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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
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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