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If F is a probability distribution function, the associated quantile function Q is essentially an inverse of F. The quantile function is defined on the unit interval (0,1). For F continuous and strictly increasing at t, then Q(u)=t iff F(t)=u. Thus, if u is a probability value, t=Q(u) is the value of t for which P(X≤t)=u. As an application, one may generate independent classes with prescribed distributions. Matlab procedures are developed for a variety of problems involving quantile functions.

The quantile function

The quantile function for a probability distribution has many uses in both the theory and application of probability. If F is a probability distribution function, the quantile function may be used to “construct” a random variable having F as its distributions function. This fact serves as the basis of a method of simulating the“sampling” from an arbitrary distribution with the aid of a random number generator . Also, given any finite class

{ X i : 1 i n } of random variables, an independent class { Y i : 1 i n } may be constructed, with each X i and associated Y i having the same (marginal) distribution. Quantile functions for simple random variables maybe used to obtain an important Poisson approximation theorem (which we do not develop in this work). The quantile function is usedto derive a number of useful special forms for mathematical expectation.

General concept—properties, and examples

If F is a probability distribution function, the associated quantile function Q is essentially an inverse of F . The quantile function is defined on the unit interval ( 0 , 1 ) . For F continuous and strictly increasing at t , then Q ( u ) = t iff F ( t ) = u . Thus, if u is a probability value, t = Q ( u ) is the value of t for which P ( X t ) = u .

The weibull distribution ( 3 , 2 , 0 )

u = F ( t ) = 1 - e - 3 t 2 t 0 t = Q ( u ) = - ln ( 1 - u ) / 3
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The normal distribution

The m-function norminv , based on the MATLAB function erfinv (inverse error function), calculates values of Q for the normal distribution.

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The restriction to the continuous case is not essential. We consider a general definition which applies to any probability distribution function.

Definition : If F is a function having the properties of a probability distribution function, then the quantile function for F is given by

Q ( u ) = inf { t : F ( t ) u } u ( 0 , 1 )

We note

  • If F ( t * ) u * , then t * inf { t : F ( t ) u * } = Q ( u * )
  • If F ( t * ) < u * , then t * < inf { t : F ( t ) u * } = Q ( u * )

Hence, we have the important property:

(Q1) Q ( u ) t iff u F ( t ) u ( 0 , 1 ) .

The property (Q1) implies the following important property:

(Q2) If U uniform ( 0 , 1 ) , then X = Q ( U ) has distribution function F X = F . To see this, note that F X ( t ) = P [ Q ( U ) t ] = P [ U F ( t ) ] = F ( t ) .

Property (Q2) implies that if F is any distribution function, with quantile function Q , then the random variable X = Q ( U ) , with U uniformly distributed on ( 0 , 1 ) , has distribution function F .

Independent classes with prescribed distributions

Suppose { X i : 1 i n } is an arbitrary class of random variables with corresponding distribution functions { F i : 1 i n } . Let { Q i : 1 i n } be the respective quantile functions. There is always an independent class { U i : 1 i n } iid uniform ( 0 , 1 ) (marginals for the joint uniform distribution on the unit hypercube with sides ( 0 , 1 ) ). Then the random variables Y i = Q i ( U i ) , 1 i n , form an independent class with the same marginals as the X i .

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Questions & Answers

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Damian Reply
Introduction about quantum dots in nanotechnology
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what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
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.
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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.
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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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
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.
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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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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