



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
$h(X)=\frac{1}{2}\lg (2\pi \sigma ^{2})+\frac{1}{2}\lg e=\frac{1}{2}\lg (2\pi \times e\sigma ^{2})$
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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
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
Do somebody tell me a best nano engineering book for beginners?
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
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
what's the easiest and fastest way to the synthesize AgNP?
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is system testing?
AMJAD
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field .
1Electronicsmanufacturad IC ,RAM,MRAM,solar panel etc
2Helth and MedicalNanomedicine,Drug Dilivery for cancer treatment etc
3 Atomobile MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:
OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
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