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Statistical Computing and Simulation Spring 2010 Assignment 6, Due June 18/2010 1. Experiment with as many variance reduction techniques as you can think of to apply the problem of evaluating P(X > 2.5) for X ~ N(0,1). 2. Evaluate the following quantities by both numerical and Monte Carlo integration, and compare their errors with respect to the numbers of observations used. Also, propose at least two simulation methods to reduce the variance of Monte Carlo integration and compare their variances. 1 (a) sin 2 (1 / x) dx 0 (b) The volume of x 2 y 2 / 2 z 2 / 3 1 in the unit cubic (0,1) (0,1) (0,1). 3. Let X i , i 1,,5 be independent exponential random variables each with mean 5 1, and consider the quantity defined by P{ iX i 21.6}. Propose at i 1 least three simulation methods to estimate and compare their variances. 4. First, simulate 100 observations from a mixed distribution of N(2,1) and N(2,1), each with probability 0.5. Then, use at least 3 density estimating methods to smooth the observations. You need to specify the parameters in the smoothing methods, and compare the results. 5. Let x be 100 equally spaced points on [0,2] and let yi sin xi i with i ~ N (0,0.09) . Apply at least 3 linear smoothers and compare the differences, with respect to mean squares error (i.e., bias2 and variance) from 1,000 simulation runs. 6. Visit the webpage of Department of Statistics, Ministry of Interior of the Taiwan Government (www.moi.gov.tw/stat) and download the age-specific death records of year 2008. Use the smoothing techniques introduced in class to revise the age-specific mortality rates and compare with the values from 2008 Taiwan abridged life table. (Note: You only need to do the smoothing for the male or for the female, depending on your gender.)