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Two binary random variables X and Y are distributed according to the joint distributions p(x = y = 0) = p(x = 0, y = 1) = p(x = y = 1) = 1/3 . Then the entropy
H(x) is
  • a)
    0.9183                 
  • b)
    0.9327                
  • c)
    0.9651                    
  • d)
    0.9832
Correct answer is option 'A'. Can you explain this answer?
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The marginal probabilities are given by
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Solution:

Given, p(x = y = 0) = p(x = 0, y = 1) = p(x = y = 1) = 1/3.

To find the entropy H(x), we need to find the marginal probabilities of X.

P(X=0) = P(X=0,Y=0) + P(X=0,Y=1) = 1/3 + 1/3 = 2/3

P(X=1) = P(X=1,Y=0) + P(X=1,Y=1) = 1/3 + 1/3 = 2/3

Now, we can calculate the entropy as follows:

H(X) = -[P(X=0) log2 P(X=0) + P(X=1) log2 P(X=1)]

= -[(2/3) log2 (2/3) + (1/3) log2 (1/3)]

= -[(-0.9183) + (-1.5849)]

= 0.9183

Therefore, the entropy H(X) is 0.9183, which is option A.
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