Homework Problems 5, STAT 705, Fall 2015.

Assigned 9/21/2015,  Due Friday 10/2/2015

(1) Exercise 11.1.(b), p.431  in J. Gentle book:

Find the integral of  x^2*cos(x*y)  over  (x,y) in [0,1] x [-2,2]. You should 
find the integral using Monte Carlo Integration over the 2-dimensional domain, 
for which you should provide the R code, including an error estimate (Confidence Interval).

You should also check your answer using the R function "integrate" (with error 
bound), or you may alternatively check the answer analytically by re-expressing 
the integral before using (univariate) quadrature formulas if you like.

YOUR SOLUTION SHOULD INCLUDE BOTH A MONTE CARLO INTEGRATION CONFIDENCE INTERVAL 
AND AN ANALYTICAL CHECK VIA NUMERICAL QUADRATURES (via one of the two indicated 
methods).

(2) Do exercises Sim.1 and Sim.3 on pp.3-4 of the handout

http://www.math.umd.edu/~evs/s400/RNG2.pdf

after reading both that handout and http://www.math.umd.edu/~evs/s400/RNG1.pdf

In Sim.3, find both a theoretical and computational justification for your answers.
(This means, a theoretically based computed answer and also one based on simulation.