# PROVA FINALE 3 LDS -- febbraio 2025

# ES. 1
rm(list = ls())
al=5
be=3
n=20
sn=12
al.p=al+sn
be.p=be+n-sn
M=10000
th.mc=rbeta(M,al.p,be.p)
lam.mc=log(th.mc/(1-th.mc))
mean(lam.mc)
# 0.44
var(lam.mc)
# 0.15



# Es. 2
rm(list = ls())
al=4
be=2
n=8
sn=20
al.p=al+sn
be.p=be+n
gamma=0.1
M=10000
th.mc=rgamma(M,al.p,rate=be.p)
psi.mc=1/sqrt(th.mc)
L=quantile(psi.mc,gamma/2)
U=quantile(psi.mc,1-gamma/2)
c(L,U)
# 0.56 - 0.78


# Es. 3
rm(list = ls())
al=1
be=5
delta=1.7
lambda=1.3
M=10000
w.mc=rgamma(M,al,rate=be)
mean(w.mc)
loss.mc=abs(w.mc-delta)
# mean(loss.mc)  
mean(loss.mc>lambda)
# 0.88



# Es. 4
rm(list = ls())
al=3
be=4
n=12
M=10000
th.mc=rgamma(M,al,rate=be)
cn=(2*n-1)/(n^2)
R.mc=cn*th.mc^2
mean(R.mc)
# 0.12



# Es. 5
rm(list = ls())
al=5
be=4
M=100000
x.mc=rgamma(M,al,rate=be)
E.x=mean(x.mc)
E.x
mu3.mc=mean((x.mc-E.x)^3)
mu3.mc
# 0.15



# Es. 6
rm(list = ls())
a=0
b=1.3
n=10
M=10000
theta=4
x.matrice=runif(M*n,min=0,max=theta)
x.matrice=matrix(x.matrice,M,n)
X.max=apply(x.matrice,1,max)
L=X.max+a
U=X.max+b
mean(X.max)
mean(L<=theta&U>=theta)
# 0.98



# Es. 7
rm(list = ls())
M=50000
th.0=3
th.1=2
n=10
k=2.2
# Per il calcolo di alpha (simulare sotto H0)
x.matr.H0=rnorm(n*M,mean=0,sd=sqrt(th.0))
x.matr.H0=matrix(x.matr.H0,M,n)
S.H0=apply(x.matr.H0,1,var)
b=(n-1)/(n+1)
d=b*S.H0
mean(d<k)
# 0.47
# Per il calcolo di 1-beta (simulare sotto H1)
x.matr.H1=rnorm(n*M,mean=0,sd=sqrt(th.1))
x.matr.H1=matrix(x.matr.H1,M,n)
S.H1=apply(x.matr.H1,1,var)
b=(n-1)/(n+1)
d=b*S.H1
mean(d<k)
# 0.79



# Es. 8
rm(list = ls())
M=100000
th.0=0.9
th.1=1.1
th.vero=1
x.mc=rexp(M,rate=1/th.vero)
NUM=dexp(x.mc,rate=1/th.0)
DEN=dexp(x.mc,rate=1/th.1)
lambda=NUM/DEN
E=mean(lambda)
E
# 1.02