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 a 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'

Find E(Z^4), where Z ~ N(0,1).  Hint:  Either use (7.28), noting that
the integrate() function accepts the arguments -Inf, Inf for negative
and positive infinity; or get the exact answer via the material in Sec.
7.33.

print(                        )  # 3 (maybe with some roundoff error)

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

Find the median (value v for which P(X < v) = 0.5) for a random variable
X having an exponential distribution with parameter lambda.

expMdn <- function(lambda) 
{

}

# test case
print(expMdn(2.5)))  # about 0.28

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

Say f_X(t) = 4 t^3) for 0 < t < 1, 0 elsewhere.  Write a function to
generate n random numbers from this distribution.

r43 <- function(n) 
{

}

# test case:
print(mean(r43(10000)))  # about 0.8

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

Consider the setting of Sec. 7.21.  Use simulation to find 
E(N | N >= 25) (do not use a correction for continuity).

print(                  )  # about 29.4 or 29.5

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

The code below (7.49) evaluates that equation via pgamma().  Show
alternate code, using one or more of dpois(), ppois() and qpois(), to
compute that same probability.

print(                   )