%% (a) generate normal distributed random numbers:
n = 10000;
x = randn( n, 1 );

%% (b)
nsig = sum((x>=-1.0)&(x<=1.0));
Psig = nsig/length(x);
fprintf('%d of %d data elements, i.e. %.2f%% are contained in the interval -1 to +1\n\n', ...
        nsig, length(x), 100.0*Psig );
    
%% (c)
dx = 0.01;
xx = -10:dx:10;                   % x-values
pg = exp(-0.5*xx.^2)/sqrt(2*pi);  % y-values Gaussian pdf
% integral over the whole range:
P = sum(pg)*dx;
fprintf( 'Integral over the Gaussian pdf is %.3f\n', P );
% integral from -1 to 1:
P = sum(pg((xx>=-1.0)&(xx<=1.0)))*dx;   % we need to use xx, not the random numbers x!
fprintf( 'Integral over the Gaussian pdf from -1 to 1 is %.4f\n', P );

%% (d)
P = sum(pg((xx>=-2.0)&(xx<=2.0)))*dx;
fprintf( 'Integral over the Gaussian pdf from -2 to 2 is %.4f\n', P );
P = sum(pg((xx>=-3.0)&(xx<=3.0)))*dx;
fprintf( 'Integral over the Gaussian pdf from -3 to 3 is %.4f\n\n', P );

%% (e) probability of small ranges
nr = 50;
xmax = 3.0
xs = zeros(nr, 1);  % size of integration interval
Ps = zeros(nr, 1);  % storage
for i = 1:nr
    % upper limit goes from 4.0 down to 0.0:
    xupper = xmax*(nr-i)/nr;
    xs(i) = xupper;
    % integral from 0 to xupper:
    Ps(i) = sum(pg((xx>=0.0)&(xx<=xupper)))*dx;
end
plot( xs, Ps, 'linewidth', 3 )
xlim([0 xmax])
ylim([0 0.55])
xlabel('Integration interval')
ylabel('Probability')
fprintf('The probability P(0.1234) = %.4f\n\n', sum(x == 0.1234)/length(x) );
savefigpdf(gcf, 'normprobs.pdf', 12, 8);