Specify the parameters of the distribution, and try a couple of dierent paramete

Question

Specify the parameters of the distribution, and try a couple of dierent parameter sets. In each case, compute the mean and variance from the sampled data, and compare it with the theoretical mean and variance. Also plot a histogram of the samples. For sampling from the Beta-Binomial distribution, use the Polya Urn Model.

1.2 Matlab Code Write Matlab code to generate samples from the following distributions: (A) Binomial (B) Poisson (C) Normal (D) Beta (E) Beta-Binomial Specify the parameters of the distribution, and try a couple of different parameter sets. In each case, compute the mean and variance from the sampled data, and compare it with the theoretical mean and variance. Also plot a histogram of the samples. For sampling from the Beta-Binomial distribution use the Polya Urn Model. 1.3 Polya Urn Model Suppose that we have an Urn, which initially contains a red balls and B green balls. We select a ball at random from the Urn. If we observe a red ball, then we return two red balls to the Urn. Likewise, if we observe a green ball, then two green balls are returned to the Urn. If the experiment is repeated n times, then the probability of observing a red balls, has a beta-binomial distribution, with parameters n. a and B. 1.4 Example Code An example code SamplingDistributionsExample.m is posted on the Blackboard. It generates sam ples from the beta distribution, by utilizing the betarnd(..) functio Figure 1 shows some plots generated from this code. Study this code and the results. 1.5 Matlab functions Table 1 gives some of the Matlab functions you will be utilizing for the project. For further information on any of these functions, type help name, where name is the function name. You can also access Matlab Help by clicking on the question mark, then browse or search for information.

Explanation / Answer

clc;
close all;
clear all;

%BINOMIAL DISTRIBUTION
N = 10;
X = 0:1:N-1;
P = 1/N;
Y1 = binopdf(X,N,P);
[M1,V1] = binostat(N,P) %Theoretical mean = 1 and Theoretical variance = 0.9
figure
h = histfit(Y1);
%plot(X,h);

%POISSON DISTRIBUTION
Lambda = 0.01;
N = 10;
X = 0:1:N-1;
Y2 = poisspdf(X,P);
[M2,V2] = poisstat(P) %Theoretical mean and Theoretical variance = 0.01
figure
h = histfit(Y2);

%NORMAL DISTRIBUTION
mean = 0.01;
sd = 4;
X = 0:1:N-1;
Y3 = normpdf(X,mean,sd);
[M3,V3] = normstat(mean,sd) %Theoretical mean = 0.01 and Theoretical variance = 16
figure
h = histfit(Y3);

%BETA DISTRIBUTION
alpha = 0.25;
beta = 0.25;
X = 0.01:0.01:0.99;
Y4 = betapdf(X,alpha,beta);
[M4,V4] = betastat(alpha,beta) %Theoretical mean = 0.2 and Theoretical variance = 0.1667
figure
h = histfit(Y4);

%BETA BINOMIAL DISTRIBUTION
alpha = 2;
beta = 2;
N = 10;
Y5 = [];
for X = 0:1:N-1;
    Y5 = [Y5,bbinopdf(X,N,alpha,beta)];
end
M5 = N*pi %Theoretical mean = n*pi = 31.41
rho = 1/(alpha + beta + 1);
V5 = N*pi*(1-pi)*(1-((N-1)*rho))
figure
h = histfit(Y5);

%SAVE BELOW CODE AS A FUNCTION named "bbinopdf"

function y = bbinopdf(x,n,a,b)
%BBINOPDF Beta-binomial probability distribution function.
% Y = BBINOPDF(X,N,A,B) returns the beta-binomial probability
% density function with parameters N, A and B at the values in X.
% Note: The density function is zero unless N, A and B are integers.
%
% The Beta-binomial distribution is used to model the number of successes
% in n binomial trials when the probability of success p is a Beta(a,b)
% random variable. The extreme flexibility of the shape of the Beta
% distribution means that it is often a very fair representation of the
% randomness of p.
%
% A variable with a Beta-binomial distribution is distributed as a binomial
% distribution with parameter p, where p is distribution with a beta
% distribution with parameters a (alpha) and b (beta). For n trials, it has
% probability density function:
%
%                                 B(x+a,n-x+b)
%                   p(x) = n_C_x ---------------
%                                     B(a,b)
%
% where B(a,b) is a beta function and n_C_x is a binomial coefficient.
% (http://mathworld.wolfram.com/BetaBinomialDistribution.html)
% (http://en.wikipedia.org/wiki/Beta-binomial_model)
%
% The probability of success varies randomly, but in any one scenario that
% probability applies to all trials. For example, you might consider using
% the Beta-binomial distribution to model:
%
% --The number of life insurance policy holders who will die in any one
% year, where some external variable (e.g. highly contagious disease,
% extreme weather) moderates the probability of death of all individual to
% some degree.
% --The number of cars that crash in a race of n cars, where the predominant
% factor is not the skill of the individual driver, but the weather on the
% day.
% --The number of bottles of wine from a producer that are bad where the
% predominant factor is not how each bottle is treated, but something to do
% with the batch as a whole.
%
% The Beta-binomial is a two-dimensional multivariate Polya distribution,
% as the binomial and beta distributions are special cases of the
% multinomial and Dirichlet distributions, respectively.
%
% Syntax: function y = bbinopdf(x,n,a,b)
%    
% Inputs:
% x - number of success
% n - number of trials
% a - alpha parameter of Beta
% b - beta parameter of Beta
%
% Output:
% y - beta-binomial probability value
%
% Example. Suppose we make 5 trials on sample with a Beta-binomial
% distribution with parameters alpha = 3, and beta = 4, and we are
% interested to get the probability of get exactly 3 successes.
%
% Calling on Matlab the function:
%             y = bbinopdf(3,5,3,4)
%
% Answer is: (in format long)
%
% y =
%
%   0.21645021645022
%
% Created by A. Trujillo-Ortiz, R. Hernandez-Walls, F.A. Trujillo-Perez
%            and N. Castro-Castro
%            Facultad de Ciencias Marinas
%            Universidad Autonoma de Baja California
%            Apdo. Postal 453
%            Ensenada, Baja California
%            Mexico.
%            atrujo@uabc.mx
% Copyright. September 28, 2009.
%
% ---Con cariño para Ney.----
%
% To cite this file, this would be an appropriate format:
% Trujillo-Ortiz, A., R. Hernandez-Walls, F.A. Trujillo-Perez and
%   N. Castro-Castro (2009). bbinopdf:Beta-binomial probability
%   densiy function. A MATLAB file. [WWW document]. URL
%   http://www.mathworks.com/matlabcentral/fileexchange/25454
%

if nargin < 4,
    error('bbinopdf:TooFewInputs','Requires four input arguments.');
end

[errorcode x n a b] = distchck(4,x,n,a,b);

if errorcode > 0
    error('bbinopdf:InputSizeMismatch',...
        'Requires non-scalar arguments to match in size.');
end

if (length(x)~=1) || (fix(x) ~= x) || (x < 0),
   error('bbinopdf:InvalidData',...
       'BBINOPDF requires that X must be a non-negative and integer.')
end

if (length(n)~=1) || (fix(n) ~= n) || (n < 0),
   error('bbinopdf:InvalidData',...
       'BBINOPDF requires that N must be a non-negative and integer.')
end

if (length(a)~=1) || (fix(a) ~= a) || (a < 0),
   error('bbinopdf:InvalidData',...
       'BBINOPDF requires that A must be a non-negative and integer.')
end

if (length(b)~=1) || (fix(b) ~= b) || (b < 0),
   error('bbinopdf:InvalidData',...
       'BBINOPDF requires that B must be a non-negative and integer.')
end

y = exp(gammaln(n + 1)-gammaln(x + 1)-gammaln(n - x + 1))*...
    beta((a + x),(b + n - x))/beta(a,b);

%alternatively, using the Gamma function, it can be expressed as:
%
%y = exp(gammaln(n + 1)-gammaln(x + 1)-gammaln(n - x + 1))*(gamma(a + b)*...
%    gamma(a + x)*gamma(b + n - x))/(gamma(a)*gamma(b)*gamma(a + b + x))

return,

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