Due dates of assignments and projects; exam dates:
http://www.math.dartmouth.edu/~doyle/docs/60.2010/dates/dates.html
http://www.math.dartmouth.edu/~m46s09/res.html
The Computer Help Desk in Baker-Berry can help you with issues you have getting Matlab going on your machine. Bring your laptop to class so we can play explore Matlab together. To get a head start, check out Alex Barnett's `bare essentials of Matlab':
http://www.math.dartmouth.edu/~m46s09/intro46.m
Look here for some simple probability demonstrations:
http://www.math.dartmouth.edu/~doyle/docs/60.2010/matlab/ch1/prob.m
http://www.math.dartmouth.edu/~doyle/docs/60.2010/pretestans.pdf
Can you see why these are the correct answers? Note that the approved way to answer question 6 is described in the reading (see below). Don't worry about 11-13.
Carefully read section 1.1 of G&S (Grinstead and Snell). Look at and think about problems 1-5, 8-10, 14-16. We'll be doing all of these problems, and some subset will due as early as Monday.
http://www.math.dartmouth.edu/~doyle/docs/60.2010/prog/prog.html
Look these programs over, run them, and figure out how they work. That is, assuming they actually do work. I hope they do, but there could always be bugs.
Assignment 1:
Read section 1.2 of G&S. If you are puzzled by their approach to sample spaces, random variables, and distributions, take a look at this attempt to straighten it out: http://www.math.dartmouth.edu/~doyle/docs/rv/rv.pdf
Take a preliminary look at the problems that will be due Friday (see below).
Here's a model for solutions to the problems on Assignment 1:
http://www.math.dartmouth.edu/~doyle/docs/60.2010/hw/m60hw1.pdf
Please look this over carefully. If you have questions, send me email. We'll be having our first short exam Friday 16 April. It will include problems like those on this assignment, together with other material from G&S chapters 1-3.
Read Chapter 2 of G&S. In reading 2.1, bear in mind the problems that have been assigned for Friday. For section 2.2, the problems I have in mind (not assigned yet, though) are 5, 6, 7 and 19, 8, 12 and 20, 15, 16 and 21. Also, play around with and ponder the following Matlab script:
n=10000
r=sqrt(rand(1,n));
theta=2*pi*rand(1,n);
x=r.*sin(theta);
y=r.*cos(theta);
scatter(x,y)
Roger Flynn, `Tactics and Strategy in Squash':
The International Rules of Squash Racquets only allow us to score a point when we are Hand-in (serving) at which time our opponents (Hand-out) are not permitted to score. If that is the case then ...when Hand-in, our sole objective should be to attack our opponents with as much gusto as we are able and to take whatever risks are necessary in order to create the most difficult shots for our opponents to return (ie. hit the ball as far away from our opponent as possible). When we are Hand-out however, only our opponent is permitted to score a point and ...our obligation is to stop him/her from scoring. This is best achieved by changing our attack to the back of the court so that our risk of error is minimal and we also push our opponent into the most difficult part of the court from which to play a winner.
Entropy:
http://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution
Cantor's diagonal argument:
http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
Non-measurable set:
http://en.wikipedia.org/wiki/Non-measurable_set
Solovay's model:
http://en.wikipedia.org/wiki/Solovay%27s_model
The axiom of choice:
http://en.wikipedia.org/wiki/Axiom_of_choice
The Banach-Tarski paradox:
http://en.wikipedia.org/wiki/Banach_Tarski_paradox
Assignment 2:
G&S section 1.1: 14,15,16
G&S section 1.2: 12,13,16,17,18,19,28
G&S section 2.1: 5,6,7,9
For problems with a Matlab component, please include printouts of programs, along with printouts of program runs when appropriate (e.g. when the result is a graph; you don't need a printout if the result is just a few numbers).
Read Chapter 3 of G&S. A lot will be review from the `Chapter 3' handout for HW1. The main new element is the binomial distribution.
For Exercise 2.2.5, you are meant to take for granted the memoryless property of the exponential distribution.
Office hours/help session 9:00-9:50 in 105 Kemeny.
Assignment 3:
I'm going to start collecting assignments here:
http://www.math.dartmouth.edu/~doyle/docs/60.2010/dates/dates.html
In problem 3.2.34(b), note that the program Grinstead and Snell are asking for is to compute the probability using the formula of part (a), rather than estimating it by simulation. You may get warnings from matlab about possibly inaccurate results when n is 100 or bigger: Don't fret about this.
Of course, I think you should do a simulation as well. This can be done in various ways. One approach is illustrated in this snippet, which generates 20 random numbers between 1 and 10 and reports how many distinct numbers occur:
length(unique(ceil(rand(1,20)*10)))Here's another snippet to compute the k in problem 3.1.23. Note the use of `break' to jump out of the loop:
for k=1:n-1
s=sum(1./[k:n-1]);
if s<=1
break
end
end
Review session for Exam 1 during the X-hour, 9:00-9:50 in 105 Kemeny.
Good matlab functions to know about: randperm, nnz, unique.
Test:
http://www.math.dartmouth.edu/~doyle/docs/60.2010/tests/test1.pdf
Answer key:
http://www.math.dartmouth.edu/~doyle/docs/60.2010/tests/test1ans.pdf
Go over the answers to the test (link above).
Begin reading G&S Chapter 4 in preparation for HW4 (link below).
Help session for HW 4 9:00-11:00 in Kemeny 105/120.
See here:
http://www.math.dartmouth.edu/~doyle/docs/60.2010/dates/dates.html
Things to ponder:
1. What happens in the Polya urn model if you put in *two* new balls of the chosen color, instead of just one? What about if you put in a ball of the opposite color?
2. In the Monty Hall problem, why can't you win more than 2/3 of the time, even if Monty has a bias about which door he will open when he has a choice? If there are n doors, and he opens one door, show that under some rule for which door he opens, you can win a fraction (n-1)/n of the time. I'm sure this is the bast you can hope for, but how to prove it?
Here's the expository paper on frustration solitaire:
http://www.math.dartmouth.edu/~doyle/docs/rank/rank.pdf
This is an excellent example of inclusion-exclusion in a case where the probability of the intersection of k events doesn't depend only k.
The variant of Bernoulli's principle that makes a spinning baseball curve is called the `Magnus effect':
http://en.wikipedia.org/wiki/Magnus_effect
Check out this maritime application of the Magnus effect:
http://en.wikipedia.org/wiki/Rotor_Ship
HW5 help session 8:30-11:00 in 120 Kemeny--note the room!
Regarding the `bus paradox':
You don't know a thing
About suffering and pain
If you haven't tried to catch
A taxi in the rain
And you can wait for a bus
Half an hour or more
'Cause they like to travel
In gangs of four
- P.D.Q. Bach, `Classical Rap'
http://tinyurl.com/classicalrap
Matlab has trouble with the Poisson distribution if you just plug in the formula. Instead you can use the function `poisspdf'. Tail probabilities are given by the (regularized) incomplete gamma function: http://tinyurl.com/poissontail
% Compute Poisson tail probabilities using the incomplete gamma function. format long lambda=2000 n=lambda k=[0:n]; pnaive=sum(exp(-lambda)*lambda.^k./factorial(k)) pmatlab=sum(poisspdf(k,lambda)) pgamma=gammainc(lambda,1+n,'upper')
Here's the output:
lambda =
2000
n =
2000
pnaive =
NaN
pmatlab =
0.505946700451550
pgamma =
0.505946700451784
Help session for HW6, 12:00-1:50 in 120 Kemeny.
Links to explore:
http://tinyurl.com/cauchyschwarz
http://en.wikipedia.org/wiki/Jensen's_inequality
http://en.wikipedia.org/wiki/Algebraic_number
http://mathworld.wolfram.com/Pentagram.html
http://mathworld.wolfram.com/GoldenRatio.html