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Statistical Computing and Simulation
Spring 2010
Assignment 6, Due June 18/2010
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).
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.
(a)    sin 2 (1 / x) dx
(b) The volume of x 2  y 2 / 2  z 2 / 3  1 in the unit cubic (0,1)  (0,1)  (0,1).
Let X i , i  1,,5 be independent exponential random variables each with mean
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.
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.
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.
Visit the webpage of Department of Statistics, Ministry of Interior of the Taiwan
Government ( 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.)