Assume X is N(0; 1). We want the right tail

probability = PfX zg. For z 0, is very small, and estimating this small

probability accurately is not easy. Now to improve accuracy, we samples from another

Gaussian random variable mean 1 = z and variance equal to 1.

(a) (10 points) Derive the important ratio.

(b) (5 points) Write down the importance sampling algorithm.

(c) (10 points) Now assume 0 = 1, z = 10. Using N = 100 Monte Carlo trials. Evaluate

the tail probability using direct Monte Carlo bI1, and using importance sampling using

bI2. Now repeat this 500 times, to compare the variance of bI1 and bI2.