# Appendix a to applied probability: directory of m-functions and m  (Page 9/24)

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trialmatch.m Estimates the probability of matches in n independent trials from identical distributions. The sample size and number of trials mustbe kept relateively small to avoid exceeding available memory.

% TRIALMATCH file trialmatch.m Estimates probability of matches % in n independent trials from identical distributions% Version of 8/20/97 % Estimates the probability of one or more matches% in a random selection from n identical distributions % with a small number of possible values% Produces a supersample of size N = n*ns, where % ns is the number of samples. Samples are separated.% Each sample is sorted, and then tested for differences % between adjacent elements. Matches are indicated by% zero differences between adjacent elements in sorted sample. X = input('Enter the VALUES in the distribution ');PX = input('Enter the PROBABILITIES '); c = length(X);n = input('Enter the SAMPLE SIZE n '); ns = input('Enter the number ns of sample runs ');N = n*ns; % Length of supersample U = rand(1,N); % Vector of N random numbersT = dquant(X,PX,U); % Supersample obtained with quantile function; % the function dquant determines quantile% function values for random number sequence U ex = sum(T)/N; % Sample averageEX = dot(X,PX); % Population mean vx = sum(T.^2)/N - ex^2; % Sample varianceVX = dot(X.^2,PX) - EX^2; % Population variance A = reshape(T,n,ns); % Chops supersample into ns samples of size nDS = diff(sort(A)); % Sorts each sample m = sum(DS==0)>0; % Differences between elements in each sample % -- Zero difference iff there is a matchpm = sum(m)/ns; % Fraction of samples with one or more matches d = arrep(c,n);p = PX(d); p = reshape(p,size(d)); % This step not needed in version 5.1ds = diff(sort(d))==0; mm = sum(ds)>0; m0 = find(1-mm);pm0 = p(:,m0); % Probabilities for arrangements with no matches P0 = sum(prod(pm0));disp('The sample is in column vector T') % Displays of results disp(['Sample average ex = ', num2str(ex),]) disp(['Population mean E(X) = ',num2str(EX),]) disp(['Sample variance vx = ',num2str(vx),]) disp(['Population variance V(X) = ',num2str(VX),]) disp(['Fraction of samples with one or more matches pm = ', num2str(pm),]) disp(['Probability of one or more matches in a sample Pm = ', num2str(1-P0),])

## Distributions

comb.m function y = comb(n,k) Calculates binomial coefficients. k may be a matrix of integers between 0 and n . The result y is a matrix of the same dimensions.

function y = comb(n,k) % COMB y = comb(n,k) Binomial coefficients% Version of 12/10/92 % Computes binomial coefficients C(n,k)% k may be a matrix of integers between 0 and n % result y is a matrix of the same dimensionsy = round(gamma(n+1)./(gamma(k + 1).*gamma(n + 1 - k)));

ibinom.m Binomial distribution — individual terms. We have two m-functions ibinom and cbinom for calculating individual and cumulative terms, $P\left({S}_{n}=k\right)$ and $P\left({S}_{n}\ge k\right)$ , respectively.

$P\left({S}_{n}=k\right)=C\left(n,\phantom{\rule{0.166667em}{0ex}}k\right){p}^{k}{\left(1-p\right)}^{n-k}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left({S}_{n}\ge k\right)=\sum _{r=k}^{n}P\left({S}_{n}=r\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0\le k\le n$
For these m-functions, we use a modification of a computation strategy employed by S. Weintraub: Tables of the Cumulative Binomial Probability Distribution for Small Values of p , 1963. The book contains a particularly helpful error analysis, written by Leo J. Cohen. Experimentation with sums and expectations indicates aprecision for ibinom and cbinom calculations that is better than ${10}^{-10}$ for $n=1000$ and p from 0.01 to 0.99. A similar precision holds for values of n up to 5000, provided $np$ or $nq$ are limited to approximately 500. Above this value for $np$ or $nq$ , the computations break down. For individual terms, function y = ibinom(n,p,k) calculates the probabilities for n a positive integer, k a matrix of integers between 0 and n . The output is a matrix of the corresponding binomial probabilities.

function y = ibinom(n,p,k) % IBINOM y = ibinom(n,p,k) Individual binomial probabilities% Version of 10/5/93 % n is a positive integer; p is a probability% k a matrix of integers between 0 and n % y = P(X>=k) (a matrix of probabilities) if p>0.5 a = [1 ((1-p)/p)*ones(1,n)]; b = [1 n:-1:1]; c = [1 1:n]; br = (p^n)*cumprod(a.*b./c);bi = fliplr(br); elsea = [1 (p/(1-p))*ones(1,n)];b = [1 n:-1:1];c = [1 1:n];bi = ((1-p)^n)*cumprod(a.*b./c); endy = bi(k+1);

#### Questions & Answers

what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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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