N<-100000

x0<-4
true.p<-1-pnorm(x0)

(pnorm(x0))^N

set.seed(24)
X<-rnorm(N)

Ihat<-cumsum(X>x0)/c(1:N)
plot(Ihat,type="l",xlab="N")
abline(h=true.p,lwd=2,col="red")


lam<-10
X<-x0-log(runif(N))/lam
p0.x<-lam*exp(-lam*(X-x0))
w.x<-dnorm(X)/p0.x
Ihat.p0<-cumsum(w.x)/c(1:N)
plot(1:N,Ihat,type="l",xlab="N",ylab=expression(hat(I)[N](g)))
lines(1:N,Ihat.p0,lty=3,lwd=2)
abline(h=true.p,lwd=2,col="red")

lam<-100
X<-x0-log(runif(N))/lam
p0.x<-lam*exp(-lam*(X-x0))
w.x<-dnorm(X)/p0.x
Ihat.p0<-cumsum(w.x)/c(1:N)
plot(1:N,Ihat,type="l",xlab="N",ylab=expression(hat(I)[N](g)))
lines(1:N,Ihat.p0,lty=3,lwd=2)
abline(h=true.p,lwd=2,col="red")

########################################################
N<-1000000

#Reproduce plot from notes
set.seed(24);nu<-5

#Alternately
#set.seed(25764);nu<-2

X<-rnorm(N)

Ihat<-cumsum(X>x0)/c(1:N)
X<-rt(N,df=nu)
p0.x<-dt(X,df=nu)
w.x<-dnorm(X)*(X>x0)/p0.x
Ihat.p0<-cumsum(w.x)/c(1:N)
plot(1:N,Ihat,type="l",xlab="N",ylab=expression(hat(I)[N](g)))
lines(1:N,Ihat.p0,lty=3,lwd=2)
abline(h=true.p,lwd=2,col="red")