rm(list = ls())
set.seed(2026)

# Une uniforme et une normale

n = 100000

X = runif(n, min = 0, max = 5)
Y = rnorm(n, mean = 0, sd = sqrt(2))

h = 5*sqrt((X+Y)*(Y>=2))*sqrt(4*pi)*sin(Y^4)*exp(-1.5*X)

delta = mean(h)
var_delta = var(h)

delta
var_delta

# Une exponentielle et une normale

X = rexp(n, 1.5)
Y = rnorm(n, mean = 0, sd = sqrt(2))

h = 2/3*sqrt((X+Y)*(Y>=2)*(X<=5))*sqrt(4*pi)*sin(Y^4)

delta = mean(h)
var_delta = var(h)

delta
var_delta

# Une exponentielle et une normale tronquée

X = rexp(n, 1.5)

p = 0.0786
U = runif(n,0,1)
Y = sqrt(2)*qnorm(p*U-p+1, mean = 0, sd = 1)

h = 2/3*sqrt((X+Y)*(X<=5))*sqrt(4*pi)*sin(Y^4)*p

delta = mean(h)
var_delta = var(h)

delta
var_delta