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EntropyPower Inequality
Given a continuous random variable
with differential entropy ,
the entropy power of
is given by
We have the following two properties.
Property 1.77. Let
and
be independent continuous random variables of finite variance. Then the
differential entropy satisfies



(1.14) 
with equality iff
and
are gaussian.
Property 1.78. Let
and
be two independent continuous vector valued random variables with finite
variance. Then
Note 1.8. The inequality (1.14) is famous in the literature as
"Power
inequality". Some studies on it can be seen in Stam (1959) [96],
Blachman (1965) [12] and Costa (1985) [28].
21062001
Inder Jeet Taneja
Departamento de Matemática  UFSC
88.040900 Florianópolis, SC  Brazil