Let X and Y be random variable with the following distributions respectively :X 1 2 3 4 5 P 0.1 0.2 0.4 0.2 0.1 Y 10 20 35 50 60 P 0.1 0.2 0.4 0.2 0.1
Entropy of distribution of X and Y are same, Coefficient of Variations are not same
Entropy \(H = -\sum p_i \log p_i\) depends only on the probabilities, not on the values the variable takes. The coefficient of variation \(CV = \sigma/\mu\) depends on the values themselves, so two variables with the same probability vector can have equal entropy and a different spread relative to the mean.
Step 1 — Both X and Y have probabilities 0.1, 0.2, 0.4, 0.2, 0.1, so their entropies are equal.
Step 2 — For X: \(\mu_X = 3\), \(\sigma_X^2 = 0.1(4) + 0.2(1) + 0 + 0.2(1) + 0.1(4) = 1.2\), so \(CV_X = \sqrt{1.2}/3 \approx 0.365\).
Step 3 — For Y: \(\mu_Y = 1 + 4 + 14 + 10 + 6 = 35\), \(\sigma_Y^2 = 0.1(625) + 0.2(225) + 0 + 0.2(225) + 0.1(625) = 215\), so \(CV_Y = \sqrt{215}/35 \approx 0.419\).
The coefficients of variation differ because Y is not a simple rescaling of X (its values are not evenly spaced), so any claim of equal coefficients fails, and any claim of unequal entropy fails because the probability vectors are identical.
Entropy of distribution of X and Y are same, Coefficient of Variations are not same option is correct.
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