Introduction to Probability Theory

There are TWO sections of AMS131 being offered in Spring 14.  The two sections will run parallel.  The intention is for the same material to be covered in the same weeks for the two sections.  This page will provide information for both sections.

General Class Information:

 

Lecture Times

Section 1: Tuesday and Thursday, 10-11:45am, J Baskin Auditorium 101

Section 2: Tuesday and Thursday, 2-3:45pm, N. Sci Annex 101

 

Instructor

Robin Morris

email: rdm @ soe.ucsc.edu - please put "AMS 131" in the subject line

phone: email is preferred; (408) 821 8932 if desperate

Office: Baskin Engineering 357b

Office Hours: Tuesdays and Thursdays 12:30-1:30pm, or by appointment

 

TAs

Section 1: Chelsea Lofland, clofland@soe.ucsc.edu and Sisi Song, song@soe.ucsc.edu

Section 2: Cheng-Han Yu, cheyu@soe.ucsc.edu

TA office hours are

  • Sisi Song - Monday 1-2pm, Wednesday 2-3pm
  • Chelsea Lofland - Monday 5.30-6.30pm, Thursday 3-4pm
  • Cheng-Han Yu - Monday 2-4pm

All the office hours will be held in Baskin Engineering 358.

 

Sections

Section 1: Tu 2-3:10pm, E2-194; Th 6-710pm, E2-194; Th 7:30-8:40pm, E2-194

Section 2: M 11am-12:10pm, Phys Sciences 130; W 3:30-4:40pm, Phys Sciences 130

The first section will meet on Wednesday April 2nd, and the last section will meet on Tuesday June 3rd.

There is one holiday this quarter, Memorial Day on Monday May 26th.  Sections will not meet on that day.

 

Tutoring

Tutoring may be available, if a suitable tutor can be found.  If you know anyone who might be a good tutor for this course, please let me know.

 

Discussion Forum

There is a piazza discussion forum available at https://piazza.com/ucsc/spring2014/ams131 

Please address all technical issues with the discussion forum to Vladimir Furman, vfurman @ ucsc.edu

 

Textbook

Probability and Statistics, DeGroot and Schervish, 4th Edition, Pearson (2002)

 

Schedule

The proposed schedule is below.  Note that this may (will?) change depending on how the quarter progresses

Date Topic Chapter Homework Lecture Notes
Tue April 1st

Introduction; 3 views of probability; sample spaces; naive definition;

counting; axioms of probability

 1.1-1.8

 1.4, Q 6, 7, 8

1.5, Q 5, 6, 8, 9, 10

1.6, Q 3, 4, 8

1.8, Q 1, 6, 9

 notes
Th April 3rd

Birthday problem; properties of probability; inclusion-exclusion;

matching problem; independence; conditional probability; Bayes' rule

1.7,1.10

2.1-2.3 

 1.7, Q 1, 2, 10

1.10, Q 2, 7, 10

2.1, Q 2, 3, 4, 7

2.2, Q 2, 4, 6, 7, 10, 11, 19

2.3, Q 4, 5

 notes 
Tu April 8th

Law of total probability; conditional probability examples; 

conditional independence; Monty Hall problem; Simpson's Paradox.

2.1, 2.3

10.5 

2.1 Q 14

2.3 Q 1, 4, 5, 6

10.5 Q 1, 7

 notes

Please read the textbook section on Simpson's paradox.

Th April 10th

Gambler's ruin; random variables; Bernouli; Binomial; Hypergeometric;

CDFs; PMFs

2.4, 3.1

3.3 

2.4 Q 1, 2, 7 

3.1 Q 2, 3, 4, 7

3.3 Q 2, 3, 5, 6, 16

notes 
Tu April 15th Class will not meet.      
Th April 17th

Independence; Geometric distribution; expected values; indicator RVs;

linearity; Negative Binomial; examples

QUIZ 1

2.2, 5.5

4.1, 4.2 

5.5 Q 5, 6 

4.1 Q 1, 3, 4, 5, 8, 11

4.2 Q 2, 3, 8, 9

 

notes

 (5.5 Q5 - we haven't covered the negative binomial yet, but we will)

(4.1 Q 8, 11 - come back to these once we've covered continuous RVs)

(4.2 Q 3 - come back to these once we've covered continuous RVs)

 

Solutions to Quiz 1

 

Tu April 22nd

Poisson distribution; Poisson approximation; discrete vs. continuous;

PDFs; variance; standard deviation; Uniform distribution; universality

3.1, 3.2,

3.8

3.3, 4.3, 5.4 

 

3.2 Q 2, 10

3.8 Q 3, 4, 8, 13

4.3 Q 4, 5, 7

5.4 Q 4, 5, 6, 12 

notes

We only covered up to Poisson distribution/Poisson approximation today; We'll catch up later.

Th April 24th

Standard Normal Distribution; Normal normalizing constant;

Normal distribution; standardization; Law of the unconscious statistician

5.6, 4.1 

5.6 Q 3, 5, 8, 10, 11, 14, 16

4.1 Q 1, 3, 8, 11 

notes

 

Tu April 29th

Midterm Review    

review notes

 Review Problems:

1.12 Q 3, 4, 6, 9

2.5 Q 3, 5, 13, 16, 20, 23, 28

3.11 Q 1, 4, 9

4.1 Q 7

4.3 Q 9

4.9 Q 4

5.11 Q 5, 11, 12, 13, 18, 20

solutions to review problems

Th May 1st

Midterm Exam (in class)    Read "a mathematician's lament" and join the discussion on Piazza

I will post solutions to the midterm soon. 

solutions to the midterm

Tu May 6th

Exponential distribution; memoryless property; MGFs;

Bayes rule; Laplace's rule of succession

5.7, 4.4   

We finished off the Normal distribution today.  I'll figure out soon what we're going to cover going forwards.

 notes

Th May 8th

Use of MGFs; moments of Exponential and Normal; Sums of Poissons;

joint, conditional and marginal distributions; 2-D LOTUS; examples

4.4, 5.6

3.4-3.6 

5.6  - see above

4.1, 4.2, 4.3 - see above

5.2 Q 6, 7, 10

5.4 - see above

 

 

 

Today we covered the proof of LOTUS, the variances of the Poisson and Binomial distributions, and Laplace's rule of succession.

notes 

Tu May 13th

Expected distance between Normals; Multinomial; Cauchy; covariance;

correlation; variance of a sum; variance of Hypergeometric 

1.9, 4.1

4.6, 5.3 

7.2 Q 2, 3, 6

Today we covered Bayesian inference for the success parameter of a Binomial distribution, and derived the normalizing constant for a Beta distribution with integer parameters.

 notes

Th May 15th

Transformations; Log Normal; convolutions; Beta distribution;

Bayes' Billiards

QUIZ 2

5.6, 5.8

3.8, 3.9 

3.4 Q 2, 4, 8

3.5 Q 2, 3, 7

3.6 Q 2, 3, 9 

Today we looked at two examples of Bayesian inference (including online spelling correction), and then covered joint, marginal and conditional pdfs.  We also covered 2D LOTUS.

notes

 NOTE: quiz 2 has been moved to Tuesday May 20th

Tu May 20th

Gamma distribution; Poisson process; Beta-Gamma; order statistics;

conditional expectation

 

QUIZ 2

 

5.7, 5.4

7.8, 4.7 

 4.6 Q 5, 9, 13

5.3 Q3, 4

 

Today we looked at covariance and correlation, and derived the variance of a Hypergeometric distribution.

 notes

solutions to quiz 2

Th May 22nd

Conditional expectation (cont); waiting times  4.7 

 3.8 Q 3, 4, 8, 13

3.9 Q 2, 3

 Today we covered transformations of random variables, convolution (sums of RVs) and waiting times when tossing coins.

notes

Tu May 27th

Sum of random number of RVs; Inequalities; 

Law of large numbers; central limit theorem

6.1-6.4 

 6.2 Q 2, 3, 8

6.3 Q 2, 3, 8

6.4 Q 1

 Today we covered inequalities, law of large numbers and the central limit theorem, and the normal approximation.

notes

Th May 29th

Chi-squared; Student-t; multivariate normal;

Markov chains; transition matrix; stationary distribution

QUIZ 3

8.2, 8.4

5.10, 3.10 

 3.10 Q 2, 4, 9, 12

Today we started Markov Chains,  covering multi-step transition matrices and the evolution of the PMF over the states.

notes

solutions to quiz 3

Tu June 3rd

Markov chains (cont). 3.10     notes

Th June 5th

Review    

Review Questions

1.12 Q 4, 11

2.5 Q 2, 5, 17, 20, 21, 24, 26, 36

3.11 Q 4, 5, 8, 10, 16, 21, 27, 28, 29

4.9 Q 5, 8, 11, 14, 22

5.11 Q 7, 8, 12, 20

6.5 Q 9, 10, 12

Solutions to the review questions are now available.

 

Review notes.

In-class review notes, AMS 131-01.

In-class review notes, AMS 131-02

 

Monday June 9th

Section 1 Final Exam, 8-11am, Baskin Auditorium

Section 2 Final Exam, 8-11am, Baskin Auditorium

Section 2 Final Exam, 1-4pm, BE 358,  for those who have a conflict with the morning session.

   

Some practice final questions are available.  Solutions are now posted.  Please do try to solve the problems yourself before looking at the solutions

 The table of distributions is now available.

Thursday June 12th

Section 2 Final Exam, noon-3pm      
   
   

 

Additional Information

DRC

If you qualify for classroom accomodations because of a disability, please submit your Accomodation Authorization from the Disability Resource Center (DRC) to me during my office hours in a timely manner, preferably within the first two weeks of the quarter. Contact the DRC at 459-2089 V, 459-4806 TTY.

Academic Integrity

You are reminded of the University's Policy on Academic Integrity.  I hope not to have to remind any of you individually about this policy.

Some Thoughts About Lectures

"In Praise of Lectures" gives some ideas about the purpose of lectures, note-taking, and not being afraid to ask questions. It's target audience is more advanced mathematics students, but everything it says applies here. Think about the ideas it presents, and you will have a better time in AMS7 lectures. In particular

  • Lectures complement reading the textbook. In lectures I can spend extra time explaining ideas that students find confusing or difficult. I can try to judge from your behaviour your level of comprehension and adjust what I say accordingly.
  • I am not, however, a mind-reader. If you have questions, please ask them. If you don't understand something, chances are there are others who don't understand either, but are more inhibited than you are.
  • If you don't want to concentrate on the lecture, you're not required to attend. Please be considerate of those who do want to concentrate.
  • The lectures will present the material, but you will only know if you truly understand it if you try the homework problems. Only by applying the ideas yourself will you know that you have mastered them. “I went to a lecture on the violin, but when I tried playing one it sounded horrid. The lecturer can't have been any good.”
  • If you are having difficulties, please come and see me during office hours. Do this early in the quarter, rather than a week before the final exam. My goal is for everyone to understand and be comfortable with the material. If this is also your goal, I'm willing to do what's needed to help you achieve that goal.

 

Instructors and Assistants