close all
% generator slucajnih celih brojeva po uniformnog raspodeli
help randi 

% [1,6] celi brojevi u opegu od 1 do 6
% promenljiva k ce biti matrica dimenzija:
% 1 - jedan red
% 10 - 10 kolona
k = randi ([1,6],1,10) % simulira 10 bacanja kockice 

% ako je ; na kraju reda ne ispisuje se rezultat na ekranu
k = randi ([1,6],1,600000); % 600000 bacanja 

% crtanje histograma
help hist
help histc
help bar

% zakon raspodele za kockicu
Hk = histc (k,[1,2,3,4,5,6]);
Pk = Hk / length (k);
figure (1)
bar (1:6,Pk,'hist')
l=line([1,6],[1/6,1/6]);
set(l,'Color','r')
set(l,'LineWidth',4)
title('Zakon raspodele')
xlabel('elemntrani ishod')
ylabel('verovatnoca ishoda')
grid

% gustina verovatnoce za Gausovu slucajnu promenljivu
n = 1000000; % broj elemenata uzorka
g = randn(1,n); % generisanje Gausove raspodele - rezultat matrica dimenzija 1 x n
[Hg,xg] = hist (g,100); % histogram u 100 tacaka
dx = xg(2) - xg(1); % dx je sirina klase (been)
fg = Hg / (n*dx); % izracunavanje funkcije gustine verovatnoce
figure(2)
bar(xg,fg,'hist')
x = -4.5:1/100:4.5;
y = 1/sqrt(2*pi)*exp(-x.^2/2);
hold on
plot(x,y,'Color','r','LineWidth',3)
title('Gausova raspodela')
hold off
grid

% linearna transformacija Gausove raspodele
X = g;
a = 4;
b = 10;
Y = a*X + b;
[h,x] = hist (Y,100);
dx = x(2) - x(1);
f = h / (n*dx);
figure(3)
bar(x,f,'hist')
title('Gausova raspodela posle transformacije')
grid
Xsr = mean(X)
Xvar = var(X)
Xstd = std(X)
Ysr = mean(Y)
Yvar = var(Y)
Ystd = std(Y)

% kvadratna transformacija Gausove raspodele
Y=X.^2;
[hy xy]=hist(Y,100);
fy=hy/(n*(xy(2)-xy(1)));
figure(4)
subplot(221),plot(X(1:1000))
subplot(222),bar(xg,fg,'hist')
subplot(223),plot(Y(1:1000))
subplot(224),bar(xy,fy,'hist')

% sinusna transformacija uniformne raspodele
n=100000;
X=2*pi*rand(1,n);
[hx xx]=hist(X,100);
fx=hx/(n*(xx(2)-xx(1)));
Y=sin(2*pi*1000+X);
[hy xy]=hist(Y,100);
fy=hy/(n*(xy(2)-xy(1)));
figure(5)
subplot(221),plot(X(1:1000))
subplot(222),bar(xx,fx)
subplot(223),plot(Y(1:1000))
subplot(224),bar(xy,fy)

% transformacija Dekartovih koordinata (x, y) u polarne koordinate (r,w)
% transformacija jakobijanom 2 x 2 promenljive
n=100000;
X=randn(1,n);
Y=randn(1,n);
R=sqrt(X.^2+Y.^2);
W=atan(X./Y);
[hr xr]=hist(R,100);
fr=hr/(n*(xr(2)-xr(1)));
[hw xw]=hist(W,100);
fw=hw/(n*(xw(2)-xw(1)));
figure(6)
subplot(211),bar(xr,fr)
title('Raspoela promenljive W')
subplot(212),bar(xw,fw)
title('Raspoela promenljive R')

% transformacija raspodele kroz limiter
% y = x, za x>-a i x<a
% y = -a, za x<-a
% y = +a, za x>+a
n=100000;
X=randn(1,n);
Y=zeros(size(X));
a=2;
for i=1:n
  if(X(i)>=a)
    Y(i)=a;
  elseif(X(i)<=-a)
    Y(i)=-a;
  else
    Y(i)=X(i);
  end
end
[hx xx]=hist(X,100);
fx=hx/(n*(xx(2)-xx(1)));
[hy xy]=hist(Y,100);
fy=hy/(n*(xy(2)-xy(1)));
figure(7)
subplot(211),bar(xx,fx)
title('Raspoela promenljive X')
subplot(212),bar(xy,fy)
title('Raspoela promenljive Y')

% raspodela zbira dve nezavisne slucajne 
n=100000;
X=rand(1,n);
Y=rand(1,n);
Z=X+Y;
[h,x]=hist(Z,100);
dx=x(2)-x(1);
f=h/(n*dx);
figure(1)
bar(x,f)
title('Raspodela zbira')

% raspodela zbira dve nezavisne slucajne 
n=100000;
X=rand(1,n);
Y=X;
Z=X+Y;
[h,x]=hist(Z,100);
dx=x(2)-x(1);
f=h/(n*dx);
figure(9)
bar(x,f)
title('Raspodela zbira')