Question: We would like to calculate the integral 0 log(1 + x) exp(x) dx. a. Simulate 10,000 exponential random variables with parameter 1 and use these

We would like to calculate the integral ∫∞0 log(1 + x) exp(−x) dx.
a. Simulate 10,000 exponential random variables with parameter 1 and use these to estimate the integral. Also, find the simulation standard error of your estimator.
b. Simulate 10,000 gamma random variables with parameters 1.5 and 1 and use these to estimate the integral (importance sampling). Find the simulation standard error of the estimator. (In case you do not have the gamma function available, Γ(1.5) = √π/2.)
c. Which of the two methods appears to be more efficient? Can you explain why?

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