close all
clear all


T = 10000;
mu =2 ;
sigma = 0.4 ;


x = (randn(T,1)* sigma)  +mu  ;  %% X distributed (mean = mu and std =  sigma)
Z = (x -mu)./sigma ;
xPos =  1.5;


figure
subplot(4,2,1)
hist(x)
title('a) X')
subplot(4,2,2)
hist(Z)
title('b) Z')



subplot(4,2,3)
xsort = sort(x);
PDF_X =  normpdf(xsort,mu,sigma );
plot(xsort,PDF_X)
title(strcat('c) pdf(X),  integral  =',num2str( trapz(xsort,PDF_X))))
hold on
y1=get(gca,'ylim');
plot([xPos xPos],y1);
hold off


subplot(4,2,4)
zsort = sort(Z);
PDF_Z =  normpdf(zsort,0,1 );
plot(zsort,PDF_Z)
title(strcat('d) pdf(Z),  integral  =',num2str(trapz(zsort,PDF_Z))))
hold on
y1=get(gca,'ylim');
plot([xPos xPos],y1);
hold off


subplot(4,2,5)

plot(xsort,PDF_Z)
title(strcat('e) pdf(Z) wrong domain,  integral  =',num2str(trapz(xsort,PDF_Z))))
hold on
y1=get(gca,'ylim');
plot([xPos xPos],y1);
hold off


subplot(4,2,7)
PDF3 = PDF_Z./sigma ;
plot(xsort,PDF3)
title(strcat('f) pdf(Z)/sigma,  integral  =',num2str(trapz(xsort,PDF3))))
hold on
y1=get(gca,'ylim');
plot([xPos xPos],y1);
hold off