Show that ef hx can be computed in closed form and derive

Question

For the computation of the expectation E_f [h(X)] when f is the normal pdf and h(x) = exp(-(x - 3)²/2) + exp(-(x - 6)²/2):

a. Show that Ef [h(X)] can be computed in closed form and derive its value.

b. Construct a regular Monte Carlo approximation based on a normal N (0, 1)sample of size Nsim=10^3 and produce an error evaluation.

c. Compare the above with an importance sampling approximation based on an importance function g corresponding to the U(-8, -1) distribution and a sample of size Nsim=10^3.

(Warning: This choice of g does not provide a converging approximation of E_f [h(X)]!)