Which of the following statements is/are TRUE?
and , where .
Let denote the entropy of (in bits), parameterized by .
Which of the following statements is/are TRUE?
Correct answer
(B) g(0.3) g(0.4); (C) g(0.3) g(0.25)
Solution
The entropy is given by:
To find the maximum, we differentiate with respect to :
Setting gives .
Since , the entropy is maximum at . The function increases for and decreases for .
To find the maximum, we differentiate with respect to :
Setting gives .
Since , the entropy is maximum at . The function increases for and decreases for .
- Comparing and : is closer to the peak () than . Calculating values: and . Thus . (B is TRUE)
- Comparing and : Both are less than . Since is increasing in this range and , . (C is TRUE)
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