set.seed(1974)
N <- 10000
x <- rexp(N)
h <- function(x) 1+x^2+sin(x)*abs(x)
thetahat <- mean(h(x))
tauhat <- var(h(x))
sqrt(tauhat/N)
c(thetahat-sqrt(tauhat/N)*qnorm(0.975),thetahat+sqrt(tauhat/N)*qnorm(0.975))
z <- function(x) h(x)*exp(-x)
curve(z,0,20)
integrate(z,0,20)

N <- 10000
thetahat <- rep(0,1000)
for (j in 1:1000)
{
  x <- rexp(N)
  thetahat[j] <- mean(h(x))
}
hist(thetahat,nclass=20,prob=TRUE)
curve(dnorm(x,3.5,0.04),from=0,to=12,n=1000,add=TRUE,col="red")
luglu <- cumsum(h(x))/(1:N)
plot(1:N,luglu,type="l")