# Prova finale LIS 1 - 2023-2024 - 30 maggio 2024


# ES 1
rm(list=ls())
al=3
be=4
pgamma(1.5,al,rate=be)-pgamma(0.5,al,rate=be)
# RIS 0.614


# ES 2
rm(list=ls())
dati=c(1.88, 1.72, 3.79, 4.66, 4.12, 2.86, 3.14, 3.34, 2.02, 3.74)
n=length(dati)
xmed=mean(dati)
xmed
sig2=1
sig=sqrt(sig2)
# Lq
q=0.8
k=sqrt(-2*log(q))
L.lik=xmed-k*sig/sqrt(n)
U.lik=xmed+k*sig/sqrt(n)
c(L.lik,U.lik)
# RIS    2.915 3.338
# IC
alpha=0.1
z=qnorm(1-alpha/2)
L=xmed-z*sig/sqrt(n)
U=xmed+z*sig/sqrt(n)
c(L,U)
# RIS    2.606 3.647


# ES 3
rm(list=ls())
dati=c(1.88, 1.72, 3.79, 4.66, 4.12, 2.86, 3.14, 3.34, 2.02, 3.74)
t.test(dati,mu=2.5, alternative = "two.sided", conf.level = 0.95)
# RIS 
# stima puntuale 3.127, IC  2.410 3.843
# p-value = 0.07911 --> si accetta H0


# ES 4
rm(list=ls())
theta=3
M=100000
x.mc=rgamma(M,1,scale=theta)
E.mc=mean(x.mc)
y.mc=(x.mc-E.mc)^3
mu.3=mean(y.mc)
mu.3
# RIS   55.950


# ES 5
rm(list=ls())
theta=3
n=7
lambda=(theta-1)/2
M=100000
x.mc=rgamma(M*n,1,scale = lambda)
x.mc=matrix(x.mc,M,n)
stim.mc=2*apply(x.mc,1,mean) + 1
# per MSE
mean((stim.mc-theta)^2)
# RIS    0.57


# ES 6
rm(list=ls())
theta=3
a=1
b=1.2
n=8
M=100000
x.mc=runif(M*n,0,theta)
x.mc=matrix(x.mc,M,n)
max.mc=apply(x.mc,1,max)
L=a*max.mc
U=b*max.mc
copertura=mean(L<=theta&U>=theta)
copertura
# RIS   0.765


# ES 7
rm(list=ls())
n=15
theta=5
M=100000
x.mc=rpois(M*n,theta)
x.mc=matrix(x.mc,M,n)
mediane.mc=apply(x.mc,1,median)
L=mediane.mc-1/sqrt(n)
U=mediane.mc+1/sqrt(n)
copertura=mean(L<=theta&U>=theta)
copertura
# RIS   0.500


# ES 8
rm(list=ls())
th0=3.5
th1=2.5
n=9
k=2
M=100000
# sotto H0
x.mc.H0=rnorm(n*M,mean=0,sd=sqrt(th0))
x.mc.H0=matrix(x.mc.H0,M,n)
S2.H0=apply(x.mc.H0,1,var)
prob.err.tipo1=mean(S2.H0<k)
prob.err.tipo1    # alpha
# RIS  alpha=0.197
# sotto H1
x.mc.H1=rnorm(n*M,mean=0,sd=sqrt(th1))
x.mc.H1=matrix(x.mc.H1,M,n)
S2.H1=apply(x.mc.H1,1,var)
prob.err.tipo2=mean(S2.H1>k)
prob.err.tipo2   # beta
potenza=1-prob.err.tipo2
potenza   # 1-beta
# RIS   beta=0.603     1-beta=0.396