Shannon's entropy though defined for a discrete random variable can be extended to situations when the random variable under consideration is continuous.
Let X be a continuous random variable with probability density function on I, where, then the entropy is given by
whenever it exists. The measure (1.13) sometimes called "differential
entropy". It has many of the properties of discrete entropy but unlike
the entropy of a discrete random variable that of a continuous random variable
may be infinitely large, negative or positive (Ash, 1965 ).
The entropy of a discrete random variable remains invariant under a change
of variable, however with a continuous random variable the entropy does
not necessarily remain invariant.