/////// Exercice 1

// Question 1
n=200;
x=grand(n,1,"nor",0,1);
y=grand(n,1,"nor",0,1);
correl(x,y,eye(n,n))

// Question 2
a=0.2
u=a*x+sqrt(1-a**2)*y;

mean(u)
variance(u)

clf();histplot(20,u,normalization=%t);
absc=linspace(-10,10,100);
plot(absc,exp(-(absc)^2/2)/sqrt(2*%pi));

// Question 3

covar(x,u,eye(n,n))
correl(x,u,eye(n,n))

// Question 4

clf();plot(x,u,'p',-3:3,-3:3);

for a2 = -100:100
    a=a2/100;
    u=a*x+sqrt(1-a**2)*y;
    clf();
    plot([-3,3],[0,0])
    plot([0,0],[-3,3])
    plot(x,u,'p');
end

// Question 5
v=x^2;

// Questions 6 et 7

clf();
histplot(20,v,normalization=%t)
absc=linspace(0.01,10,100);
plot(absc,exp(-absc/2)./(sqrt(2)*gamma(0.5)*sqrt(absc)))

// Questions 8 et 9

clf();
absc=linspace(-4,4,100);
plot(absc,exp(-(absc)^2/2)/sqrt(2*%pi))
k=1
ord=gamma((k+1)/2)/(gamma(k/2)*sqrt(k*%pi))*((1+absc.^2/k).^(-(k+1)/2));
plot(absc,ord,'r')

k=10
ord=gamma((k+1)/2)/(gamma(k/2)*sqrt(k*%pi))*((1+absc.^2/k).^(-(k+1)/2));
plot(absc,ord,'g')

k=100
ord=gamma((k+1)/2)/(gamma(k/2)*sqrt(k*%pi))*((1+absc.^2/k).^(-(k+1)/2));
plot(absc,ord,'m')

/////// Exercice 2

// Question 1

x=[3.41 2.85 7.57 9.73 10.19 4.58 2.62 5.90 4.57 4.52];
sigma=2;
n=length(x)

// Question 2

3+sigma*cdfnor("X",0,1,0.95,0.05)/sqrt(n)

// Question 3

mean(x)

// Question 4

[P,Q]=cdfnor("PQ",mean(x),3,sigma/sqrt(n))

// Question 5

n=1000
x=grand(n,1,"nor",4.5,sigma);

3+sigma*cdfnor("X",0,1,0.95,0.05)/sqrt(n)
mean(x)
[P,Q]=cdfnor("PQ",mean(x),3,sigma/sqrt(n))

// Question 6

n=1000
x=grand(n,1,"nor",2.8,sigma);

3+sigma*cdfnor("X",0,1,0.95,0.05)/sqrt(n)
mean(x)
[P,Q]=cdfnor("PQ",mean(x),3,sigma/sqrt(n))

// Question 7

N=1000
n=100
resultats=zeros(N,1);

for i=1:N
    x=grand(n,1,"nor",3,sigma);
    resultats(i)=mean(x);
end

// Question 8

clf(); histplot(20,resultats,renormalization=%f)
plot([cdfnor("X",3,sigma/sqrt(n),0.95,0.05),cdfnor("X",3,sigma/sqrt(n),0.95,0.05)],[0,2.5])

// Question 9

sum(resultats>cdfnor("X",3,sigma/sqrt(n),0.95,0.05))/N

// Question 10

N=1000
n=100
resultats=zeros(N,1);

for i=1:N
    x=grand(n,1,"nor",4,sigma);
    resultats(i)=mean(x);
end

clf(); histplot(20,resultats,renormalization=%f)
plot([cdfnor("X",3,sigma/sqrt(n),0.95,0.05),cdfnor("X",3,sigma/sqrt(n),0.95,0.05)],[0,2.5])

sum(resultats>cdfnor("X",3,sigma/sqrt(n),0.95,0.05))/N