####################################################
#
ESERCITAZIONE 1
####################################################
#####################################################
# ESERCIZIO 1
# Calcolo di quantili
#####################################################
## Normale media=0 varianza=1
help(qnorm)
# P(X<x)=0.95
qnorm(0.95, mean = 0, sd = 1)
qnorm(0.95, 0, 1, lower.tail = TRUE)
qnorm(0.95)
# P(X>x)=0.07
qnorm(1-0.07,0,1)
qnorm(0.07,0,1, lower.tail = FALSE)
## Normale media=5 varianza=9
# P(X<x)=0.95
qnorm(0.95, mean = 5, sd = sqrt(9))
# P(X>x)=0.07
qnorm(0.07,5,sqrt(9),FALSE)
## t di student df=9
help(qt)
# P(X<x)=0.95
qt(0.95, df = 9)
# P(X>x)=0.07
qt(0.07,9, , lower.tail = FALSE)
#---------------------------------------## Chi quadro df=12
help(qchisq)
# P(X<x)=0.95
qchisq(0.95, df=12)
# P(X>x)=0.07
qchisq(0.07, 12, , lower.tail = FALSE)
## F df1 = 4 df2 = 7
help(qf)
# P(X<x)=0.95
qf(0.95, df1 = 4, df2 = 7)
# P(X>x)=0.07
qf(0.07, df1 = 4, df2 = 7, , FALSE)
#####################################################
# ESERCIZIO 2
# Generazione di numeri casuali
#####################################################
## Normale media=0 varianza=1
rnorm(n = 1, mean = 0, sd = 1)
rand_Nst <- rnorm(n = 1000, mean = 0, sd = 1)
mean(rand_Nst)
sd(rand_Nst)
## Binomiale n.prove=20, p=0.3
help(rbinom)
rand_B <- rbinom(n = 1000, size = 20, prob = 0.30)
head(rand_B)
#----------------------------------------## Normale media=5 varianza=9
rand_N <- rnorm(n = 1000, mean = 5, sd = sqrt(9))
mean(rand_N)
sd(rand_N)
## t di student df=9
rand_t <- rt(n = 1000, df = 9)
mean(rand_t)
## Chi quadro df=12
rand_chi <- rchisq(n = 1000, df = 12)
mean(rand_chi)
var(rand_chi)
## F df1=4 df2=7
rand_F <- rf(n = 1000, df1 = 4, df2 = 7)
head(rand_F)
#####################################################
# ESERCIZIO 3
# Calcolo delle funzioni di densità e
# di probabilità cumulate
#####################################################
## Normale media=0 varianza=1
# f(1.64)
dnorm(x=1.64, mean=0, sd=1)
# P(X<2.3)
pnorm(q = 2.3, mean = 0, sd = 1, lower.tail = TRUE)
# P(X>1.5)
pnorm(q = 1.5, mean = 0, sd = 1, lower.tail = FALSE)
# P(-2.4<X<3.2)
a <- pnorm(q = 3.2, mean = 0, sd = 1)
b <- pnorm(q = -2.4, mean = 0, sd = 1)
c <- a-b
c
## Binomiale n.prove=20 e p=0.30
# P(X=6)
dbinom(x = 6, size = 20, prob = 0.30)
# P(X<=5)
pbinom(q = 5, size = 20, prob = 0.30, TRUE)
# P(X>=13)
pbinom(q = 13, size = 20, prob = 0.30, FALSE)
# P(X>13)
pbinom(q = 14, size = 20, prob = 0.30, FALSE)
#----------------------------------------# Normale media=0 varianza=1
# P(X>1.5)
1-pnorm(q = 1.5, mean = 0, sd = 1)
# P(X<-2.7)
pnorm(q = -2.7, mean = 0, sd = 1)
# P(X>-1)
pnorm(q = -1.0, mean = 0, sd = 1, FALSE)
## Binomiale n.prove=20 e p=0.30
# P(2<=X<=7)
d <- pbinom(q = 7, size = 20, prob = 0.30)
e <- pbinom(q = 2, size = 20, prob = 0.30)
f <- d - e
f
## Altre distribuzioni
# t di student df=9
dt(x = 2, df = 9)
pt(q=2, df=9)
# Chi quadro df=12
dchisq(x = 3, df = 12)
pchisq(q = 3, df = 12)
# F df1 = 4, df2 = 7
df(x = 5, df1 = 4, df2 = 7)
pf(q = 5, df1 = 4, df2 = 7)
#####################################################
# ESERCIZIO 4
# Rappresentazioni grafiche
#####################################################
## Normale media=0 varianza=1
# funzione di densità di probabilità
yNst <- dnorm(x = rand_Nst, mean = 0, sd = 1)
plot(rand_Nst, yNst, type = 'p')
plot
hist(rand_Nst)
# funzione di ripartizione
yNst_c <- pnorm(q = rand_Nst, mean = 0, sd = 1)
plot(rand_Nst, yNst_c, type = 'p')
## Binomiale n.prove=20, p=0.30
# funzione di probabilità
yBin <- dbinom(x = rand_B, size = 20, prob = 0.30)
hist(rand_B)
plot(rand_B, yBin, type = 'p')
# funzione di ripartizione
yBin_c <- pbinom(q = rand_B, size = 20, prob = 0.30, TRUE)
plot(rand_B, yBin_c, type = 'p')
# -------------------------------------------yChi <- dchisq(x = rand_chi, df = 12)
yChi_c <- pchisq(q = rand_chi, df = 12)
plot(rand_chi, yChi, type = 'p')
hist(rand_chi)
plot(rand_chi, yChi_c, type = 'p')
#####################################################
# ESERCIZIO 5
# Combinazione lineare di variabili casuali
#####################################################
# V.C. Normale(5, 2)
rand_X <- rnorm(n = 1000, mean = 5, sd = sqrt(2))
# V.C. Normale(7, 4)
rand_Y <- rnorm(n = 1000, mean = 7, sd = sqrt(4))
# Z come combinazione lineare di X e Y
Z <- 2*rand_X + 3*rand_Y
mean(Z)
var(Z)
#####################################################
# ESERCIZIO 6
# Calcolo probabilità marginali e condizionate
####################################################
# Matrice di frequenze assolute osservate:
X = matrix(round(runif(12, min=0, max=100)), nrow = 3, ncol = 4)
# Distribuzione delle freq. marginali assolute di colonna
rowSums(X)
# Distribuzione delle freq. marginali assolute di riga
colSums(X)
# Matrice delle prob. condizionate di riga
prop.table(X,1)
rowSums(prop.table(X,1)) # Ok
# matrice delle prob. condizionate di colonna
prop.table(X,2)
colSums(prop.table(X,2)) # Ok
# Matrice delle prob. congiunte
prop.table(X)
######################################################
# ESERCIZIO 7
# Normale multivariata
######################################################
# Istallare il pacchetto mvtnorm
install.packages("mvtnorm")
require(mvtnorm)
# Fissiamo media e varianza
mu = c(1,-1)
Si = matrix(c(4,1,1,1),2,2)
# Calcoliamo funzione di densità in un punto
f = dmvnorm(c(0,0),mu,Si)
# Rappresentiamo grafico densità
x = seq(-6,9,0.25); nx = length(x)
y = seq(-6,9,0.25); ny = length(y)
Z = matrix(0,nx,ny)
for(i in 1:nx) for(j in 1:ny) Z[i,j] = dmvnorm(c(x[i],y[j]),mu,Si)
persp(x,y,Z)
# phi, theta
persp(x,y,Z,phi=20)
persp(x,y,Z,phi=-30)
persp(x,y,Z,theta=10)
persp(x,y,Z,phi=20,theta=10)
persp(x,y,Z,phi=20,theta=40)
persp(x,y,Z,phi=20,theta=40, col = "lightblue")
contour(x,y,Z)
#----------------------------------------------------------------# Provare altra varianza
Si2 = matrix(c(4,0,0,1),2,2)
Z = matrix(0,nx,ny)
for(i in 1:nx) for(j in 1:ny) Z[i,j] = dmvnorm(c(x[i],y[j]),mu,Si2)
contour(x,y,Z)
#-----------------------------------------------------------------# Estrazione numeri casuali da una normale doppia
Si2 = matrix(c(4,0,0,1),2,2)
X2 = rmvnorm(1000,mu,Si2)
colMeans(X2)
cov(X2)
cor(X2)
######################################################
# ESERCIZIO 8
# Verosimiglianza
######################################################
## Rappresentazione verosimiglianza Bernoulli
# Estraiamo 100 osservazioni
y = 1*(runif(100)>0.5)
p = 0.45
f = dbinom(y,1,p)
L = prod(f)
# Rappresentiamo likelihood
pv = seq(0,1,0.001); np = length(pv)
Lv = rep(0,np)
for(i in 1:np) Lv[i] = prod(dbinom(y,1,pv[i]))
plot(pv,Lv,type="l")
# Rappresentiamo log-likelihood
lv = log(Lv)
plot(pv,lv,type="l")
# massimo vicino a 0.5
phat = sum(y)/100
## Rappresentazione verosimiglianza Normale
# Estraiamo 100 osservazioni
y = rnorm(50)
mu = -0.5
f = dnorm(y,mu)
L = prod(f)
# Rappresentiamo likelihood
muv = seq(-1,1,0.01); np = length(muv)
Lv = rep(0,np)
for(i in 1:np) Lv[i] = prod(dnorm(y,muv[i]))
plot(muv,Lv,type="l")
# Rappresentiamo log-likelihood
lv = log(Lv)
plot(muv,lv,type="l")
# Massimo vicino a 0
muhat = sum(y)/50
si2hat = sum((y-muhat)^2)/50
