



In this module we consider differential entropy.
Consider the entropy of
continuous random variables. Whereas the (normal)
entropy is the entropy of a
discrete random variable, the differential entropy is the entropy of a continuous random variable.
Differential entropy
Differential entropy
 The differential entropy
$h(X)$ of a continuous random variable
$X$ with a pdf
$f(x)$ is defined as
$h(X)=\int_{()} \,d x$∞
∞
f
x
f
x
Usually the logarithm is taken to be base 2, so that the unit of the differential entropy is bits/symbol. Note that is the discrete case,
$h(X)$ depends only on the pdf of
$X$ . Finally, we note that
the differential entropy is the expected value of
$\lg f(x)$ , i.e.,
$h(X)=E(\lg f(x))$
Now, consider a calculating the differential entropy of some random variables.
Consider a normal distributed random variable
$X$ , with mean
$m$ and
variance
$\sigma ^{2}$ .
Then its density is
$\sqrt{\frac{1}{2\pi \sigma ^{2}}}e^{\left(\frac{(xm)^{2}}{2\sigma ^{2}}\right)}$ .
We can then find its differential entropy as follows, first calculate
$\lg f(x)$ :
$\lg f(x)=\frac{1}{2}\lg (2\pi \sigma ^{2})+\lg e()\frac{(xm)^{2}}{2\sigma ^{2}}$
Then since
$E((Xm)^{2})=\sigma ^{2}$ ,
we have
$h(X)=\frac{1}{2}\lg (2\pi \sigma ^{2})+\frac{1}{2}\lg e=\frac{1}{2}\lg (2\pi \times e\sigma ^{2})$
Got questions? Get instant answers now!
Properties of the differential entropy
In the section we list some properties of the differential entropy.
 The differential entropy can be negative

$h(X+c)=h(X)$ , that is translation
does not change the differential entropy.

$h(aX)=h(X)+\lg \lefta\right$ , that is scaling
does change the differential entropy.
The first property is seen from both
and
. The two latter can be shown by using
.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
nano basically means 10^(9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Source:
OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
Google Play and the Google Play logo are trademarks of Google Inc.