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Chapter 10: Q. 10.5 (page 432)
Let X and Y be independent exponential random variables with mean 1.
(a) Explain how we could use simulation to estimate E[eXY].
(b) Show how to improve the estimation approach in part (a) by using a control variate.
(a) Estimator
(b) Redefine estimator and it is defined below.
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Present a method for simulating a random variable having distribution function
Use the inverse transformation method to present an approach for generating a random variable from the Weibull distribution
Let X be a random variable on (0, 1) whose density is f(x). Show that we can estimate # 1 0 g(x) dx by simulating X and then taking g(X)/f(X) as our estimate. This method, called importance sampling, tries to choose f similar in shape to g, so that g(X)/f(X) has a small variance.
In Example 2c we simulated the absolute value of a unit normal by using the rejection procedure on exponential random variables with rate 1. This raises the question of whether we could obtain a more efficient algorithm by using a different exponential density鈥攖hat is, we could use the density g(x) = 位e鈭捨粁. Show that the mean number of iterations needed in the rejection scheme is minimized when 位 = 1.
Give a technique for simulating a random variable having the probability density function
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