5.4 Differential entropy

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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 uniformly distributed random variable $X$ from $c$ to $c+\Delta$ . Then its density is $\frac{1}{\Delta }$ from $c$ to $c+\Delta$ , and zero otherwise.

We can then find its differential entropy as follows,

$h(X)=-\int_{c}^{c+\Delta } \frac{1}{\Delta }\lg \left(\frac{1}{\Delta }\right)\,d x=\lg \Delta$
Note that by making $\Delta$ arbitrarily small, the differential entropy can be made arbitrarily negative, while taking $\Delta$ arbitrarily large, the differential entropy becomes arbitrarily positive.

Consider a normal distributed random variable $X$ , with mean $m$ and variance $\sigma ^{2}$ . Then its density is $\sqrt{\frac{1}{2\pi \sigma ^{2}}}e^{-\left(\frac{(x-m)^{2}}{2\sigma ^{2}}\right)}$ .

We can then find its differential entropy as follows, first calculate $-\lg f(x)$ :

$-\lg f(x)=\frac{1}{2}\lg (2\pi \sigma ^{2})+\lg e()\frac{(x-m)^{2}}{2\sigma ^{2}}$
Then since $E((X-m)^{2})=\sigma ^{2}$ , we have
$h(X)=\frac{1}{2}\lg (2\pi \sigma ^{2})+\frac{1}{2}\lg e=\frac{1}{2}\lg (2\pi \times e\sigma ^{2})$

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.
• $h(aX)=h(X)+\lg \left|a\right|$ , that is scaling does change the differential entropy.
The first property is seen from both and . The two latter can be shown by using .

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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?
?
Kyle
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
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research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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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?
absolutely yes
Daniel
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
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Abigail
for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
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NANO
what is fullerene does it is used to make bukky balls
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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
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is Bucky paper clear?
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Do you know which machine is used to that process?
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how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
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
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