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4. A random variable Z has a inverse Gaussian dist

4. A random variable Z has a inverse Gaussian distribution if it has density f(z) ? z-3/2 exp -?1z ?2 z + 2 ?1?2 + log 2?2 , z> 0 where ?1 > 0 and ?2 > 0 are parameters. It can be shown that E(Z) = 5?2 ?1 and E 1 Z = 5?1 ?2 + 1 2?2 . (a) Let ?1 = 1.5 and ?2 = 2. Draw a sample of size 1,000 using the independence-MetropolisHastings method. Use a Gamma distribution as the proposal density. To assess the accuracy, compare the mean of Z and 1/Z from the sample to the theoretical means Try different Gamma distributions to see if you can get an accurate sample. (b) Draw a sample of size 1,000 using the random-walk-Metropolis Hastings method. Since z > 0 we cannot just use a Normal density. One strategy is this. Let W = log Z. Find the density of W. Use the random-walk-MetropolisHastings method to get a sample W1,,WN and let Zi = eWi . Assess the accuracy of the simulation as in part (a)

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