3. Importance sampling. {25 points] Similar to the example we had in lecture, using importance sampling to evaluate tail probability of Gaussian random variable. Assume X is NLu, I}. We want the right tail probability or = IP{X 3-: z}. For z IE)- yu, or is verjr small, and estimating this small probability accurately is not easy. Now to improve accuracy, we samples from another Gaussian random variable mean in = z and variance equal to 1. {a} {Ill points] Derive the important ratio. {b} {5 points] Write down the importance sampling algorithm. {c} {Ill points] Now assume an = 1 z 10. Using N 1m Monte Carlo trials. Evaluate the tail probability using direct Monte Carlo I1, and using importance sampling using I2. Now repeat this Ell times to compare the variance of I1 and I2