Electronics and Communication Engineering (ECE) Exam > Electronics and Communication Engineering (ECE) Questions > A source has an alphabet {a1, a2, a3, a4, a5...
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A source has an alphabet {a1, a2, a3, a4, a5, a6} with corresponding probabilities {0.1, 0.2, 0.3, 0.5, 0.15, 0.2}. The entropy (in bits / symbol) of this source is ______. (Answer up to one decimal place)
Correct answer is '2.4'. Can you explain this answer?
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A source has an alphabet {a1, a2, a3, a4, a5, a6} with corresponding ...
Entropy Calculation
To calculate the entropy of a source, we need to use the formula:
Entropy (H) = - Σ P(x) * log2(P(x))
Where P(x) is the probability of symbol x occurring.
Given Information
The source has an alphabet {a1, a2, a3, a4, a5, a6} with corresponding probabilities {0.1, 0.2, 0.3, 0.5, 0.15, 0.2}.
Entropy Calculation Steps
1. Calculate the entropy for each symbol:
- For symbol a1: P(a1) = 0.1, entropy of a1 = - 0.1 * log2(0.1)
- For symbol a2: P(a2) = 0.2, entropy of a2 = - 0.2 * log2(0.2)
- For symbol a3: P(a3) = 0.3, entropy of a3 = - 0.3 * log2(0.3)
- For symbol a4: P(a4) = 0.5, entropy of a4 = - 0.5 * log2(0.5)
- For symbol a5: P(a5) = 0.15, entropy of a5 = - 0.15 * log2(0.15)
- For symbol a6: P(a6) = 0.2, entropy of a6 = - 0.2 * log2(0.2)
2. Sum up the entropies of all symbols:
H = entropy of a1 + entropy of a2 + entropy of a3 + entropy of a4 + entropy of a5 + entropy of a6
3. Calculate the total entropy:
H = - 0.1 * log2(0.1) - 0.2 * log2(0.2) - 0.3 * log2(0.3) - 0.5 * log2(0.5) - 0.15 * log2(0.15) - 0.2 * log2(0.2)
4. Simplify the expression and calculate the result:
H = 0.3322 + 0.4644 + 0.5211 + 0.5 + 0.4422 + 0.4644
H ≈ 2.7243
Final Answer
Rounding the result to one decimal place, the entropy of the source is approximately 2.7 bits/symbol. However, the correct answer given is '2.4'. It is possible that there was an error in the calculation or rounding.
To calculate the entropy of a source, we need to use the formula:
Entropy (H) = - Σ P(x) * log2(P(x))
Where P(x) is the probability of symbol x occurring.
Given Information
The source has an alphabet {a1, a2, a3, a4, a5, a6} with corresponding probabilities {0.1, 0.2, 0.3, 0.5, 0.15, 0.2}.
Entropy Calculation Steps
1. Calculate the entropy for each symbol:
- For symbol a1: P(a1) = 0.1, entropy of a1 = - 0.1 * log2(0.1)
- For symbol a2: P(a2) = 0.2, entropy of a2 = - 0.2 * log2(0.2)
- For symbol a3: P(a3) = 0.3, entropy of a3 = - 0.3 * log2(0.3)
- For symbol a4: P(a4) = 0.5, entropy of a4 = - 0.5 * log2(0.5)
- For symbol a5: P(a5) = 0.15, entropy of a5 = - 0.15 * log2(0.15)
- For symbol a6: P(a6) = 0.2, entropy of a6 = - 0.2 * log2(0.2)
2. Sum up the entropies of all symbols:
H = entropy of a1 + entropy of a2 + entropy of a3 + entropy of a4 + entropy of a5 + entropy of a6
3. Calculate the total entropy:
H = - 0.1 * log2(0.1) - 0.2 * log2(0.2) - 0.3 * log2(0.3) - 0.5 * log2(0.5) - 0.15 * log2(0.15) - 0.2 * log2(0.2)
4. Simplify the expression and calculate the result:
H = 0.3322 + 0.4644 + 0.5211 + 0.5 + 0.4422 + 0.4644
H ≈ 2.7243
Final Answer
Rounding the result to one decimal place, the entropy of the source is approximately 2.7 bits/symbol. However, the correct answer given is '2.4'. It is possible that there was an error in the calculation or rounding.
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Community Answer
A source has an alphabet {a1, a2, a3, a4, a5, a6} with corresponding ...
= -(0.1 log20.1 + 0.2 log20.2) + 0.3 log20.3 + 0.05 log20.05 + 0.15 log20.15 + 0.2 log20.2
= 2.4087 bits/symbol
or H(x) = 2.4 bits/symbol
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