Problem on the instrumental densities

Question

Response to the following problem:

(O Ruanaidh and Fitzgerald, 1996) For simulating random variables ´ from the density

F(X) ∝ exp {-x² √x}[sin(x)²] , 0

compare the following choices of instrumental densities on R:

g₁(x) = (1/2)e^-|x| ,     g₂(x) = (1/2π)(1/1+x²/4) ,    g₃(x) = (1/√2π)e^-x2/2

For each of the instrumental densities, estimate the number M of simulations needed to obtain three digits of accuracy in estimating Ef [X].

Deduce from the acceptance rate an estimator of the normalizing constant of f for each of the instrumental densities.