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The random variable X takes values in {-1,0,1} with probabilities P(X = -1) = P(X = 1) = α and P(X = 0) = 1 - 2α, where 0 < α < 1/2. Let g(α) denote the entropy of X (in bits), parameterized by α. Which of the following statements is/are TRUE?

1. g(0.4) > g(0.3)
2. g(0.3) > g(0.4)
3. g(0.3) > g(0.25)
4. g(0.25) > g(0.3)

Correct answer: g(0.3) > g(0.4)

Solution

The entropy function is concave, meaning that as the probabilities become more uniform, the entropy increases. Since g(α) is decreasing for values of α in the specified range, g(0.3) is greater than g(0.4) because 0.3 is less than 0.4, leading to a more uniform distribution.

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