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Use of m-procedures to compare

We have two m-procedures to make the comparisons. First, we consider approximation of the

A graph of a Gaussian Approximation to Poisson Distribution.The x-axis displays the values for t ranging from 2-16 while the y-axis represents the values of distribution functions ranging from 0-1. There are two plotted distributions. The Poisson approximation is represented stepwise with green circles present near the external right angles indicating the position of the adjusted Gaussian approximation. The Gaussian approximation is represented with a dashed blue line and corresponds roughly to the green circles. A graph of a Gaussian Approximation to Poisson Distribution.The x-axis displays the values for t ranging from 2-16 while the y-axis represents the values of distribution functions ranging from 0-1. There are two plotted distributions. The Poisson approximation is represented stepwise with green circles present near the external right angles indicating the position of the adjusted Gaussian approximation. The Gaussian approximation is represented with a dashed blue line and corresponds roughly to the green circles.
Gaussian approximation to the Poisson distribution function μ = 10 .
A graph of a Gaussian Approximation to Poisson distribution for  mu=100. The x-axis displays the values for t ranging from 80-120 while the y-axis represents the values of distribution functions ranging from 0-1. There are two plotted distributions. The Poisson approximation is represented stepwise with green circles present near the external right angles indicating the position of the adjusted Gaussian approximation. The Gaussian approximation is represented with a dashed blue line and corresponds roughly to the green circles. A graph of a Gaussian Approximation to Poisson distribution for  mu=100. The x-axis displays the values for t ranging from 80-120 while the y-axis represents the values of distribution functions ranging from 0-1. There are two plotted distributions. The Poisson approximation is represented stepwise with green circles present near the external right angles indicating the position of the adjusted Gaussian approximation. The Gaussian approximation is represented with a dashed blue line and corresponds roughly to the green circles.
Gaussian approximation to the Poisson distribution function μ = 100 .

Poisson ( μ ) distribution. The m-procedure poissapp calls for a value of μ , selects a suitable range about k = μ and plots the distribution function for the Poisson distribution (stairs) and the normal (gaussian) distribution (dash dot) for N ( μ , μ ) . In addition, the continuity correction is applied to the gaussian distribution at integer values (circles). [link] shows plots for μ = 10 . It is clear that the continuity correction provides a much better approximation. The plots in [link] are for μ = 100 . Here the continuity correction provides the better approximation, but not by as much as for the smaller μ .

A graph of a Gaussian Approximation to Poisson distribution for  mu=100. The x-axis displays the values for t ranging from 80-120 while the y-axis represents the values of distribution functions ranging from 0-1. There are two plotted distributions. The Poisson approximation is represented stepwise with green circles present near the external right angles indicating the position of the adjusted Gaussian approximation. The Gaussian approximation is represented with a dashed blue line and corresponds roughly to the green circles. A graph of a Gaussian Approximation to Poisson distribution for  mu=100. The x-axis displays the values for t ranging from 80-120 while the y-axis represents the values of distribution functions ranging from 0-1. There are two plotted distributions. The Poisson approximation is represented stepwise with green circles present near the external right angles indicating the position of the adjusted Gaussian approximation. The Gaussian approximation is represented with a dashed blue line and corresponds roughly to the green circles.
Poisson and Gaussian approximation to the binomial: n = 1000, p = 0.03.
A graph of an Approximation of Binomial by Poisson and Gaussian. The x-axis displays the values for t ranging from 15-40 while the y-axis represents the values of distribution functions ranging from 0-1. There are two plotted distributions. The Binomial approximation is represented stepwise with green circles present near the external right angles indicating the position of the adjusted Gaussian approximation. The Poisson approximation is represented with a dashed blue line and corresponds roughly to the green circles, except at the top right of the graph where the Poisson distribution falls below the Binomial. A graph of an Approximation of Binomial by Poisson and Gaussian. The x-axis displays the values for t ranging from 15-40 while the y-axis represents the values of distribution functions ranging from 0-1. There are two plotted distributions. The Binomial approximation is represented stepwise with green circles present near the external right angles indicating the position of the adjusted Gaussian approximation. The Poisson approximation is represented with a dashed blue line and corresponds roughly to the green circles, except at the top right of the graph where the Poisson distribution falls below the Binomial.
Poisson and Gaussian approximation to the binomial: n = 50, p = 0.6.

The m-procedure bincomp compares the binomial, gaussian, and Poisson distributions. It calls for values of n and p , selects suitable k values, and plots the distribution function for the binomial, a continuous approximation to the distribution function for the Poisson,and continuity adjusted values of the gaussian distribution function at the integer values. [link] shows plots for n = 1000 , p = 0 . 03 . The good agreement of all three distribution functions is evident. [link] shows plots for n = 50 , p = 0 . 6 . There is still good agreement of the binomial and adjusted gaussian. However, the Poisson distribution does nottrack very well. The difficulty, as we see in the unit Variance , is the difference in variances— n p q for the binomial as compared with n p for the Poisson.

Approximation of a real random variable by simple random variables

Simple random variables play a significant role, both in theory and applications. In the unit Random Variables , we show how a simple random variable is determined by the set of points on the real line representingthe possible values and the corresponding set of probabilities that each of these values is taken on. This describes the distribution of the random variable and makes possible calculations of eventprobabilities and parameters for the distribution.

A continuous random variable is characterized by a set of possible values spread continuously over an interval or collection of intervals. In this case, the probability is also spread smoothly. The distributionis described by a probability density function, whose value at any point indicates "the probability per unit length" near the point. A simple approximation is obtained by subdividing an interval whichincludes the range (the set of possible values) into small enough subintervals that the density is approximately constant over each subinterval. A point in each subinterval is selected and is assigned the probabilitymass in its subinterval. The combination of the selected points and the corresponding probabilities describes the distribution of an approximating simple random variable. Calculations based on thisdistribution approximate corresponding calculations on the continuous distribution.

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
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Samson Reply

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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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