Homework Set 11, Due Monday October 24, 2016.
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Assigned 10/17/2016, due 10/24


IMPORTANCE SAMPLING SIMULATION PROBLEM.

This problem was adapted from the Robert and Casella (1999) book 
on Monte-Carlo, and was originally taken from a journal paper.

PROBLEM:

Estimate by a Monte Carlo experiment, with and without 
importance sampling, the probability

P(W_j > 0 for j=1,...6)

where the (dependent) 6-vector random variables W are defined 
from  a vector Z of 6 iid N(0,1) random variables, by 

W = c(  (diag(0:5) + 1) %*% Z )

In your importance sampling approach, sample from a distribution 
in R^6 (6-dimensional Euclidean vector space) supported completely 
in the positive orthant (i.e., in the set of vectors with all 
positive coordinates. 

NOTES added 10/23: I originally thought that you should expect to 
achieve a dramatic speedup with the importance sampling approach 
with a good choice of distribution to sample from.

However, I made a mistake in my choice of parameters. 
[The matrix A = (diag(0:5) + 1) should have been replaced by its inverse.] 

With the current set of parameters, the desired event probability is not 
small, and there will be no advantage in importance sampling.

You should still have come up with an importance-sampling algorithm for 
the  problem as stated. But if the problem was too hard to do under the 
requirement that the sampling distribution be supported completely 
within the positive orthant in R^6, you were allowed to relax that requirement.