Continuing the preceding problem, suppose that instead of having a three-sided coin, you have a slightly broken random number generator, xrand, which returns a random number between 0 and 3. Seventy percent of the time it's in [0, 1], uniformly distributed; 20% of the time it's in [1, 2], uniformly distributed; and 10% of the time it's in [2, 3], uniformly distributed. You're given a single sample t generated by xrand, and you ask your friend to tell you y = f(t).

(a) Show how to use the value y to estimate the integral as before. Hint: Use importance sampling.

(b) Compute the variance of your estimate.

(c) When, qualitatively, would you expect the variance to be larger than that of the preceding problem's estimator? When would you expect it to be lower?

(d) Write a short program to confirm that the expected value and variance really are what you predicted.