Math 419/592
Chapter 4 homework that I used to assign, but not this year.
Not due at all.


1.  Problem 4.59: give an explicit, formal conditioning argument.
	If you haven't used the distributive property of multiplication
	and the fact that p+q=1, you haven't done it formally enough.
	
	Optional: if you're curious what the M values look like, you can
	use the formulas in 4.60 to graph some.  I recommend N=10 and
	p=1/2, 1/3, and 1/4.

2.  Using N=10 and p=1/2, calculate M_5 as follows:
	(a) find the CDF of the time-to-finish starting from 5.
	Make sure you're considering both types of finishing:
		hitting 0 or hitting N.
	(b) Find the PMF from the CDF, and plot it.  Why is it so jagged?
	(c) Find the mean and standard deviation of the distribution.
	The mean should match what you got in problem 1, of course.
	
	The next part is for grad students:
	(d) Find the mean by using the CDF and the formula from Problem 2.46.
	Watch out for off-by-one errors.  Of course, it should match what
	you got in part (c).

3. (did get assigned this year)

4.  Consider a simple racing game: you start on square 36, and square 1
	is "Home".  On each turn, you flip a coin, and move 2 squares
	toward Home if it is Heads, and 1 square if Tails.  Use
	Pr{Heads} = 1/2.  If you are on square 2 and you get Heads,
	you may go Home; you don't have to land exactly on it.
	Let Y be the number of turns until you get Home.  Calculate
	the mean and standard deviation of Y, using a DTMC of course.
	The results of this problem will be especially relevant in
	Chapter 7.

5.  Using the inventory problem from the last homework (Poisson demand with
	mean 3.5, etc.), suppose we start in state 10.  What are the mean
	and standard deviation of the time until we reach state 0?

6. (did get assigned this year)

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Things to read and contemplate:
Problem 4.61
Problem 4.77
The article by Howard, available on Blackboard.
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