Suppose that we wish to integrate the same function g(x) as in Example 12.4.1.
a. Use the method of control variates that was described in Exercise 13 to estimate ∫ g(x) dx. Let h(x) = 1/(1 + x2) for 0 < x <1, and k = e−0.5. (This makes h about the same size as g.) Let f (x) be the function f3 in Example 12.4.1. How does the simulation standard error using control variates compare to not using control variates?
b. Estimate the variances and correlation of the W(i)’s and V(i)’s (notation of Exercise 13) to see what a good value for k might be.