#
# Monte Carlo Integration
#
##############################



#--- int_0^1 e^(-x) dx --------
True <- 1-exp(-1); True




#--- numerical integration -----
f1 <- function(x)   {  exp(-x) }
NI <- integrate( f1, 0, 1)

str(NI)   # see what's inside
NI$value  # only the value of the integral 




#--- Monte Carlo Integration ---
n <- 10000

U <- runif(n, 0, 1)
X <- exp(-U)
MCI <- mean(X); MCI

ME <- 1.96*sqrt( var(X)/n ); ME





#--- MCI with Antithetic Variates ---

n <- 5000
U <- runif(n, 0, 1)
V <- 1-U

cor(U,V)

X <- exp(-U)
Y <- exp(-V)

cor(X,Y)

AT   <- (X+Y)/2 
MCI2 <- mean(AT); MCI2

ME2  <- 1.96*sqrt( var(AT)/n ); ME2