# Calculates m replications of the Monte Carlo
# integration in part (a) with a = y and n = n1
monte <- function(y,n1,m){
vect<-c(1:m)
for (i in 1:m)
{
x <- rnorm(n1)
total <- length(x[x > y])
vect[i] <- total/n1
}
return(vect)
}
# Calculates m replications of the importance sampling
# in part (c) with a = y1, mu = y2 and n = n1
import <- function(y1,y2,n1,m)
{
vect <- c(1:m)
for (i in 1:m)
{
x <- rnorm(n1,y2)
weight <- dnorm(x)/dnorm(x, y2)
vect[i] <- sum(weight[x > y1])/n1
}
return(vect)
}
# Calculates the importance sampling in part (d) for
# P(X > y) and q(x) the Exp(1) truncated at y
# Plots the cumulative estimate of P(X > y) with n = n1  
import1 <- function(y,n1){
x<-rexp(n1)+y
weight=dnorm(x)/dexp(x-y)
plot(cumsum(weight)/1:n1,type="l",ylab="Cumulative sum of estimate",xlab="n")
abline(a=pnorm(-y),b=0,col="red")
}