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Unified (r,s)Entropy
for Continuous Probability Distributions
In this section we extend the notion of generalized entropies to
the continuous case, and examine some properties of the resulting unified entropy
function.
Let X be an absolutely continuous random variable, that is, a random
variable having a probability density function p(x). The unified entropy
of X is defined as follows:



(3.14) 
provided the integrals exist, where , .
The contrast between continuous and discrete distribution is worth emphasising:

(i) The entropy measure of continuous distribution need not exist.

(ii) When it does exist, there is nothing to ensure that is positive because
can exceed unity. We consider the following examples:
Example 3.1. Let
be a random variable with probability density function
Then
with
when .
(iii) The unified entropy
are not limits of the unifiedentropy
of the discrete case. This we shall verify in the following example.
Example 3.2. Let
be a discrete random variable taking the values
with equal probabilities .
Then
As
increases, the distribution of
converges to a continuous uniform distribution in (0,1). If ,
we have
however,
(iv) The unified entropy
is not invariant with respect to a change of variables. We illustrate this
point with the following example:
Example 3.3. We consider a function ,
where
is a stricly increasing function of .
Since the mapping from
to
is one to one, we have
where .
Therefore,
which is different from
unless
be the identity function.
These important differences between discrete and continuous cases are
a warning that the results for the discrete distributions cannot be translated
to continuous case without independent verification. Fortunately, some
of the significant concepts rely upon differences between entropies and
for these the difficulties disappear.
21062001
Inder Jeet Taneja
Departamento de Matemática  UFSC
88.040900 Florianópolis, SC  Brazil