Suppose that we wish to estimate the integral

In parts (a) and (b) below, use simulation sizes of 1000.

a. Estimate the integral by importance sampling using random variables having a truncated normal distribution. That is, the importance function is

1/√2π[1− Φ(1)] e^−0.5x2 , for x >1.

b. Estimate the integral by importance sampling using random variables with the p.d.f. x exp(0.5[1− x2]), for x >1. Prove that such random variables can be obtained as follows: Start with a random variable that has the exponential distribution with parameter 0.5, add 1, then take the square root.

c. Compute and compare simulation standard errors for the two estimators in parts (a) and (b). Can you explain why one is so much smaller than the other?

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