HW9 SOLUTIONS     10/19/16
===========================

(1) If  X_i are independent random variables with 

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

we want to calculate or estimate via Simulation the probability

P(X_1 + X_2 + X_3 > 4)

(a) The exact answer is a convolution-type integral (over the whole line)

\int  { 1 - pt(4-t,8) } g(t) dt

where the convolution of the Unif and Expon densities give

g(t) = 0.5*exp(-t)*(exp(min(1,t))-exp(-1)) ,   all t > -1
     = 0     for t < -1

> integrate(function(x) 0.5*exp(-x)*(exp(pmin(1,x))-exp(-1))*
           (1-pt(4-x,8)), -1,Inf)

0.04233057 with absolute error < 8e-05


(b) Straightforward simulation:

> Xmat = cbind(rexp(1e6), runif(1e6,-1,1), rt(1e6,8))
  Xvec = c(Xmat %*% rep(1,3))

> mean(Xvec > 4) + (qnorm(0.995)/1e3)*c(-1,1)*sd(Xvec > 4)
[1]  0.04213349 0.04317451   ### This is a 99% Confidence Interval
                            ### with point estimate 0.042654

(c) Xmat2 = -cbind(log(1-exp(-Xmat[,1])), Xmat[,2:3])
    Xvec2 = c(Xmat2 %*% rep(1,3))

> 0.5*(mean(Xvec>4) + mean(Xvec2>4))
[1] 0.042587

### Estimated standard error:

> cor(Xvec>4, Xvec2>4)
[1] -0.04445675           ### not very negative, so not much saving
                ### as compared with just averaging with another 1e6 deviates
> sqrt(0.5*(1-0.04445675))  
[1] 0.6912103   ## factor by which the CI based on averaged data will be smaller

## New CI:
0.042587 + (qnorm(.995)/1.e3)*c(-1,1)*sd(Xvec>4)*0.69121
[1] 0.04222722 0.04294678


(d) Control-variates method:

> summary(lm(I(Xvec>4) ~ I(Xvec-1)))$coef
              Estimate   Std. Error  t value Pr(>|t|)
(Intercept) 0.04269797 0.0001722210 247.9254        0
I(Xvec - 1) 0.06461778 0.0001052741 613.8051        0

### So here the CI is

> 0.04269797 + qnorm(.995)*c(-1,1)*0.0001722210
[1] 0.04225436 0.04314158

### NOTE that the true answer is actually right at the lower end of 
###     all of these confidence intervals !!

### Repeating the same simulation again, with independent numbers, gives:

(b)    0.04168811 0.04272389

(c)    0.04188546 0.04260154

(d)    0.04180288 0.04268732