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ModifiedHandinAssignment1
RandomvariablegenerationandMonteCarlointegration
ComputerIntensiveStatisticalMethods
MajidKhorsandVakilzadeh
Lecturer:AndersSjgren
2013
Nov
03
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Table of Contents
Introduction ..................................................... ....................................................... ........... 1
Hand-in assignment 1 .......................................................................................................... 1
Task 1- Generate Random Variables ..... ...... ..... ..... ...... ..... ...... ..... ...... ...... ..... ...... ..... ...... ...... ... 1Part a- Standard normal variables from cauchy distribution ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. 1
Part b ............................................................................................................................... 4
Task 2 ................................................................................................... ........................... 7
part a ................................................. ........................................................ ...................... 7
part b .............................................................................................................................. 10
Introduction% This report provides the Matlab code written for hand-in assignment 1
% accompanied with a brief explanation of different parts which for the
% sake of simplicity are included in Matlab script in green color.
Hand-in assignment 1-------Random Variable Generation and Monte-Carlo Integration------------
clc
clear
Task 1- Generate Random Variables
Part a- Standard normal variables from cauchy
distribution% In this part two embeded accept-reject algorithm is used:
% 1. To yield Cauchy variable from uniformly distributed R.V.s
% so, in the first part f= (pi*b*(1+((x-a)/b)^2))^-1 (cauchy pdf)
% and g= U[xi,xf](defined on the support of interest),
%
% 2. Accepted variable in the previous step are used in the second
% Accept-Reject step to generate the Standard Normal Variables
% so, in the second part
% f = (sqrt(2*pi*sig^2))^-1*exp(-(x-mio)^2/2/sig^2) (Normal pdf)
% and g is obtained cauchy distribution in the previous step,
%
Ns = 100000; % Desired number of samples
a=0; % Parameter values for the standard cauchy
b=1; % Parameter values for the standard cauchy
% support ofthe uniform distribution
xi=-10;
xf=10;
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% Normalizing Factor for the 2nd accept-reject method
M2= 1.5; % Is chosen such that the peak value of the cauchy at x=0 be more
% that standard normal distribution
% Normal distribution
sig=1; mio=0;
% Algorithm
i=0;r=0;j=0;
whilej
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Part b=================Simulating from a Gamma distribution====================
==============with noninteger shape and scalling factor==================
% In this part first the exp(1) R.V.s are generated by transforming
% u~U[0,1] using -log(1-u), then we know that if Xj are i.i.d exp(1)
% variables, then Y=B*(sum(Xj)), for j=1:A, is distributed by ga(a,b)
% Note: A and B are natural numbers
%
% Finally the resulted gamma(A,B) are used in Accept-Reject step tp generate
% ga(4.3,6.2)
Ns = 10000; % Desired number of samples
A1=4; % shape factor for gamma distribution(instrumental pdf)
B1=7; % Scale factor for gamma distribution(instrumental pdf)
A2= 4.3; % shape factor for gamma distribution (Target pdf)
B2= 6.2; % scale factor for gamma distribution (Target pdf)
% Normalizing constant
M=1.2;
% Algorithm
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i=0;r=0;
whilei
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fprintf('The probability of delay is %4.2f%%\n',sum(I)/Ns*100)
% part a.c
Delay_Pr = cumsum(I)./(1:Ns);
sigma=sqrt(cumsum((I-Delay_Pr).^2))./(1:Ns);
figureplot(1:Ns,Delay_Pr,'r');hold on
plot(1:Ns,Delay_Pr+1.96*sigma,'g');hold on
plot(1:Ns,Delay_Pr-1.96*sigma,'g')
title('convergence plot for Probability of delay')
legend('Probability of delay','Based on CLT')
xlabel('sample size')
figure
forR=1:100
T1 = binornd(1,.2,1,Ns).*lognrnd(1,.1,1,Ns);
T2 = lognrnd(0,.25,1,Ns);
T3 = lognrnd(0,.3,1,Ns);
T4 = lognrnd(0,.2,1,Ns); T = T1 + T2 + T3 + T4;
I=T>4;
Delay_Pr = cumsum(I)./(1:Ns);
plot(1:Ns,Delay_Pr,'r');hold on
end
title('convergence plot for Probability of delay')
xlabel('sample size')
% As seen the variability of the expected profit is more when we repeat the
% experiment 100 times and it shows that CLT provides uncertainty bounds
% which are too confident
The expected time-to-completion is 3.64%
The 95% confidence interval for time-to-completion is 0.00%
The probability of delay is 22.91%
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part b
T1 = binornd(1,.2,1,Ns).*lognrnd(1,.1,1,Ns);
T2 = lognrnd(0,.25,1,Ns);
T3 = lognrnd(0,.3,1,Ns);
T4 = lognrnd(0,.2,1,Ns);T = T1 + T2 + T3 + T4; % Time-To-Completion
Profit = 200 + 50 * randn(1,Ns);
D=(T-4);D(D
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fprintf('Probability of loosing money is %4.2f%%\n',CI95)
Probability of profit of 100MSEK or more is 0.81%
Probability of loosing money is 0.00%
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Published with MATLAB R2013a