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 

(Text answer.)  State the support of M in Sec. 4.3.2.

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

(Code answer.)  Suppose X has a Poisson distribution with lambda = 10.6.
Find P(X >= 11 | X >= 2).  Use built-in R functions.

print(         )

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

(Code answer.)  In certain statistical methodologies, one computes
A + cI, where A is a square matrix and c is a scalar.  Write a loop-free
function to do this computation.  

addCI <- function(A,c) 
{

}

m <- rbind(1:3,7:9,c(1,0,2))
print(addCI(m,3))

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

(Code answer.)  Say we have a Markov chain in which the states are also
numbers, not just labels.  (Most of our examples are like this.) Write a
function that finds E(X_{n+1} | X_n = i) for a transition matrix P.

evalnext <- function(P,i) 
{

}

m <- rbind(c(0.2,0.2,0.6),c(0,0,1),c(0.5,0.5,0))
print(evalnext(m,3))  # 1.5

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

(Code answer.) The pi vector for a Markov chain is an eigenvector of P',
with eigenvalue 1, where P is the transition matrix. In fact, theory
shows that the other eigenvalues are less than 1.  This suggests yet
another way to compute the pi vector. You will implement that here.  Use
the R function eigen().  You can check the online help, but the key is
that the eigenvectors are contained in the 'vectors' component of the
output, one per column, in order of decreasing eigenvalues.  Note: If x
is an eigenvector of A, then so is -8.88x.

findpiEig <- function(P) 
{
   
}

p <- rbind(c(0.5,0.5,0),c(1/3,1/3,1/3),c(0,0.5,0.5)) 
print(findpiEig(p))