Homework Problem 9, Due Monday October 17, 2016.
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Assigned 10/10/2016, due 10/17

Consider the problem of estimating/calculating the probability

P(X_1 + X_2 + X_3 > 4)

where X_i are independent random variables with 

X_1 ~ Expon(1),  X_2 ~ Unif[-1,1],  X_3 ~ t_{8} (Student's t, 8df)

Do this in four ways, and compare their accuracy and speed:

(1) Convolution or integration, using either  "integrate"  or 
a direct quadrature rule like Simpson's rule, or even a suitable 
R package if you find one.

(2) Direct Monte Carlo simulation -- you choose an appropriate number
         of triples to simulate to achieve accuracy comparable to 
         the other methods.

(3) Simulation, but making use of antithetic variables

(4) Simulation using the method of control variates, i.e. via linear 
         regression of  I[X_1+X_2+X_3 > 4] on   X_1+X_2+X_3 .