Consider a random sequence of 1s and 0s, i.e., the probability of a 0 or 1 is always . The corresponding probability density function is
and the entropy is
Thus, 1 bit is required for each bit of the sequence. In other words, the sequence cannot be compressed. There is no redundancy.
If instead the probability of a 0 is 1/4 and that of a 1 is 3/4, we get
and the sequence can be compressed about .
In the degenerate case for which the probability of a 0 is 0 and that of a 1 is 1, we get
Thus, the entropy is 0 when the sequence is perfectly predictable.
Maximum Entropy Distributions
Entropy of a Probability Distribution