Homework Problem Set 6, Due Wednesday September 28, 2016.
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Assigned 9/19/2016, due 9/28:


(a) Find a numerical value for the integral of the function

g(x,y) = (3x^2 + x*y + 5y^3)/(1+x^2 + y^2)^3

over the rectangle   (-2,2) x (-3,3).

Obtain the value using Monte Carlo simulation, and assess 
the sampling error by statistical considerations (e.g., in 
the form of a confidence intervals). But also get the best 
answer you can using a numerical integration function such 
as "integrate" or such as a Simpson's rule. Can you assess 
the error that way ?


(b) Show how to generate 1000 independent random variables 
from the density equal to 

f(x) = C* max( exp(-abs(2*x)), dnorm(x),  dbeta((x+2)/4,2,1) )
                            for  -2 < x < 2
   and  = 0  for abs(x) >=2.

where the constant C is determined to make this a density.

HINT: find (analytically or numerically) the subintervals on 
   which you know the exact analytical form of the density f(x).