# Lab. 1 TSdD 2024-2025 (soluzioni) -- 24 settembre 2024

# Esercizio 1
# consultare help(dbinom) per le funzioni R legate alle v.a. Bernoulliana/Binomiale
# a)
dbinom(3,size=5,prob=0.3)
# b)
pbinom(3,size=10,prob=0.3) # oppure
sum(dbinom(0:3,size=10,prob=0.3))

# c)
qbinom(0.95,size=100,prob=0.3)

# d)
pbinom(5,10,0.2)-pbinom(2,10,0.2) # oppure
sum(dbinom(3:5,10,0.2))

# e)
rbinom(20,10,0.3)
rbinom(20,10,0.7)

# Esercizio 2
xx=seq(0,15,1)
yy=dpois(xx,3)
plot(xx,yy,type="h",lwd=2)

# Esercizio 3
# a)
curve(dnorm(x,-4,sqrt(1)),from=-10,to=12)
curve(dnorm(x,7,sqrt(1)), add=TRUE, lty=2)

# b)
aa=rnorm(1000,0,1) # qui lo facciamo con 1000 valori
hist(aa,prob=T)
curve(dnorm(x,0,1), add=TRUE)
# density estimation
dd=density(aa)
lines(dd,lty=3,lwd=2)

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

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

# Esercizio 6
# consultare help(dchisq)
curve(dchisq(x,df=3),from=0, to=20,ylim=c(0,0.5))
curve(dchisq(x,df=5),lty=2, add=TRUE)
curve(dchisq(x,df=10),lty=3, add=TRUE)

# Esercizio 7
# consultare help(dnorm)
curve(dbeta(x,shape1=0.5,shape2=0.5))