# Script TD 9
#
# Cours de probabilités appliquées Ensimag 1A
# Sujet: https://proba.pages.ensimag.fr/td/td_9/



##############
# Exercice 1 #
##############

lambda = 1.0
mean(rexp(1000000, rate = lambda) > rexp(1000000, rate = lambda/2.3))
1 / 3.3


##############
# Exercice 2 #
##############

mean(abs(runif(100000) - runif(100000)) < 1/6)
11/36

##############
# Exercice 5 #
##############

n <- 10000

u <- runif(n)
v <- runif(n)

x <- c()
y <- c()

for (i in 1:n) {
  x <- c(x, min(u[i], v[i]))
  y <- c(y, max(u[i], v[i]))
}

mean(x)
mean(y)
mean(x * y)

# Cov[X, Y] = E[XY] - E[X]E[Y]
mean(x * y) - mean(x) * mean(y)
cov(x, y)
1/36