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 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 

Consider the line with the call to mean() on p.31.  Fill in the blanks:
The R ____________________ will first ______________ the quantity q.

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

Consider the board game example, Section 2.11 of our book. Find the
probability that after one turn (including bonus, if any), the player is
at square 1.  Your answer must be R code.  Fill in the blank.

print(              )

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

Again consider the board game example. Find the probability that B is
less than or equal to 4. (Be careful; B = 0 does count as B <= 4.)  Fill
in the blank.

print(                   )

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

Here you will modify the Broken Rod example, Sec. 2.14.9.  Now the
number of pieces will be random, taking on the values 2, 3 and 4 with
probabilities 0.3, 0.3 and 0.4.  Add code appropriately to the
following:

minpiece <- function(k) {
   breakpts <- sort(runif(k-1))
   lengths <- diff(c(0,breakpts,1))
   min(lengths)
}

bkrod <- function(nreps,k,q) {
   minpieces <- 
      replicate(nreps,
         {
         minpiece(k)
         })
   mean(minpieces < q)
}

set.seed(99999)
print(bkrod(5000,5,0.02))  # may take a few seconds
# output about 0.135

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

Consider the Preferential Attachment Model, Sec. 2.13. Each time a new
vertex is added, the numbers of rows and columns in the adjacency matrix
increases by 1.  Here you will write a function to update the matrix,
based on the old one, 'oldAdj'.  The old vertex to which the new vertex
attaches is specified in the argument 'attachV'.  The return value of the
function is the new matrix.  Complete the code:

adjUpdate <- function(oldAdj,attachV)
{
   oldSize <- nrow(oldAdj)
   newSize <- oldSize + 1
   newAdj <- matrix(rep(0,newSize^2),nrow=newSize)
}

m <- rbind(c(0,1,1),c(1,0,0),c(1,0,0))
print(adjUpdate(m,2))
# output should be
# 0 1 1 0
# 1 0 0 1
# 1 0 0 0
# 0 1 0 0