Let g be a function, and suppose that we wish to compute the mean of g(X) where

Question:

Let g be a function, and suppose that we wish to compute the mean of g(X) where X has the p.d.f. f . Suppose that we can simulate pseudo-random values with the p.d.f. f . Prove that the following are the same:
• Simulate X(i) values with the p.d.f. f , and average the values of g(X(i)) to obtain the estimator.
• Do importance sampling with importance function f to estimate the integral ∫ g(x)f (x) dx.