DESCRIPTION

Students:  Please keep in mind the OMSI rules.  Save your files often,
make sure OMSI fills your entire screen at all times, etc.  Remember
that clicking CopyQtoA will copy the entire question box to the answer
box.

In questions involving code which will PARTIALLY be given to you in the
question specs, you may need add new lines.  There may not be
information given as to where the lines should be inserted.

If a question includes test code, make sure to include it in your
submission.

Do not answer any question with simulation code unless this is specified. 

MAKE SURE TO RUN THE CODE IN PROBLEMS INVOLVING CODE!  Hit the OMSI
Submit and Run button.

QUESTION  -ext .R -run 'Rscript omsi_answer1.R'

(R answer, 20 points.)  Here and in the remaining questions, consider
the following variant of Hwk III.C.  Buses arrive at random times, with
interarrival times being i.i.d.  Consider a nonrandom time t, e.g. t =
5:00 pm, and let W be the amount of time that passes from t until the
next bus arrives (i.e. the bus arrives at time t+W).

Let X denote an interarrival time.  There is a limit theorem (not the
CLT) that says that, for large t, EW is approximately E(X^2)/(2EX).

Suppose X has a U(0,1) distribution.  Find the approximate value of EW.

QUESTION -ext .R -run 'Rscript omsi_answer2.R'

(R code answer, 20 points.)  Same as Question 1, but with X having a
Poisson distribution with lambda = 2.2.   (Hint:  Make use of a mailing
tube for variance.)

QUESTION -ext .R -run 'Rscript omsi_answer3.R'

(R code answer, 20 points.)  Again in the context of Question 1, let L
denote the length of time from the last arrival before t and the first
arrival after t.  It can also be shown that for large t,

f_L(s) approx s f_X(s) / EX

(no 2 this time).  Suppose X has the density in the example in Section
7.4.  Find the approximate value of P(L > 3.2).

QUESTION -ext .R -run 'Rscript omsi_answer4.R'

(R code answer, 40 points.)  Here you will check the formula
given in Question 1, via simulation.

sim <- function(t,nreps) 
{

  sumw <- 0.0
  for (rep in 1:nreps) {
  }
  sumw / nreps
}

set.seed(9999)
print(sim(50.0,5000))