Math 419/592
Homework for Chapter 2
Due Friday, January 25th at the start of class.
Remember that all homework must be typed.


Everybody:
Problem 2.16
	Follow-up to Problem 2.16: Suppose the airline wants
	Pr(no bumped passengers) >= 85%
	How many tickets should they sell?  I encourage you to use
	your favorite software to do this problem.
	The general name for this kind of problem is Overbooking or
	"Yield Management" or "Revenue Management".  More complicated versions
	might make a good project, if you're interested.
Problem 2.30

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The Pareto Problem:

Everybody: Let's practice doing little simulations in Excel (or Matlab/Scilab,
or some other package, if you wish).

A "Pareto" distribution is defined as follows: if
X ~ Pareto(parameters a,b),
then the CDF is
Pr{X<=t} = F(t) =  1-(b/t)^a    for t>=b, and zero if t<b
the mean is a*b/(a-1), sometimes.
the variance is b^2 * a/[ (a-1)^2 * (a-2) ], sometimes.  
Why is it just sometimes?  We'll discover that below.
Pareto distributions are sometimes used as the distribution of someone's
service time in a queueing system.

Part 1: derive the PDF from the CDF.  Draw graphs of the PDF using
	b=1, a=3.5 and
	b=1, a=2.5 and
	b=1, a=1.5 all on one graph, if possible.
	Also, give the mean and standard deviation for each.
	(note: this is partly a trick question.  Tell me why.)

Part 2: Figure out how to invert the CDF so we can simulate Pareto variables.
	Recall: let X = F^{-1}( U ), where U is Uniform(0,1) just like RAND()

Part 3: Simulate 1000 variables with b=1, a=3.5; make sure their mean and
	standard deviation match what you expect.  Do a histogram to
	make sure it matches what you saw in part 1.  Choose your buckets for
	the histogram with some care.

Part 4: Let Xn be IID Pareto random variables with b=1, a=2.5, for n=1..10
	Let W = X1 + X2 + ... + X10
	Simulate W about 1000 times.
	How well do the observed mean and variance of W match
	what theory would predict?
	Also, do a histogram and say whether or not you think it is
	close to a bell curve.

To "turn in" parts 3 and 4, don't print out all 1000 variables (please!).
Print out a single formula, and then give the observed mean and standard
deviation, and the histogram as a graph.

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The Newsvendor Problem:

I encourage you to use Excel, Matlab, Scilab, etc. for this problem.

Let the random variable X be the demand for newspapers at a news stand. X is distributed as follows:

x    10     11    12    13    14    15    16    17    18    19    20
f(x) .05   .05    .1    .1    .1    .2    .1    .1    .1    .05  .05

At the start of the day the newsboy buys papers for 10 cents a piece.  He sells papers for 50 cents
a piece.  If there are any papers left over at the end of the day, he may sell the papers to the 
recycling company for 5 cents a piece.  

(a) Compute E[X]
(b) Compute StdDev(X).  I suggest you use the formula Var(X)=E[X^2] - (E[X])^2.
You can compute E[X^2] in Excel by using something like
=sumproduct(F2:F12,F2:F12,G2:G12)
That is, a sumproduct with 3 vectors instead of 2.
In Scilab, you do something like 
xvec=10:20;
probvec=[.05 .05 .1 .1 .1 .2 .1 .1 .1 .05 .05];
mymean = sum(xvec .* probvec);
rawmoment2=sum(xvec .* xvec .* probvec)

(c) Suppose he buys 14 papers, and demand is 12 papers.  What is his profit?
(d) Suppose he buys 14 papers, and demand is 17 papers.  What is his profit?
(e) Suppose he buys 14 papers.  What is the expected value of his profit?
(f) Make a table: if he buys “q” papers, his expected profit is ____ for q=10..20
(g) How many papers should the newsboy buy at the start of the day to maximize expected profit?
Hint: it's probably not just plain E[X].  Compare your answer to E[X].

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Grad students: also do
Problem 2.46
Problem 2.58
	Follow-up to Problem 2.58: also write a similar formula
	involving 1-F_{Y}(x)
	Then, specialize that to the case where Y ~ Exp(rate=s).
	Have you ever seen this formula in other contexts?
(undergrads: at least contemplate the formulas in those two problems.)


Everybody: read and ponder the following problems:
Problem 2.23 
Problem 2.24 
Problem 2.28 (if you're actually doing the problem, then for part (b),
	compute Pr{X=1} to show whether it is fair or not.)