# LIS 24-25
# Laboratorio 2 - 06 marzo 2025
# Variabili aleatorie; funzione  di verosimiglianza (fdv)


# Esercizio 1
curve(dexp(x,rate=1/0.5),from=0,to=5)
curve(dexp(x,rate=1),add=T,lty=2)
curve(dexp(x,rate=1/2),add=T,lty=3)


# Esercizio 2
curve(dexp(x,rate=0.5),from=0,to=2,ylim=c(0,2))
curve(dexp(x,rate=1),add=T,lty=2)
curve(dexp(x,rate=2),add=T,lty=3)


# Esercizio 3
curve(dgamma(x,shape=1,scale=2),from=0,to=10)
curve(dgamma(x,shape=2,scale=2),add=T,lty=2)
curve(dgamma(x,shape=3,scale=2),add=T,lty=3)
curve(dgamma(x,shape=5,scale=1),add=T,lty=4)


# Esercizio 4
curve(dnorm(x),from=-3,to=3)
pnorm(-1)


# Esercizio 5
pnorm(3,1,sqrt(2))-pnorm(-2,1,sqrt(2))    # 0.904


# Esercizio 6
# help(dchisq)
curve(dchisq(x,0.5),xlim=c(0,20),ylim=c(0,0.5))
curve(dchisq(x,3),add=TRUE, lty=2)
curve(dchisq(x,5), add=TRUE, lty=3)
curve(dchisq(x,10), add=TRUE, lty=4)


# Esercizio 7
xn=c(1,1,0,0,0,1,1)
fi=dbinom(xn,size=1,prob=0.4)
prod(fi)
# oppure
tn=sum(xn)
n=length(xn)
th=0.4
th^tn*(1-th)^(n-tn)
# 0.0055
# per n=15 e tn=10
tn=10
n=15
th=0.4
th^tn*(1-th)^(n-tn)
# circa 0


#Esercizio 8
xn=c(0.5,2,1)
n=length(xn)
fi=dnorm(xn,mean=1,sd=sqrt(2))
prod(fi)
# 0.016


# Esercizio 9
xn=c(1,1,0,1,0)
n=length(xn)
th1=0.1
th2=0.9
tn=sum(xn)
Lik1=(th1^tn)*(1-th1)^(n-tn)
Lik1
# 0.0008
Lik2=(th2^tn)*(1-th2)^(n-tn)
Lik2
# 0.0079
# rapporto
Lik1/Lik2
# 0.11  


# Esercizio 10
n=50
tn=30
th1=0.1
th2=0.9
Lik1=(th1^tn)*(1-th1)^(n-tn)
Lik1
Lik2=(th2^tn)*(1-th2)^(n-tn)
Lik2
# rapporto
Lik2/Lik1
# 3486784401