GATE 2025 EC – Question 54
The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)=\alpha$ and $P(X=0)=1-2\alpha$, where $0<\alpha<\frac12$.
Let $g(\alpha)$ denote the entropy of $X$ (in bits), parameterized by $\alpha$.
Which of the following statements is/are TRUE?
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Correct answer: (B) $g(0.3)>g(0.4)$; (C) $g(0.3)>g(0.25)$
Explanation
$g(\alpha)=-2\alpha\log_2\alpha-(1-2\alpha)\log_2(1-2\alpha)$, which increases up to its maximum at $\alpha=1/3$ (uniform distribution) and decreases afterwards. $g(0.3)=0.6(1.737)+0.4(1.322)=1.571$; $g(0.4)=0.8(1.322)+0.2(2.322)=1.522$; $g(0.25)=0.5(2)+0.5(1)=1.5$. So $g(0.3)>g(0.4)$ (B true) and $g(0.3)>g(0.25)$ (C true); A and D are false.