## Math 60 Log

Week 1 (March 27-29) :

Wednesday :For the first couple weeks we will be introducing some probabilistic concepts via experiment. You will need to simulate probabilistic phenomena to do this. I recommend using maple. Here is some help regarding down loading maple 7 to a Mac and here is some help for loading it onto a PC . Here are the templates from the first lecture. First we looked at the flipping of a fair coin, then we analyzed the Expected Value , and hopefully by Friday we will explore the Variance . Your first assignment is essentially to redo these experiments for an unfair "coin", one whose probability of turning up heads is not 1/2 (you should pick your own probability of the coin turning up heads and perhaps experiment with a few different values). The assignment is to analyze the random variable which is one if the coin turns up heads and -1 if it turns up tails. Assignment: first make claim about what the expected value of the random variable should be in this setting and then experimentally justify your claim via a computer simulation. This assignment also must include a paragraph or so describing the sense in which your experiments help justify your claim!

For an example of an unfair "coin" I recommend going to the Math 40 class (at 1:45 in Bradley 105) where pennies will be spun rather than flipped.

Friday: Today we explored the notion of expected value a little more carefully and in particular the expect length of a streak when flipping a coin. We also explored the First Fundamental Mystery of Probability .

Homework. Problem 1. (To be turned in) Work out approximately how many shots a basketball player attempts in a season and a approximate the average percentage of these attempted shots that are made. Modify the streak program to simulate the streaks that a "random basketball" players will experience. Write a paragraph arguing how you might determine whether basketball players behave randomly or not.

Problem 2. (To be turned in) Perform the survey described in class on 5 people.

Week 2 (April 1-5) :

X-Session in Berry Starr 274 form 12:00-12:50 on Tuesday April 2nd. We will talk about Monday's homework which is the first example of modifying a template in a meaning full way. Attendance is optional, but if you have limited or no programming experience you will probably want to attend this X session.

Monday. Today we explored the notion of variance and in particular explored the Second Fundamental Mystery of Probability and the Third Fundamental Mystery of Probability .

Homework: 1.(to be turned in) Examine the standardized sum of an unfair coin toss (as we did for the standardized fair coin toss - see the standardized sum program). How does the empirical distribution appear to compare to the normal curve as the number of tosses in each experiment gets larger? Recall: this will require observing that the variance of the underlying random variable is 4*p*(1-p) which can be experimentally verified.

2. (to be turned in) Simulate the expected value of the longest streak of successes as opposed to the expected value of the longest streak. Compare the expected length of a success streak with that of a streak and formulate a conjecture about the relationship between these two expected values. Discuss the empirical evidence that lead you to your for your conjecture.

Wednesday : Today we saw the the Fundamental Theorem of Probability and explored a bit about what it means to statisticians. We did this via the maximal streak random variable.

Your homework will be to perform the same analysis we performed for the streak random variable for the following random variable: imagine that each of the plus and minus ones that we see after our fair coin toss corresponds to a dollar amount that we receive or owe. We play this game N times. After each time we play we either owe some amount, are breaking even or we are ahead. For example N=4 and we "flipped" the payoffs {-1,-1,-1,1} in that order then after the second toss we owe 2 and after the third toss we owe 3. Simulate the random variable which corresponds to counting the number of times during the experiment when you are ahead. For example for with our {-1,-1,-1,1} flips we are never ahead and the result would be 0. Pick an M and look at the standardized sum representing the performing this experiment M times. Experimentally verify that this random variable is becoming normal as M is getting larger.

The homework should be turned in on MONDAY!!!!! . If it is turned in via email please call it LastnameHW4, or example I would call my maple assignment LeibonHW4 and if I had a PC I would send my professor the file LeibonHW41.html and a bunch of gif images that were located in the image folder after saving my program as html.

A Question: The assignment says: Simulate the random variable which corresponds to counting the number of times during the experiment when you are ahead. Does this mean the total number of times you are ahead, or the number of streaks when you are ahead?

Friday : Today we explored how a good understanding of the big square root can lead to a dramatically improved version of the calculus.

Week 3 (April 8-12) :

X-Session in Berry Starr 274 form 12:00-12:50 on Tuesday April 9th

Monday. Please read chapters 1 and 2 of the book and due and turn in problem 17 from chapter 1.1 (it should be turned in on Friday - please call it LastnameHW5 if you email it to me). Also by Friday please do (but turn in only if you'd like) problems 4,5,11,14,18, and 21 from section 1.2. Today in class we learned how make a finite sample space and place a notion of probability on it.

Wednesday. Here are the simulations that the book uses. They can be run in maple which in particular gives you a plethora of new templates to work with. Today in class we explored an infinite sample space and attempted to place a notion of probability on it.

Friday. Please read chapter 2 of the book and turn by X-session on Tuesday numbers 12 and 13 from section 2.2. Also do problems 2 and 8 of section 2.2 (turn them in if you'd like).

Week 4 (April 15-19) :

Come and ask questions about the homework (or anything else as time permits) at the Optional X-Session in Bradley 104 from 12:00-12:50 on Tuesday April 16. First exam date: the in class exam will be administered on Wednesday May 1st at which point the take exam will be handed out and due Friday May 3rd at class time.

Monday . Here are some examples showing how to use maple in order to estimate the values of various probability distribution functions (pdf), cumulative distribution functions (cdf), and inverse cumulative distribution functions (icdf). Please read chapter 3 as soon as possible. For Friday please do the following problems: from section 2.2 problems 5,6,9 10, and 14 (and turn in numbers 9 and 10) and from section 3.1 problems 4, 8 and 12.

Wednesday. We learned about the binomial distribution and performed a to hypothesis test.

Friday. Please read chapter 3 of the book and turn by X-session on Tuesday number 22 from 3.1 and number 27 from 3.2. Also do problems 13 and 18 from 3.1 and problems 6 and 10 from 3.2.

Week 5 (April 22-26) :

Come and ask questions about the homework (or anything else as time permits) at the Optional X-Session in Bradley 104 from 12:00-12:50 on Tuesday April 23.

Monday: Please read chapter 3 of the book and turn in by Friday numbers 22 and 34 from 3.2. Also do problems 3,4,5, 10 and 18 from from 4.1.

Wednesday. Today we learned about conditional probability and its relationship to goats .

Friday. Please read chapter 4 of the book and turn by X-session on Tuesday numbers 27 and 63 from 4.1 and do and turn if you'd like numbers 22,28,34, and 51 from 4.1.

Week 6 (April 29 - May 3) :

Come and ask questions about the homework (or anything else as time permits) at the Optional X-Session in Bradley 104 from 12:00-12:50 on Tuesday April 30.

Monday. Today we finished up chapter 4.

Wednesday. Exam: comprehensive in-class exam administered and comprehensive take-home exam handed out. After receiving some emails form confused students I have augmented the exam with some potentially helpful clarifications . If you find something confusing you may want to look at the revised exam and see if the issue has now been clarified. Friday. Please read chapter 5 of the book and turn by X-session on Tuesday numbers 9 and 18 from 5.1 and do and turn if you'd like numbers 2 and 7 from 5.2.

Week 7 (April 6 - May 10) :

We will go over the exam at the Optional X-Session in Bradley 104 from 12:00-12:50 on Tuesday May 7.

Monday. Today we saw that the the exponential and geometric distributions are the memoryless distributions. Do Friday's homework for tomorrow and look over chapters 6 and 8.

Wednesday Extra credit announcement. Up to ten extra credit points will be added to your exam scores provided you turn in a "mini-project". A "mini-project" might consist of the following three parts. 1. questions 2. simulations designed to explore your questions and a 3. conjectured answer based on your simulations and any other form of reasoning. You may work with other people on this "mini-project" if you'd like. (A VERY terse example of a "mini-project": 1. It is clear to me that if I take the bus waiting problem and make the 100 minute wait longer then I should get closer to the "integral value". But what if if make the time smaller? When it is 0 I know the answer is ten, but how does it decrease to its eventual integral value as time gets larger?... 2. Now I simulate this experiment with the uniform on [5,15] distribution for various times from 0 to 100 and I plot the expected waiting times.... 3. From my simulation, and especially my plot, it appears that one approaches the integral value very quickly. In fact it looks like an exponential curve with a negative exponent....)

Please read chapter 6 carefully and look over chapter 8, turn on Friday numbers 14 and 20 from 6.1 and do and turn in if you'd like numbers 6 and 19 form 6.1.

Friday. Please read chapters 6 and 8 of the book and turn by next Wednesday numbers 15 of 6.2 and 9 from 8.1 and do and turn if you'd like number 10 from 6.3 and numbers 5, 6 and 7 from 8.1.

Week 8 (May 13 - May 17) :

No X-Session this week . In fact my Tuesday office hours have been moved to 4:00 on Monday.

Monday: Please read chapter 7. Note Friday's homework is due Wednesday. Here is a neat mini-project where you can witness Benford's distribution in action!

Friday. Please read chapter 9. Turn by next X-session number 3 in 7.1 and 5 in 7.2. Also go through and rederive on all the properties of the Fourier transform discussed on Friday, you need not turn this in (but it is a really good idea and a question or two about this might show up on the "in class" part of the final).

Week 9 (April 20 - May 24) :

I will hand back and go over a whole lot of homework at the Optional X-Session in Bradley 104 from 12:00-12:50 on Tuesday May 21.

Monday. Today we saw the Fourier transform in action and began exploring how to use it to prove the central limit theorem.

Tuesday/Wednesday. Please turn in on Friday 6.2 #18, 9.2 #5 and 9.3 #14. Also due but turn in only if you'd like chapter 9.1 1,3,4 and chapter 6.2 17.

Friday Please turn by next Wednesday 6.3 #17,18, 7.2 #15, 9.3 #10. Also due but turn in only if you'd like chapter 6.3 23,24,25 (not easy but worth it!)

Week 10 (May 28 - May 29) :

I will go over a whole lot of homework at the Optional X-Session in Bradley 104 from 12:00-12:50 on Tuesday May 28.

The exam will be handed out in class on Wednesday and must be returned to me by Tuesday June 4 at 5:00 P.M.