This problem introduces a variation on the Monte Carlo integration technique of Example...

This problem introduces a variation on the Monte Carlo integration technique of Example A of Section 5.2. Suppose that we wish to evaluate

Let g be a density function on [a, b]. Generate X₁, · · · , X_n from g and estimate I by

a. Show that .

b. Find an expression for .

c. Show that if a = 0, b = 1, and g is uniform, this is the same Monte Carlo estimate as that of Example A of Section 5.2. Can this estimate be improved by choosing g to be other than uniform? (Hint: Compare variances.)