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% clean up workspace
clear; close all; clc;
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% define input parameters
% IMPORTANT: always define the inputs in the beginning in order to make the
% script as flexible as possible
NumSimulations=10000; % scalar
mu=3;
sigma_square=5;
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% generate normal sample with above parameters
sigma=sqrt(sigma_square);
X=normrnd(mu,sigma,NumSimulations,1);
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% plot sample
figure % open new figure
ObservationTime=[1:NumSimulations];
plot(ObservationTime,X,'.');
grid on
title('normal sample vs observation time')
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% plot histogram
NumBins=round(10*log(NumSimulations));
figure % open new figure
hist(X,NumBins);
grid on
title('histogram of normal sample')
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% plot empirical cdf
figure % open new figure
[f,x]=ecdf(X);
plot(x,f);
grid on
% plot exact cdf
F=normcdf(x,mu,sigma);
hold on % superimpose graph
h=plot(x,F);
set(h,'color','r');
legend('empirical','exact','location','best')
title('cdf of normal distribution')
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% plot empirical quantile
u=[.01 : .01 : .99]; % range of quantiles (values between zero and one)
q=prctile(X,u*100);
figure % open new figure
plot(u,q);
grid on
% plot exact quantile
Q=norminv(u,mu,sigma);
hold on
h=plot(u,Q);
set(h,'color','r');
legend('empirical','exact','location','best')
title('quantile of normal distribution')
xlabel('Grade')
ylabel('Quantile')
