Differential entropy differs from normal or absolute entropy in that the random variable need not be discrete. Given a continuous random variable 
with a probability density function 
, the differential entropy 
is defined as
 |
(1)
|
When we have a continuous random vector 
that consists of 
random variables 
, 
, ..., 
, the differential entropy of 
is defined as the 
-fold integral
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is the joint probability density function of 
.
Thus, for example, the differential entropy of a multivariate Gaussian random variate 
with covariance matrix 
is
 |  | ![1/2ln[(2pie)^n|det(P)|]](/images/equations/DifferentialEntropy/Inline23.svg) |
(4)
|
 |  | ![1/2n[1+ln(2pi)]+1/2ln|det(P)|.](/images/equations/DifferentialEntropy/Inline26.svg) |
(5)
|
Additional properties of differential entropy include
 |
(6)
|
where 
is a constant and
 |
(7)
|
where 
is a scaling factor and 
is a scalar random variable. The above property can be generalized
to the case of a random vector 
premultiplied by a matrix 
,
 |
(8)
|
where 
is the determinant of matrix 
.
