Homework 3: Expectations

due Tuesday, Feb. 5th

Your homework should be typed in Word or something similar, with
appropriate spreadsheet segments copied and pasted in.  

1. a) If the occurence of A makes B more likely, can you show that
the occurrence of B makes A more likely?  Prove it formally
using math notation.

b) Provide an applied example.  Cards, dice, etc. do not count as
applied.

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2. 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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3. Suppose you have $15000 in your savings account, and you're considering how
to invest some of it.  Which of the following two investments would you prefer?
Explain why.  Different people may have different final conclusions.

Investment A: costs $10000 now.  Two years from now, it might pay you $12000
(for a net increase of $2000) with probability 0.7, or it might pay you
$9000 (a net loss of $1000) with probability 0.3.

Investment B: costs $10000 now.  Two years from now, it might pay you $15000
(for a net increase of $5000) with probability 0.7, or it might pay you
$4000 (a net loss of $6000) with probability 0.3.


Also, what if all dollar values (other than your savings account) 
were reduced by a factor of 10000?
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In past years, I assigned the following problems, but this year
you are not required to do them.  You might want to look at them,
though.

A hint: Venn diagrams can be very helpful on some of these problems.
Problem numbers are from the 8th edition.
Problem 1.8
Problem 1.18
Problem 1.29
Problem 1.48

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

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.)