Homework Problem 7, Due Monday October 3, 2016.
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Assigned 9/26/2016, due 10/3 [But note that there is no class 
that day, so you may turn it in along with the next HW on Wed. 10/5.]


Give R code for a function that generates


(1) 10,000 random points from the region in the plane containing (0,0) 
and bounded by the lines

     y = 3 + 2x,     y = 2 -7x

and  y= 2,    y=-5.



(2) 10,000 independent random variable variates on (0,5) with 
univariate density 

     g(x) = C * (2 + 0.2*sin(3x^2))   ,   for  x in (0,5)
          = 0                             elsewhere

for a constant C which makes this a proper density.

HINT: use an accept-reject method.


(3) 1,000 independent random pairs (X[i],Y[i]) on (0,Inf) x (0,Inf)
with joint density

     f(x,y) = K y/(1+x^2+y^2)^4      for x, y > 0

where K is a constant which makes this a proper density.

For each of the functions you write, check that they perform 
as required. The checks could involve moments of coordinates, relative 
frequencies with which points are drawn from various subsets of the 
indicated value spaces for the random vectors generated, a measurement 
of properties of the set into which your generated points ever fall, or 
a combination of methods.

The checks could also involve mathematical proofs related to the density
of a function of random variables in terms of other random variables.