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.  DON'T FORGET TO SAVE AND SUBMIT YOUR WORK, even work in progress.

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.  Do not
modify the lines already there.

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

Do not answer question via simulation code unless this is specified. 

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

QUESTION 

We discussed a term from probability theory that is heavily used but is
misleading.  What term was it?

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

In the context of the example in Sec. 3.6.2, find the variance of X+2Y,
expressing your answer in R:

print(             )

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

Consider the ALOHA example, using two nodes, both of which start out
active, with p = 0.4, q = 0.8.  Find the expected value of the number of
attempted transmissions (successful or not) during the first epoch,
answering using R:

print(               )

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

Consider the board game example, Section 2.11 of our book.  

Instead of assuming we start at square 0, as before, we now assume we
start on any of the 8 squares, with probability 1/8 each.  Also, we now
want to find P(X = 7) and EB, where X is the position after one turn
(including bonus, if any), and B is the bonus (taking on the values
0,1,...,6).  Add lines to the simulation code appropriately.

boardsim <- function(nreps) {
   for (i in 1:nreps) {
      position <- sample(1:6,1)
      bonus <- 0
      if (position == 3) {
         bonus <- sample(1:6,1)
         position <- (position + bonus) %% 8
   }
   c(prbsq7,eb)
}

set.seed(99999)
print(boardsim(1000))

QUESTION -ext .R -run 'Rscript omsi_answer5.R'

Write a function that returns E[g(X)] for a given discrete random
variable X having finite support.  The latter is specified via its
support and probabilities on that support, given in the top and bottom
rows of a matrix.  Complete the code below:

egx <- function(xInfo,g) 
{

}

print(egx(rbind(1:6,rep(1/6,6)),function(x) x^2))  # about 15.17