DESCRIPTION

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DON'T FORGET TO SAVE AND SUBMIT YOUR WORK, even work in progress.
The network may get busy at the end of the exam period, making  
it difficult to submit then.  The server will shut down at the 
appointed time, and NO further work will be accepted.         

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In questions involving code which will PARTIALLY be given to you in the
question specs, you need to add new lines.  There may not be information
given as to where the lines should be inserted.  Do not modify the lines
already there unless instructed to do so.  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. 

If you finish early, please leave the exam room.  If you stay, close
your laptop and sit quietly.

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SHOW YOUR WORK in problems involving numerical answers.  Either do    
so via comment lines or multiline code with variable names, or in a  
detailed numerical expression if it clearly shows your thinking. You
need not show ALL details, but enough to demonstrate your insight. 
Single-number answers will likely get 0 credit.                   

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

Fill in the blank below so that the matrix

1 4 7
2 5 8
3 6 9

is printed.

print(matrix(                 ))

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

Consider the die game, p.103.  Suppose on each turn we roll two dice
instead of one.  We still have 10 states, 1 through 10, but the
transition matrix P changes.  If for instance we are now in state 6 and
roll (3,2), we win a dollar and our next state is 1.  Fill in the blank
so that it prints out the first four elements of row 1 of P, which will
consist of the probabilities p_{1,j}, j = 1,2,3,4.

print(c(            ))

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

Suppose Z is an indicator random variable with P(Z = 1) = w.  Write a
function to calculate the skewness of Z,

E[ {(Z - EZ)/ sqrt(Var(Z))}^3 ]

skInd <- function(w)
{


}

print(skInd(0.25))  # about 1.15

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

Consider the setting of the binomial distribution family, with one
change:  The probability of success varies from trial to trial.  Let p_i
denote the probability of success on trial i, and as before let X denote
the number of successes out of n trials.  Write a function to calculate
Var(X).  For full credit, do not use any loops.

# pvec is the vector (p_1,...,p_n)
varx <- function(pvec)  # n inferred, no need for a parameter n
{

}

print(varx(c(0.25,0.45,0.30))