# Prova finale LIS 22 -- 31 maggio 22

# Es. 1
rm(list=ls())

n=7
# theta=0.9
# dati=rgamma(n,1,scale=theta)
dati=c(3.0826826, 0.8573664, 2.4893457, 0.3870199, 1.0520483, 0.3786478, 1.5411024)
dati=round(dati,2)
dati
sn=sum(dati)
smv=sn/n
smv
theta.0=0.9
Lik.0=(1/(theta.0^n))*exp(-sn/theta.0)
Lik.smv=(1/(smv^n))*exp(-sn/smv)
# Veros relativa
Lik.rel=Lik.0/Lik.smv
Lik.rel
# Ris.: 0.452...


# Es. 2
rm(list=ls())

n=10
theta=3
sig2=2
sig=sqrt(sig2)
# dati=rnorm(10,mean=theta,sd=sig)
dati=c(4.846913, 1.037077, 4.800176, 3.260488, 4.063884, 3.836854, 1.609754, 2.609587, 1.768431, 4.016410)
dati=round(dati,2)
dati
smv=mean(dati)
# Insieme L.q
smv
q=0.8
k.q=sqrt(-2*log(q))
k.q
L=smv-k.q*sig/sqrt(n)
U=smv+k.q*sig/sqrt(n)
c(L,U)   
# 2.88,3.48
# Lungh
U-L
# 0.59


# Es. 3
rm(list=ls())

n=10
theta=2.3
sig2=2
sig=sqrt(sig2)
dati=c(4.846913, 1.037077, 4.800176, 3.260488, 4.063884, 3.836854, 1.609754, 2.609587, 1.768431, 4.016410)
dati=round(dati,2)
dati

# (a)
smv=mean(dati)
smv
# 3.18
alpha=0.1
t.test(dati,conf.level=1-alpha)$conf.int
# 2.39 3.97

# (b)
alpha=0.04
t.test(dati, alternative = "greater", conf.level=1-alpha,mu=2.30)
# p-value = 0.03489 < 0.04 ---> RIF. H0 (con alpha=0.04)

(c)
# t = 2.0573, df = 9


# Es. 4
rm(list=ls())

n=15
M=10000
mu=3
sig2=4
sig=sqrt(sig2)
x.matr=rnorm(n*M,mean=mu,sd=sig)
x.matr=matrix(x.matr,M,n)
S2.MC=apply(x.matr,1,var)
L=S2.MC/2
U=2*S2.MC
mean(L<=sig2&U>=sig2)
# 0.92


# Es. 5
rm(list=ls())

n=10
M=10000
theta0=4
theta1=3

# simulazione sotto H0
x.matr.H0=rbeta(n*M,shape1=theta0,shape2=1)
x.matr.H0=matrix(x.matr.H0,M,n)
xmed.H0=apply(x.matr.H0,1,mean)
smom.H0=xmed.H0/(1-xmed.H0)

# simulazione sotto H1
x.matr.H1=rbeta(n*M,shape1=theta1,shape2=1)
x.matr.H1=matrix(x.matr.H1,M,n)
xmed.H1=apply(x.matr.H1,1,mean)
smom.H1=xmed.H1/(1-xmed.H1)

k=3.5

# (a) P(R|H0)
mean(smom.H0<=k)
# 0.3159

# (b) P(A|H1)
mean(smom.H1>=k)
# 0.3428

# (c) P(R|H1)=1-P(A|H1)
1-mean(smom.H1>=k)
# 0.6572