# Math 3160-003, Spring 2012

## Schedule

Week Sections in text (estimated) Quiz/Test Administrivia
Week 1: Jan. 17-20 2.1, 1.1
Week 2: Jan. 23-27 1.1, 1.3, 1.4
Week 3: Jan. 30 - Feb. 3 1.4, 1.5, 1.6 Monday: last day to drop without a "W"
or choose the P/F option
Week 4: Feb. 6-10 2.2, 2.3
Week 5: Feb. 13-17 2.4 Midterm #1
Week 6: Feb. 20-24 3.1, 3.2, 3.3
Week 7: Feb. 27 - March 2 3.4, 5.1
Week 8: March 5-9 5.1, 5.2, 5.3
Week 9: March 12-16 Spring break!
Week 10: March 19-235.4 Midterm #2
Week 11: March 26-30 5.5, 6.1 Monday: last day to drop
or choose to get a letter grade
Week 12: April 2-6 6.2, 6.3
Week 13: April 9-13 6.4, 6.5
Week 14: April 16-20 6.6 and moment-generating functions
Week 15: April 23-27 moment-generating functions and review

#### January 17

At the end of class, we were talking about why C_(n,k) = C_(n-1,k-1) + C_(n-1,k). Think about this a little before next time!

You might also want to think about why the formula for C_(n,k) is true. Let me summarize what I said in class again:

Suppose you want to choose 3 out of the 7 movies you own to watch over a weekend. You could start by pulling 3 of them out of a bag and setting them down in a row. There are P_(7,3) ways to do that. Suppose that what you ended up with was the following order:

1. E.T.
2. Ghostbusters
3. Das Boot
But how many other ways could you have gotten this particular set of movies? There are five other ways because there are five other orders you could have chosen them in: E/D/G, G/E/D, G/D/E, D/E/G, and D/G/E. This means that you've overcounted by a factor of 6 (or 3!), so the true number of ways to choose that set of movies (not the set of movies in that order) is P_(7,3)/3!. I hope this helps!

#### January 19

Here are the two main tips I talked about today for solving combinatorial problems.
1. Often, it's easier to count the number of ways to get the kind of thing you don't want than the kind of thing you do. Instead of counting all the things you want, you can find the total number of possibilities and subtract out the ones you don't. For instance, if I want to know all the ways to roll two different-colored standard six-sided dice and get a sum greater than 3, I can count like this: There are 36 possible rolls. Of these, 3 give sums less than or equal to 3: 1-1, 1-2, and 2-1. Therefore, there are 36-3=33 ways to roll these dice and get a sum bigger than 3.
2. If you've calculated the number of poker hands (for instance) and you want to check your calculation, think about whether you're overcounting. Are there two different ways to get the same hand using the method you used? If so, then you need to think of a new way to get the answer.

#### January 24

I recommend that you become familiar with unions, intersections, and complements of sets if you aren't already. Events are just sets of possible outcomes, and we might want to find the probability of A OR B happening: in this case, we take the union. If we want to find the probability of of A AND B happening, we'll take the intersection of A and B. If we want to find the probability of A NOT happening, we calculate the probability of A happening and then subtract it from 1.

#### January 26

Try to come up with new examples of (1) events that are not disjoint but that are independent and (2) events that are not disjoint and are not independent!

I've been asked if I could recommend another book as a supplement. Some of my colleagues suggest A First Course in Probability by Sheldon Ross (it used to be used for this course at UConn). I would personally suggest Introduction to Probability by Charles M. Grinstead and J. Laurie Snell. You can get it free online here. If you use either, please let me know how helpful it is to you so I can make better recommendations for future classes (and if you find another useful source, please let me know about it, too).

#### January 31

If you're interested in reading more about Benford's Law, check out MathWorld's page. It includes analyses of some actual data sets, and you can see how well Benford's Law models each of them.

#### February 2

For a geometric distribution in action, you might want to watch this clip from "Rosencrantz and Guildenstern Are Dead."

We talked in class today about the limitations of only knowing a random variable's expected value and not its variance. What kind of information do you get if you know a random variable's variance but not its expected value, and what would you not know?

#### February 7

It would probably be a good idea for you to start making a list of the different distributions we know. For each one, you might want to include
• the general situation it models,
• the possible outcomes (for a geometric distribution, that's {1,2,3,4,...}; for binomial(n,p), that's {0,1,2,...,n}),
• the formula (if there is one),
• formulas for the expected value and variance, and
• a fully worked out example.
So far, we have the uniform distribution, the geometric distribution, the binomial distribution, and the multinomial distribution.

#### February 9

Now we can add the Poisson distribution to the list. If you'd like to see the original Prussian cavalry horse-kick data, here it is!

Remember the rule of thumb I gave you in class today: At this point, if there is nothing that clearly triggers the event happening (like flipping a coin, rolling a die, drawing a poker hand, or gathering people in a room), the Poisson distribution is probably the best model we have.

#### February 14

Here's a little bonus on test day: some people have asked what can be done in Maple or MATLAB with respect to probability. If you'd like to download some simulations, you can do it here. I can provide Maple support.

Just play around with them. They're meant to go with a different textbook, but they're still fun and useful. Let me know if there's something you find in them that you'd like to talk about!

#### February 16

Today we talked about approximating a binomial distribution with a Poisson distribution and the hypergeometric distribution (your book calls it "Urn Problems" and puts it in Section 2.4). If you want to go over a capture-recapture problem like the one I did in class, take a look at Example 2.42.

#### February 21

My favorite applet for the Monty Hall problem lives here. You might want to think about variations on this problem. For instance, what if there are 4 doors and he opens one of them, offers you the chance to switch, and then opens another and makes the same offer?

#### February 23

I promised you I'd post quite a few things today...

Craps: Here are Frank Sinatra and Dean Martin and Marlon Brando (believe it or not) singing a couple songs from "Guys and Dolls."

Bayesian probabilities: There's a very nice discussion of conditional probabilities in this New York Times article. Take a look at the notes at the end, too!

#### February 28

The problem that I misstated so badly in class is this:

You are a general being advised by two of your lieutenants. One of them tells you that in general, the probability of an attack on your left flank is 1/5, the probability of an attack on your right flank is 3/10, and the probability of a frontal attack is 1/2. The other one has been keeping track of the enemy's communication channels and tells you that if they're going to attack on the left, the probability of this chatter is 1/5, if they're going to attack on the right, the probability of this chatter is 7/10, and if they're going to attack frontally, the probability of this chatter is 1/10. Given that you are, in fact, hearing this chatter, can you work up a probability profile for the attack?

#### March 1

We've moved into a new situation as of today. Until now, we've been talking about experiments with either a finite set of outcomes with positive probability (like flipping a coin 10 times) or experiments with a discrete list of possible outcomes (like flipping a coin until it comes up heads). Now we're talking about experiments with outcomes that range over intervals in the real numbers (like measuring your heights: I think that if we measure your heights in inches, they will all fall in the interval [48,78]). We'll see that the same techniques work to describe probabilities from both kinds of experiments, but now we'll integrate instead of taking finite sums.

#### March 6

Today we calculated the expected value and variance for three continuous distributions. It would be a good exercise for you to go back and calculate E(X^2) for these distributions: I just left them as an exercise in class today. You may also want to explicitly write down distributions that have the following properties:
• an expected value and a variance,
• an expected value but no variance, and
• neither an expected value nor a variance.

#### March 8

I calculated the distribution function for the uniform distribution and just told you what it was for the exponential distribution and power law. Verify that the answers I gave you are correct!

I also left you with a puzzle at the end of class: find a discrete distribution with no median. Enjoy your break!

#### March 22

You might be interested in an article that one of your classmates sent me a link to about coincidences.

#### March 27

At the end of class, we were working through Example 5.23: finding the joint distribution function for two random variables uniformly distributed on (0,1). We worked out the cases for x<0 or y<0 and 0<=x,y<=1. Can you work out the rest without peeking at the book?

#### March 29

The theorem I mentioned in class today but didn't state formally was Theorem 5.5: It just says that if you can write f(x,y) as a product of functions g(x)h(y) and g(x) and h(y) are related to the marginal densities in a certain way, X and Y must be independent. It's a nice result, but in general, it's not easier to use it than it is to see if the product of the marginals is equal to the joint density function.

#### April 3

Please keep in mind the errata I mentioned for the gamma distribution in Example 6.6: they're described on the errata page.

#### April 5

Think about the difference between the formulas for E(X+Y), var(X+Y), and cov(X,Y). When does each apply?

#### April 10

Try applying Chebyshev's inequality to the data about your homework sets in this class. Try estimating the probability that a student's score differs from the class average by 1, 2, or 3 points. When does the estimate become useless?

#### April 12

If you'd like to run a Galton board simulation for yourself, you can do it by clicking here. If you'd like to see one in action, go to the Boston Museum of Science and visit the Mathematica exhibit. It's the CLT in action!

#### April 17

Remember when to use the histogram correction!

#### April 19

The Weldon dice data and a followup experiment can be found here, and the data from the Franklin Institute on penny-flipping can be found here. Enjoy!

#### April 24

I went over moment-generating functions today. As I mentioned in my e-mail, there are two places I'd recommend you go to read a little bit more about them: pagse 24-25 of Professor Bass's notes and Chapter 10 of Grinstead and Snell. The latter is more detailed.

#### April 26

Today was just review: we went over Chebyshev's inequality, joint distribution problems, and sums of random variables. Good luck studying!