# MC
N <- 1e6
X <- runif(N)
Z.MC <- mean(X^2)
RMSE.MC <- sqrt(var(X^2) / N)

# antithetic variables
X <- runif(N/2)
X.prime <- 1 - X
rho <- cor(X^2, X.prime^2)
Z.AV <- (sum(X^2) + sum(X.prime^2)) / N
RMSE.AV <- sqrt(var(X^2)*(1+rho) / N)

#####
# new topic

U <- runif(10000)
X <- -log(U)
X2 <- -log(1-U)
hist(X)
hist(X2)
plot(X, X2)
cor(X, X2)

####

x <- seq(-3, 3, by=0.01)
plot(x, cos(x), type="l")
lines(x, 1 - x^2/2, col="red")

# MC
N <- 1e6
X <- rnorm(N)
Z.MC <- mean(cos(X))
RMSE.MC <- sqrt(var(cos(X)) / N)

# CV
g <- function(x) { 1 - x^2/2 }
mean(cos(X) - g(X)) + 1/2
RMSE.CV <- sqrt(var(cos(X)-g(X)) / N)
c <- cov(cos(X), g(X)) / var(g(X))

mean(cos(X) - c*g(X)) + c/2
RMSE.CV2 <- sqrt(var(cos(X)-c*g(X)) / N)