Question

Exercise 3 For the computation of the expectation Ef[h(x)] when f is the normal pdf and h(x) - exp A. Show that Ef[h(x)] can be computed in closed form and derive its value. ( 2p) + exp (-(xur) -(x-2)2 (x-4)2 Construct a regular Monte Carlo approximation based on a normal N (0,1) sample of size Nsim-10A4 and produce an error evaluation. Compare the above with an importance sampling approximation based on an importance function g corresponding to the U(-6 - 1) distribution and a sample of size Nsim-10A4 (Warning: This choice of g does not provide a onverging approximation of Ef[h(x)]!) B. C.

Answer 1

ら e.

R Program

## B.
nsim=10^4
x=rnorm(nsim,0,1)
pr=exp(-(x-2)^2/2-(x-4)^2/2)
mean(pr)
sd(pr)

OUTPUT

value=0.01049353
SD= 0.03898224

SD=

## C.
nsim=10^4
x=runif(nsim,-6,-1)
pr1=5*exp(-x^2/2-(x-2)^2/2-(x-4)^2/2)/sqrt(2*pi)
mean(pr1)
sd(pr1)

OUTPUT

value= 1.016398e-09
SD=5.040679e-09