Homework Problem 9, Due Monday Oct 12.
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(a) Generate 1000 independent batches of 20 i.i.d. 
random 2-vectors (X_i,Y_i), i=1,..,20 for which

X has a standard logistic distribution, 

and conditionally given X

Y/X ~ Unif(3/4, 5/4) 

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NOTE: this means that Y = XU  where 
U ~ Unif(3/4, 5/4) and X is independent of U.
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Give some distributional checks or graphical evidence 
to show persuasively that you have simulated the 
joint (X,Y) distribution correctly.

(b) The objective of this simulation is to evaluate

E( (Y_(10) - X_(10))^2 )               (*)

where the subscript _(10) refers to the 10th order statistic 
(ie the 10th smallest observation) within the batch.

One method of evaluating the expectation (*) is by a direct 
simulation: perform (and time!) a simulation of 10000 batches 
and find (*) as a direct empirical average of 10000 
quantities  (Y_(10) - X_(10))^2  that you simulate.

Explore whether you can speed up the simulation, while 
obtaining accuracy at least as good as with your first
method, by using fewer simulation replications and 
applying the "method of control variates" with 
control variate  (X_(10))^2 .

RECALL: THE METHOD OF CONTROL VARIATES IS ESSENTIALLY A 
REGRESSION METHOD. This order statistic distribution
(the 10th order statistic from a Unif[0,1] sample of size 20)
is known by a standard exercise to be Beta(10,11).

The desired expectation cannot easily be evaluated using a 
numerical integral. But since each Y is proportional to X, 
we might expect Y_(10) to be roughly proportional to X_(10), 
so that (Y_(10) - X_(10))^2  will be roughly proportional 
to (X_(10))^2.

So the method [see Sec 4.2 of Sec3NotF09.pdf] involves 
simulating pairs  

         (V,W)  =   (X_(10))^2, (Y_(10)-X_(10))^2)

for each batch of 20 pairs (X_i,Y_i), noting that we can 
calculate E(X_(10)^2) = c   quickly and to high precision 
by a numerical integral, and then fitting a regression model

          W_r = a + b*(V_r-c) + e_r ,      r=1,...,R

via least squares (where e_r) is independent of V_r and has 
mean 0, with the idea that the desired expectation (*) is 
the estimated value of a. The variance of the estimate should 
be approximately (1/R) times the estimated variance of the 
errors e_r , and use the confidence interval based on this 
variance in assessing the precision of your Method 2 
simulation.