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Bivariate Cases
In this subsection, we shall give properties of the generalized
conditional entropies for the bivariate case.
Property 6.1. For
,
we have
or
Property 6.2. We have
and
Property 6.3. If
and
are independent random variables, then

Property 6.4. For any
and
,
we have
Property 6.5. The following inequalities hold:
-
(i)

-
(ii)

-
(iii)

-
(iv)

Note 6.2. The conditions
are equivalent to either
or
,
,
and the conditions
are equivalent to either
,
or
.
Property 6.6. We have
-
(i)

-
(ii)
for
=2
and 3;
-
(iii)

The equality sign holds iff
and
are independent random variables.
As a consequence of properties 6.5 and 6.6, we have the following property:
Property 6.7. (i) For
,
we have

(ii) For
,
we have

(iii) For
,
we have

(iv) For
,
we have

21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil