Homework Problem 8, STAT 705, Fall 2015.

Assigned 10/7/2015,  Due Friday 10/16/2015

Give R code for a function that generates

(1) a random point from the parallelogram in the plane bounded by 
the lines

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

and  y= 1,    y=5.


(2) a 3-dimensional random vector with density

f(x,y,z) =  C x y^3 exp(-x*y) (z/(y+z))^2 , x >0, 0 < y < 1, 0 < z < 1

for a constant C that you must find.


For each of the two 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.