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figure(1);
clc
clear
dx=0.01;
x=-5:dx:5;
mu =0 ;
sigma=1;
num=exp(-0.5*((x-mu)/sigma).^2);
den=sigma*sqrt(2*pi);
y=num/den;
plot(x,y,'r','linewidth',3);
hold on;
sigma1=sqrt(5);
num1=exp(-0.5*((x-mu)/sigma1).^2);
den1=sigma1*sqrt(2*pi);
y1=num1/den1;
plot(x,y1,'Color',[0.7,0.4,0],'linewidth',3);
sigma2=sqrt(0.2);
num2=exp(-0.5*((x-mu)/sigma2).^2);
den2=sigma2*sqrt(2*pi);
y2=num2/den2;
plot(x,y2,'b','linewidth',3);
sigma3=sqrt(0.5);
mu1=-2;
num3=exp(-0.5*((x-mu1)/sigma3).^2);
den3=sigma3*sqrt(2*pi);
y3=num3/den3;
plot(x,y3,'g','linewidth',3);
hold off;
legend('mu=0,sigma^2=1','mu=0,sigma^2=5','mu=0,sigma^2=0.2','mu=-2,sigma^2=0.5');

figure(2);
N = 1e5;
X = randn(N,1);
size(X)
hist(X,31);
[nelements,centers] = hist(X,31);
Prob = nelements/N;
dx = centers(2) - centers(1);
pdf = Prob/dx;
figure(3);
plot(centers,pdf,'r');
hold on;
X=sqrt(0.2)*randn(N,1);
[nelements,centers] = hist(X,31);
Prob = nelements/N;
dx = centers(2) - centers(1);
pdf = Prob/dx;
figure(3);
plot(centers,pdf,'b');
X=sqrt(0.5)*randn(N,1)-2;
[nelements,centers] = hist(X,31);
Prob = nelements/N;
dx = centers(2) - centers(1);
pdf = Prob/dx;
figure(3);
plot(centers,pdf,'g');