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The central limit theorem (CLT) asserts that the sum of a large class of independent random variables, each with reasonable distributions,is approximately normally distributed. Various versions of this theorem have been studied intensively. On the other hand, certain common forms serve as the basis of an extraordinary amount of applied work. In the statistics of large samples, the sample average is approximately normal—whether or not the population distribution is normal. In much of the theory of errors of measurement, the observed error is the sum of a large number of independent random quantities which contribute additively to the result. Similarly, in the theory of noise, the noise signal is the sum of a large number of random components, independently produced. In such situations, the assumption of a normal population distribution is frequently quite appropriate

The central limit theorem

The central limit theorem (CLT) asserts that if random variable X is the sum of a large class of independent random variables, each with reasonable distributions, then X is approximately normally distributed. This celebrated theorem has been the object of extensive theoretical research directed toward the discoveryof the most general conditions under which it is valid. On the other hand, this theorem serves as the basis of an extraordinary amount of applied work.In the statistics of large samples, the sample average is a constant times the sum of the random variables in the sampling process . Thus, for large samples,the sample average is approximately normal—whether or not the population distribution is normal. In much of the theory of errors of measurement,the observed error is the sum of a large number of independent random quantities which contribute additively to the result. Similarly, in the theory of noise, thenoise signal is the sum of a large number of random components, independently produced. In such situations, the assumption of a normal population distribution is frequentlyquite appropriate.

We consider a form of the CLT under hypotheses which are reasonable assumptions in many practical situations. We sketch a proof of this version of the CLT,known as the Lindeberg-Lévy theorem, which utilizes the limit theorem on characteristic functions, above, along with certain elementary facts from analysis. Itillustrates the kind of argument used in more sophisticated proofs required for more general cases.

Consider an independent sequence { X n : 1 n } of random variables. Form the sequence of partial sums

S n = i = 1 n X i n 1 with E [ S n ] = i = 1 n E [ X i ] and Var [ S n ] = i = 1 n Var [ X i ]

Let S n * be the standardized sum and let F n be the distribution function for S n * . The CLT asserts that under appropriate conditions, F n ( t ) Φ ( t ) as n for all t . We sketch a proof of the theorem under the condition the X i form an iid class.

Central Limit Theorem (Lindeberg-Lévy form)

If { X n : 1 n } is iid, with

E [ X i ] = μ , Var [ X i ] = σ 2 , and S n * = S n - n μ σ n

then

F n ( t ) Φ ( t ) as n , for all t

IDEAS OF A PROOF

There is no loss of generality in assuming μ = 0 . Let φ be the common characteristic function for the X i , and for each n let φ n be the characteristic function for S n * . We have

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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