Suppose the random variable has an exponential distribution, fX (x) = exp (€“ x) u (x). We wish to estimate the probability of the event A = {X >xo} via simulation. We will compare the standard Monte Carlo estimate,

Where the random variables Xi are chosen according to the exponential distribution specifed by PDF fX (x), with an importance sampling estimate,

Where the random variables Yi are chosen from a suitable distribution specified by its PDF, fY (y). Note that both estimators are unbiased, so we will compare these estimators by examining their variances.
(a) Find the variance of the Monte Carlo estimate.
(b) Find the variance of the IS estimator assuming that the random variables Yi are chosen from a scaled exponential distribution, fY (y) = aexp (€“ ay) u (y).
(c) Assuming that xo = 20, find the value of that minimizes the variance of the IS estimator using the scale exponential distribution.
(d) How much faster do you expect the IS simulation to run as compared to the MC simulation?

71 0 Pa.is = 1 u(,) k- 1