Let (X1, X2) have the bivariate normal distribution with both means equal to 0, both variances equal to 1, and the correlation equal to 0.5. We wish to estimate θ = Pr(X1 ≤ 2, X2 ≤ 1) using simulation.
a. Simulate a sample of 10,000 bivariate normal vectors with the above distribution. Use the proportion of vectors satisfying the two inequalities X1 ≤ 2 and X2 ≤ 1 as the estimator Z of θ. Also compute the simulation standard error of Z.
b. Use the method described in Example 12.4.3 with 10,000 simulations to produce an alternative estimator Z' of θ. Compute the simulation standard error of Z' and compare Z' to the estimate in part (a).