The following is from Asmussen and Glynn:
Suggest some variance reduction methods for evaluating this integral:
$\int _{0}^{\infty} (x + 0.02x^{2})\text{exp}\{0.1\sqrt{1+\text{cos}x} - x\}dx$
I found a solution here:
However, I do not understand a few things here. For the IS method, how did they calculate the ratio of standard deviations? Generally speaking, how do you know when a particular choice for a proposal density is a good one?
Secondly, why is the selected control variate good? I know that they are supposed to have high correlation with the term in the original expectation, but how do they calculate it here? Is there also a structured approach to selecting a good CV?