Question

What is the code for running the Monte Carlo integration technique for the integral of the standard normal distribution at 2? Please include a graph of sample size vs relative error in the solution.

Answer 1

The following is the R-code for the MonteCarlo Integration:

#Code for MC Integration for a Single Instance

Nsim <- 100 #Number of simulations

x <- rnorm(Nsim) #Drawing a random sample

mc_value <- mean(x<=2) #Value of MC Estimator

pnorm(2) #Actual Value

#Code for MC Integration for Increasing Sample Sizes

Nsim <- 10^4

x <- rnorm(Nsim)

mc_valuearray <- cumsum(x<=2)/(1:Nsim) #Array of MC estimates

rel_error <- (mc_valuearray-pnorm(2))/pnorm(2)

plot(1:Nsim,rel_error,type="l",main="Relative Error vs Sample Size",xlab="Sample Size",ylab="Relative Error")

Important Note:

Here inputing "x<=2" in the R console returns a logical array containing TRUEs and FALSEs. As you may know, R treats TRUE as a 1 and FALSE as a 0, and they can be mathematically added. So, the commands "mean" and "cumsum" can be applied to logical arrays.

Here is the plot: