Concepts of Probability

Lecture is M-F, 11-12, in 150 GSPP (Goldman School of Public Policy, across the street from campus to the north). Textbook: Probability, by Jim Pitman. We will be following this fairly closely.

Office hours are Mondays 1-2pm in the Student Learning Center; also Tuesdays 1-2pm and Thursdays 9:30-10:30am, both in 335 Evans. For appointments at other times, email me: (my three initials) at stat.berkeley.edu. Also, the Student Learning Center offers free drop-in tutoring for Stat 134 during the summer, Mondays-Thursdays, 11-4, from June 25 to August 13; also at the SLC are Mike Leong's office hours specifically for Stat 134 on Mondays, Wednesdays, and Thursdays from 12-1.

Course Description: This is an introductory course to the ideas of probability. The focus is on gaining familiarity with those tools, through examples, calculations, and intuition. There will be a few proofs, but they will not be the focus of the course.

The goals of the course are to:

• Understand the basis of probability: notions of distribution, expectation, and conditioning.
• Learn probabilistic problem solving skills: stating models simply, calculating and interpreting simple properties (mean, variance), manipulating random variables.
• Become familiar with properties of and relationships between common distributions: binomial, geometric, Poisson, Gaussian, exponential, gamma, beta, hypergeometric, negative binomial.
• Understand and use limit theorems (law of large numbers, central limit theorem).

The culmination of the first half of the course is the Central Limit Theorem -- the remarkable fact that a large class of sums of independent random numbers tends to be near its average in a way described by a single, univeral, limiting law. To reach this point, we will build up a good bit of general probabilistic techniques and intuition. In the second half of the course, we embark on serious study of continuous random quantities, and range more widely into the ubiquitous probability distributions and their often suprising properties.

Course responsibilities: There will be two homework assignments weekly, due Tuesdays and Fridays by 11am, either in class or in the envelope outside 335 Evans. These will be graded by a grader, so no late homework will be accepted, barring major illness or the like. Also, each Thursday students are expected to hand in reading questions: one question, comment, or clarification from the reading for the following week, on a half-sheet of scrap paper. These will be checked off by me, and will inform the following week's focus. Solutions to the homeworks will be posted in the glass cases in the center hallway of the 3rd floor of Evans Hall, the afternoon after they are due.

Exams will be held in class on Friday, July 18, and Friday, August 15. There will also be a takehome section of the final assigned Friday, August 8, and due Wednesday, August 13. The written exams will be graded by me, and partial credit will be given. Students will be allowed one (or, three for the final) two-sided pages of notes.

Grading: the different pieces of the course will contribute to the final grade as follows: either (reading questions: 10%; homeworks: 25%; midterm: 20%; takehome final: 25%; in-class final: 20%) or (reading questions: 5%; homeworks: 15%; midterm: 25%; takehome final: 25%; in-class final 30%), whichever is greater. The lowest two homework scores will be dropped. The course will not be graded on a curve; rather, final grades will be determined based on completion of the course goals, as demonstrated by the assignments and exams.

Schedule: (which will be revised as we proceed) All references are to the textbook, Jim Pitman's Probability.

• HW 1: (due June 24) calculus background
• June 23-25 Sections 1.1, 1.3, 1.4, 1.5, 1.6 -- basics of probability, events as sets, probability distributions, histograms, conditional probability, independence, Bayes' rule.
• HW 2: (due June 27) Section 1.1: 1,2,7; Section 1.3: 2,5,6,8,11 .
• June 26-27 Sections 2.1, 2.2 -- binomial and geometric distributions, Gaussian ("normal") approximation (omitting skewness correction).
• HW 3: (due June 31) Section 1.4: 4,8; Section 1.5: 1,5; Section 1.6: 8; Chapter 1 Review: 16; Section 2.1: 1,6 .
• June 30 - July 1 Sections 2.4, 2.5 -- Poisson approximation, random sampling.
• HW 4: (due Thursday July 3) Section 2.1: 10; Section 2.2: 5,9,13; Section 2.3: 1.
• July 2-3 Section 3.1, 3.2 -- random variables, joint distributions, conditional distributions.
• July 4 no class today.
• HW 5: (due July 8) Section 2.4: 5; Section 2 Review: 7,13; Section 3.1: 4,5,12,14,18.
• July 7-9 Section 3.2 -- expectation, indicators, Markov's inequality.
• HW 6: (due July 11) Section 3.2: 2,5,8,16,18.
• July 9-11 Section 3.3 -- standard deviation, Gaussian (normal) approximation/Central Limit Theorem.
• HW 7: (due Thursday July 17) Section 3.3: 21,25; Section 3.4: 2,5,6,9; Section 3 Review: 19, 30(a,b).
• July 14-17 Sections 3.4, 3.5 -- more discrete distributions, Poisson distribution, Poisson scatter.
• July 18 Review/example problems.
• July 21 Midterm (in class) -- Chapters 1-3 -- study guide, grade distribution.
• July 22-23 Section 4.1 -- continuous probability densities.
• HW 8: (due July 25) Section 3.5: 6, 8, 12, 17; Section 4.1: 3 (a-e).
• July 24-25 Section 4.2 -- exponential and gamma distributions.
• HW 9: (due July 29) Section 4.1: 8, 13; Section 4.2: 6,9, 10(a); Section 4 Review: 5, 13, 14.
• July 28-30 Sections 4.4, 4.5, 4.6 -- change of variables, cumulative distribution function, distribution of maxima, order statistics.
• HW 10: (due August 1) Section 4.4: 1, 2, 6; Section 4.5: 1, 5; Section 4.R: 19, 21.
• July 31 - August 1 Sections 5.1, 5.2 -- continuous joint distributions and densities.
• HW 11: (due August 5) Section 4.6: 1, 2, 4; Section 4.R: 22; Section 5.1: 6, 9; Section 5.2: 1, 9.
• August 4-7 Sections 5.3, 6.4, 6.5 -- independent Gaussians and linear combinations, chi-squared distribution, correlation and covariance.
• HW 12: (due August 8) (Section 5.2: 1,9 if not completed), Section 5.3: 3, 7(c), Section 6.4: 4; Section 6.5: 6,8.
• August 8 Sections 6.1, 6.2 -- conditional expectation, discrete case.
• FINAL HW: (assigned Friday, August 8; due Wednesday August 13) Takehome section of final exam. Here is a pdf of the final.
• August 10: Review Session by Mike Leong at the SLC, from 2-4pm. Enter on the upper mezzanine -- go to the left of the Golden Bear Cafe on upper Sproul plaza, along the balcony.
• August 11-14 Sections 6.1-6.3 -- continuous conditional distributions and expectation; and general review.
• August 15 Final (in class) -- cumulative, but see this study guide. Here is the grade distribution.

Links:

• Errata to Probability, thanks to Mike Leong at the SLC.
• Phillip Stark's glossary.
• Jim Pitman's Fall 2002 and Fall 2005 final exams.
• Slides from Elchanan Mossel's Stat 134 course.