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% Session 1 (script session1.m)
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clear
T=rand(1000,1000);
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% you can put your comments here...
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for i=1:10                               %...or actually anywhere
a(i,:)=mean(T(1:i*100,:));
end
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for i=1:10
s2(i,:)=var(T(1:i*100,:));
end
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figure
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% Figure 1
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hist(a(1,:),15)
figure
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% Figure 2
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hist(a(2,:),15)
figure(1)
xlabel('a')
ylabel('number of observations')
title('Histogram of a sample mean (N=100)')
figure(2)
xlabel('a')
ylabel('number of observations')
title('Histogram of a sample mean (N=200)')
set(gca,'xlim',[0.4 0.6])
figure(1)
set(gca,'xlim',[0.4 0.6])
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mina=min(min(a));                        
maxa=max(max(a));                 
da=(maxa-mina)/15;             
edgesa=[mina:da:maxa];       
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h1=histc(a(1,:),edgesa);                
whos h1
size(h1,1)                                                            
size(h1,2)
h1=h1/(size(h1,2)*mean(h1,2));                                
mean(h1)*size(h1,2)
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figure
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% Figure 3
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plot(edgesa,h1)                                        
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h10=histc(a(10,:),edgesa);                     
h10=h10/(size(h10,2)*mean(h10,2));
hold on                                            
plot(edgesa,h10,'r')                  
legend('N=100','N=1000')              
xlabel('a')
ylabel('p(a)')
title('Estimated PDF of an average of a sample of size N drawn from a uniform distribution on [0,1]')
grid on
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mean(a')                                           
var(a')                                              
N=[100:100:1000];
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figure
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% Figure 4
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plot(N,1./var(a'),'x')                                                
hold on     
plot(N,12*N,'r')                                   
legend('numerical result','theoretical prediction')
xlabel('N')
ylabel('Var{N}')
title('Numerical and theoretical estimates of the sample-mean\prime s dispersion as a function of a sample size N')
grid on
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% End of script
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