% teorema del limite centrale
% sommiamo nv variabili indipendenti con distribuzione uniforme ottenute
% con np prove e confrontiamo la frequenza relativa della distribuzione
% della loro somma con la previsione teorica, vale a dire con una gaussiana
% di media e varianza note dalla teoria

clear all;
close all;
clc;
rng('default')

np=10000;
a=0.2;
b=3.0;
media = (a+b)/2;
varianza = (b-a)^2/12;
nbins=101;
ub=b+0.3; %sqrt(varianza) + media;
lb=a-0.3; %-sqrt(varianza) + media;
x=linspace(lb,ub,nbins);
dx=(ub-lb)/(nbins-1);
updf = unifpdf(x,a,b);
figure;
hold on;
plot(x,updf,'r')
title('Uniform')


nv=1;
y=unifrnd(a,b,nv,np);
sy=sum(y,1)/nv;
freq=hist(sy,x)/(dx*np);
dist=1/sqrt(2*pi*varianza/nv)*exp(-(x-media).^2/(2*varianza/nv));
figure;
hold on;
bar(x,freq);
plot(x,dist,'r')
title('n=1')

nv=2;
y=unifrnd(a,b,nv,np);
sy=sum(y,1)/nv;
freq=hist(sy,x)/(dx*np);
dist=1/sqrt(2*pi*varianza/nv)*exp(-(x-media).^2/(2*varianza/nv));
figure;
hold on;
bar(x,freq);
plot(x,dist,'r')
title('n=2')

nv=5;
y=unifrnd(a,b,nv,np);
sy=sum(y,1)/nv;
freq=hist(sy,x)/(dx*np);
dist=1/sqrt(2*pi*varianza/nv)*exp(-(x-media).^2/(2*varianza/nv));
figure;
hold on;
bar(x,freq);
plot(x,dist,'r')
title('n=5')

nv=30;
y=unifrnd(a,b,nv,np);
sy=sum(y,1)/nv;
freq=hist(sy,x)/(dx*np);
dist=1/sqrt(2*pi*varianza/nv)*exp(-(x-media).^2/(2*varianza/nv));
figure;
hold on;
bar(x,freq);
plot(x,dist,'r')
title('n=30')

nv=50;
y=unifrnd(a,b,nv,np);
sy=sum(y,1)/nv;
freq=hist(sy,x)/(dx*np);
dist=1/sqrt(2*pi*varianza/nv)*exp(-(x-media).^2/(2*varianza/nv));
figure;
hold on;
bar(x,freq);
plot(x,dist,'r')
title('n=50')

nv=100;
y=unifrnd(a,b,nv,np);
sy=sum(y,1)/nv;
freq=hist(sy,x)/(dx*np);
dist=1/sqrt(2*pi*varianza/nv)*exp(-(x-media).^2/(2*varianza/nv));
figure;
hold on;
bar(x,freq);
plot(x,dist,'r')
title('n=100')

nv=500;
y=unifrnd(a,b,nv,np);
sy=sum(y,1)/nv;
freq=hist(sy,x)/(dx*np);
dist=1/sqrt(2*pi*varianza/nv)*exp(-(x-media).^2/(2*varianza/nv));
figure;
hold on;
bar(x,freq);
plot(x,dist,'r')
title('n=500')