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## Entropy-Power 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].

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