Math 419/592
Homework for Chapter 7

Problem 7.7. When it says mean 2, assume that is 2 months.  Yes, this
	problem is incredibly easy.

Problem 7.10.  It is tempting to write that N_C(t) = p * N(t),
	but that's not true.  Explain why not, in addition to doing
	the problem as the textbook states it.
	
	grad students: use Example 3.17 (page 118) to compute the squared
	coefficient of variation (SCV) for the inter-arrival times
	of the N_C(t) process.
	Express the new SCV using the SCV of the original process.

Problem 7.34--recall Example 5.16 which talks about M/G/infinity systems.

Problem 7.42 (part (b) may be a repeat of what we did in class, but please do
it anyway)

Problem 7.51 As a follow-up to this problem, compute the standard deviation
of the length of a tourists' stay.  Hint:
E[inspected] = (c^2 + 1) * E[stay]

Also:
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Q1: a) Simulate a Poisson process for 50 arrivals.  Do a graph of N(t) vs. t.
Do another graph of X_{n+1} vs. X_{n}.
    b) Repeat for a Poisson process with 500 arrivals.
    Use the results from this problem to help in interpreting the graphs
you do in the next problem.

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Q2: Consider the data files provided by the instructor (on the internet).
For each one, determine if it is or is not a renewal process.  Provide
your reasoning for each.  Each non-renewal process will have one critical
graph that convinces you; include that graph in your homework.  For any that
do seem like renewal processes, include all applicable graphs.  It may help
to zoom in on the X_{n+1} vs. X_{n} in some cases, or even to do a
transform like exp(-X_{n+1}) vs. exp(-X_{n})

Extra credit: also consider the sequence of infant safe-surrenders in Michigan,
data available at
http://www.michigan.gov/documents/dhs/DHS-Safe_Delivery_Fact_Sheet__226980_7.pdf
i) Is it a renewal process (as above)?
ii) Is it a Poisson process?  Explain.
iii) In theory, should it be a Poisson process?  Explain.

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Q3: for grad students.  Consider a renewal process that is split like in
	problem 7.10 but in a deterministic manner rather than a probabilistic
	manner.  That is, only every n'th arrival is counted.  Compute the SCV
	of the new inter-arrival process as a function of n and the SCV of the
	old process.

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The following stuff is optional.

Problem 7.1 (draw pictures to offer counter-examples if they exist)

Q4: Consider an approximate Backgammon game where you
start 100 spaces from home.  At each turn, you roll two dice, take the
sum, and move that many spaces toward home.

But now, suppose we use only one die instead of two.  Let R be a random
variable that is the number of rolls until you reach home.  Using the
Central Limit Theorem for Renewal Processes, what are the approximate mean
and standard deviation of R under the new, one-die, model?
How do they compare to the two-dice model?