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Let U and V be two independent and identically distributed random variables such that P(U = 0) = P(V = 0) = 1/2. The entropy H(U ⊕ V) in bits is

1. 3/4
2. 1
3. 3/2
4. log2 3

Correct answer: 1

Solution

The random variable U ⊕ V represents the XOR operation between U and V, which results in 0 if both are the same and 1 if they are different. Since U and V are independent and identically distributed with equal probabilities of being 0 or 1, the resulting distribution of U ⊕ V is uniform with equal probabilities for 0 and 1, leading to an entropy of 1 bit.

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