course problem on statistics

8/13/2019 course problem on statistics

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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);


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);



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

course problem on statistics

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