CSI 779, Spring 1995, J. E. Gentle

Assignment I

In the S code below, I have used variables for quantities that are
fixed in the problem statement.  This is a good practice in general.
In fact, it would probably be a good idea to write the S code as
functions, with these variables as dummy arguments.

In all programs involving a random number generator, it is best to set
the seed explicitly.  This allows you to check your work, and it also
provides information you may need if you want to combine simulations.

For routine exploratory work, I usually include the statement 
    set.seed(1)
If you write your code as a function, you would not put this statement
in the function, of course.  It, together with the other
initialization statements, would go before the invocation of the
function. 

1. A simple S code to estimate this integral is
     uis <- runif(mm,aa,bb)
     inthat <- (bb-aa)*sum(uis*exp(uis)*sin(uis))/mm
   where mm is the Monte Carlo sample size and aa and bb are the
   limits of integration, 1 and 3 in this case.  (And, No! you cannot
   write it in one statement:
**INCORRECT
     inthat <- (bb-aa)*
        sum(runif(mm,aa,bb)*exp(runif(mm,aa,bb))*sin(runif(mm,aa,bb)))/mm
**INCORRECT
   because of the little thing called side effects.  In random number
   generation, "side effects" cause us to get new random numbers,
   which of course, is exactly what we want.

   The estimate should be about 22.7

2. As pointed out, the weight function is logistic: exp(-x)/(1 + exp(-x))^2
   Integrating x over the full range is equivalent to estimating the
   mean of a logistic: 0.

   A Monte Carlo estimate is just the sample mean.  In S-Plus the
   statement is 
     meanhat <- mean(rlogis(mm))
   where mm is the Monte Carlo sample size.

3. Some S-Plus code is
     xx <- matrix(scan("law.dat"),byrow=T,ncol=2)

     nn <- dim(xx)[1]
     rs <- rep(0, mm)
     for (ii in 1:mm) {
          index <- sample(1:nn, replace = T)
          rs[ii] <- cor(xx[index,1], xx[index,2])
     }
     stdstar <- sqrt(var(rs))
   Where mm is the number of bootstrap replications.

   The solution comes out to be approximately 0.13.