Homework Problem Set 9, Due Friday October 13, 2017.
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Assigned  10/4/2017,  due 10/13:       16 points


(1) Consider the problem of estimating/calculating the probability

P(X_(3) + X_(7) + X_(10) > -1.5 )                             (*)

where X_i ,  i=1,...,25  are independent random variables with standard 
normal (N(0,1)) density and these variables are ordered to form the order-statistic
sequence  X_(1) < X_(2) < ... < X_{25) 

Calculate (*) above in three ways, and compare their accuracy and speed:

(A) Direct Monte Carlo simulation -- you choose an appropriate number of batches 
of 25 iid variates to simulate to achieve accuracy comparable to the other methods.

(B) Simulation, but making use of antithetic variables.

(C) Simulation using the method of control variates, i.e. via linear regression of  
I[X_(3)+X_(7)+X_(10) > -1.5]  either on   X_(3)+X_(7)+X_(10) or on  X_(3), X_(7), and X_(10).

		 
		 
(2) IMPORTANCE SAMPLING SIMULATION PROBLEM, 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( (inverse of (diag(0:5) + 0.5) ) %*% 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. 

Try to come up with an importance-sampling algorithm for a sampling distribution 
supported completely within the positive orthant in R^6; but if that requirement is 
too hard to meet, you may relax it.