﻿ Math 265

2017 November 22,

# Math 265: Probability

Carleton College, Fall 2017, Prof. Joshua R. Davis, , CMC 219, x4095

## Introduction

Probability is a beautiful subject of pure mathematics with many applications throughout the sciences. It is the theoretical basis for statistics. It finds heavy use in quantum theory, thermodynamics, finance, traffic flow, meteorology, etc. And there's gambling.

This is a first course in probability, assuming only multivariable calculus as background. Approximately, the first half of the course is discrete and the second half continuous. The materials are

• Probability with Applications and R by Robert P. Dobrow. This text is used in all current sections of Math 265.
• Errata for that textbook.
• Exam A: Practice, Exam, Solutions (Percentiles: 75th=44/52, 50th=39/52, 25th=34/52)
• Exam B: Exam, Solutions (Percentiles: 75th=36/42, 50th=31/42, 25th=26/42)
• Exam C: Exam, Solutions (Percentiles: 75th=70/80, 50th=65/80, 25th=58/80)

Our class meets in Willis 211 during period 3A (MonWed 11:10-12:20, Fri 12:00-1:00). If you want to meet with me outside class, then try to make my office hours, which are Monday 5A, Tuesday 11:00-12:00 and 1:00-2:00, and Friday 4A. If you cannot make office hours, then e-mail me, listing several possible times.

## Responsibilities

Final grades (A, B, C, etc.) are assigned according to an approximate curving process. By this I mean that there are no predetermined percentages (90%, 80%, 70%, etc.) required for specific grades. The advantage of this system is that student grades don't suffer when I write a difficult exam. The disadvantage is that you cannot compute your own grade. Visit me in my office, if you want me to estimate your current grade for you. The following elements contribute.

• Participation: You are expected to attend every class meeting promptly, take notes, and participate in discussion and group work. You can make up for a deficiency in class participation by talking with me in office hours or generally demonstrating exceptional effort and interest. Participation is used to make small adjustments to the final course grade. Additionally, a requirement for passing this course is that you visit me in my office at least once during the first two weeks.
• Assignments: Homework assignments are the core of the course; they are where you learn the material. Altogether they count for 25% of your grade.
• Exam A: The first midterm exam is given in class about one third of the way through the course. It counts for 25% of your course grade.
• Exam B: The second midterm is given in class about two thirds of the way through the course. It counts for 25% of your grade.
• Exam C: The final exam is scheduled for Saturday November 18 from 3:30 to 6:00. This exam counts for 25% of your course grade. Self-scheduled final exams are not allowed.

You are expected to spend about 10 hours per week on this course outside class. Some students need to spend more than 10 hours. If you find yourself spending more than 15 hours, then talk to me.

## Standards for Work

On homework, you are encouraged to figure out the problems with other students. However, you should always write/type your solutions individually, in your own words. You may not copy someone else's work or allow them to copy yours. Presenting someone else's work as your own is a violation of Carleton's Academic Integrity standards.

Writing is not just for English and history majors. Written and oral communication skills are essential to every academic discipline and are highly prized by employers. In this course, your written work is evaluated both for correctness and for presentation.

Although homework is assigned every day, it is collected only once a week. When handing in a week's homework, staple your pages into a single packet, in the correct order. Multi-sheet packets that are not stapled are unacceptable. I will not accept packets that are not stapled. Is there a stapler in the classroom? Often not, so staple ahead of time. Is a paper clip okay? No.

Depending on time constraints in any given week, perhaps not all of your homework will be graded. In order to ensure full credit, do all of the assigned problems.

## Special Accommodations

During the term, you have one free pass to hand in a week's homework packet late, no questions asked. Simply hand in your late packet when the next packet is due, writing "Late Pass Used" prominently at the top of the late packet.

Once you have used your late pass, no late assignments are accepted, except in extreme circumstances that typically involve interventions by physicians or deans.

If some medical condition affects your participation in class or your taking of exams, let me know in the first week of class. You may need to make official arrangements with the Disability Services.

## Schedule

To help you decode the schedule, here is an example. On Day 1 we discuss sample spaces and probability functions. Sections 1.1-1.3 of the textbook covers that material; read them before or after class, to get another treatment. You have homework called "Day 1", some of which is due on Day 1 and some of which is due on Day 2. If you wish, you can view the file coinsAndDice.R that I used in class.

M 09/1111.1-1.3sample spaces, probability functionsDay 11, 2coinsAndDice.R
W 09/1321.4-1.7properties, counting, sampling with(out) replacementDay 25replacement.R
F 09/1531.8-1.9, 2.1-2.3random variables, conditional probabilityDay 35
M 09/1842.4-2.5conditioning, Bayes' theoremDay 48
W 09/2053.1-3.2independenceDay 58
F 09/2263.3-3.5Bernoulli, binomial distributionsDay 68binomial.R
M 09/2573.6-3.7, 5.1Poisson, geometric distributionsDay 711
W 09/2784.1R lab, random variables, expected valueDay 811basicR.R
F 09/2994.2, 4.10, 4.3functions of random variables, joint distributionsDay 911
M 10/0210first exam
W 10/0411class cancelled
F 10/06124.4-4.5independence, sums of random variablesDay 1214
M 10/09134.6-4.7variance, covariance, correlationDay 1316
W 10/11144.8, 6.1, 6.3conditional distributionsDay 1416
F 10/13156.1-6.4continuous distributions, expected value, varianceDay 1516
M 10/16midterm break
W 10/18166.5-6.6exponential distribution, functions of random variablesDay 1619inverseTransform.R
F 10/20176.7joint and marginal distributionsDay 1722
M 10/23186.8independence
W 10/2519second exam
F 10/27206.9covarianceDay 2022
M 10/30216.10, 8.1, 8.3functions of a random variable, conditional distributions, expectationDay 2125
W 11/01228.3, 7.1conditional expectation, normal distributionDay 2225
F 11/03237.1, 9.1-9.2normal distribution, laws of large numbersDay 2325binomialNormal.R
M 11/06249.4-9.5central limit theorem, moment generating functionsDay 2428
W 11/08259.4-9.5R lab, central limit theorem, moment generating functionsDay 2528clt.R
F 11/10269.3, 9.4.1Monte Carlo methodsmonteCarlo.R
M 11/1327reviewReview Problems
W 11/1528review
S 11/18final exam 3:30-6:00

If the course were longer, we might have done: Poisson processes, more on conditional expectation, more on the hypergeometric, negative binomial, gamma, beta distributions.