%Cumulative normal distribution function
%We test different approximations of the cdf

%clear memory and screen
clear all; clc;

%Partition
a=-3; b=3; stepsize=0.01; x=a:stepsize:b;

%Use the function of Matlab
tic 
Matlab = normcdf(x);
toc

%Use de approximation described in Hull
tic
Hull = NORMDISTHULL(x);
toc

% Integration using built in Matlab functions
tic
lb = -100;   %lower bound
NumIntQuad = zeros(length(x),1);
f = @(y) exp(-0.5*y.*y)/sqrt(2*pi); %standard normal density function
for i=1:length(x)
    ub = x(i);       %upper bound
    NumIntQuad(i)=quad(f,lb,ub);
end
toc

%Create a plot
figure
    plot(x,Matlab,'-b');
    grid on
    hold on
    plot(x,Hull,'-.r');
    plot(x,NumIntQuad,':g');
    hold off
    xlabel('x')
    ylabel('Normal cdf')
    title(['Estimation of the cumulative normal distribution function'])
    legend('Matlab','Hull','Numerical integration')
 
%%
%-----------------------
%CLOSER LOOK TO THE TAIL
%-----------------------

%Partition
a=-10; b=-5; stepsize=0.01; x=a:stepsize:b;

%Use the function of Matlab
tic 
Matlab = normcdf(x);
toc

%Use de approximation described in Hull
tic
Hull = NORMDISTHULL(x);
toc

% Integration using built in Matlab functions
tic
lb = -100;   %lower bound
NumIntQuad = zeros(length(x),1);
f = @(y) exp(-0.5*y.*y)/sqrt(2*pi); %standard normal density function
for i=1:length(x)
    ub = x(i);       %upper bound
    NumIntQuad(i)=quad(f,lb,ub);
end
toc

%Create a plot
figure
    plot(x,Matlab,'-b');
    grid on
    hold on
    plot(x,Hull,'-.r');
    plot(x,NumIntQuad,':g');
    hold off
    xlabel('x')
    ylabel('Normal cdf')
    title(['Estimation of the cumulative normal distribution function'])
    legend('Matlab','Hull','Numerical integration')