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A source generates 4 messages. The entropy of the source will be maximum when
- a)all probabilities are equal
- b)one of the probabilities equal to 1 and two others are zero
- c)the probabilities are 1/2, 1/4 and 1/2
- d)the two of the probabilities are 1/2 each and other zero
Correct answer is option 'A'. Can you explain this answer?
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A source generates 4 messages. The entropy of the source will be maxim...
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- a)all probabilities are equal
The entropy of the source will be maximum when. all probabilities are equal.
one of the probabilities equal to 1 and two others are zero.
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A source generates 4 messages. The entropy of the source will be maxim...
Explanation:
Entropy is a measure of uncertainty or randomness associated with a random variable. In information theory, entropy is used to measure the amount of information contained in a message or a signal. The entropy of a source is a measure of the average amount of information per symbol produced by the source.
In the given question, the source generates 4 messages. Let us consider the different options and calculate the entropy for each of them.
Option A: All probabilities are equal
If all probabilities are equal, then each message has a probability of 1/4. The entropy of the source can be calculated as:
H = - (1/4) log2 (1/4) - (1/4) log2 (1/4) - (1/4) log2 (1/4) - (1/4) log2 (1/4) = 2 bits/symbol
Option B: One of the probabilities equal to 1 and two others are zero
If one of the probabilities is equal to 1, then the other three probabilities are zero. This means that only one message can be generated by the source. The entropy of the source in this case is zero because there is no uncertainty or randomness associated with the source.
Option C: The probabilities are 1/2, 1/4 and 1/2
If the probabilities are 1/2, 1/4 and 1/2, then the entropy of the source can be calculated as:
H = - (1/2) log2 (1/2) - (1/4) log2 (1/4) - (1/2) log2 (1/2) - 0 = 1.5 bits/symbol
Option D: The two of the probabilities are 1/2 each and other zero
If two of the probabilities are 1/2 each and the other two probabilities are zero, then only two messages can be generated by the source. The entropy of the source in this case is 1 bit/symbol because there is only one bit of uncertainty associated with the source.
Conclusion:
From the above calculations, we can conclude that the entropy of the source will be maximum when all probabilities are equal (Option A). This is because in this case, all messages are equally likely to occur, and there is maximum uncertainty or randomness associated with the source.
Entropy is a measure of uncertainty or randomness associated with a random variable. In information theory, entropy is used to measure the amount of information contained in a message or a signal. The entropy of a source is a measure of the average amount of information per symbol produced by the source.
In the given question, the source generates 4 messages. Let us consider the different options and calculate the entropy for each of them.
Option A: All probabilities are equal
If all probabilities are equal, then each message has a probability of 1/4. The entropy of the source can be calculated as:
H = - (1/4) log2 (1/4) - (1/4) log2 (1/4) - (1/4) log2 (1/4) - (1/4) log2 (1/4) = 2 bits/symbol
Option B: One of the probabilities equal to 1 and two others are zero
If one of the probabilities is equal to 1, then the other three probabilities are zero. This means that only one message can be generated by the source. The entropy of the source in this case is zero because there is no uncertainty or randomness associated with the source.
Option C: The probabilities are 1/2, 1/4 and 1/2
If the probabilities are 1/2, 1/4 and 1/2, then the entropy of the source can be calculated as:
H = - (1/2) log2 (1/2) - (1/4) log2 (1/4) - (1/2) log2 (1/2) - 0 = 1.5 bits/symbol
Option D: The two of the probabilities are 1/2 each and other zero
If two of the probabilities are 1/2 each and the other two probabilities are zero, then only two messages can be generated by the source. The entropy of the source in this case is 1 bit/symbol because there is only one bit of uncertainty associated with the source.
Conclusion:
From the above calculations, we can conclude that the entropy of the source will be maximum when all probabilities are equal (Option A). This is because in this case, all messages are equally likely to occur, and there is maximum uncertainty or randomness associated with the source.
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A source generates 4 messages. The entropy of the source will be maximum whena)all probabilities are equalb)one of the probabilities equal to 1 and two others are zeroc)the probabilities are 1/2, 1/4 and 1/2d)the two of the probabilities are 1/2 each and other zeroCorrect answer is option 'A'. Can you explain this answer?
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