# Probability and Statistics Spring 2019

Lecturer
Coordinator
David Pires Tavares Martins david.martins@math.ethz.ch

## Course Description

The goal of this course is to provide an introduction to the basic ideas and concepts from probability theory and mathematical statistics. In addition to a mathematically rigorous treatment, also an intuitive understanding and familiarity with the ideas behind the definitions are emphasized. Measure theory is not used systematically, but it should become clear why and where measure theory is needed.

## Learning materials

Probability: Chapters 1-12 from the book Fundamentals of Probability: A First Course by Anirban DasGupta

Statistics: Chapter 8 and 9 from the book "Mathematical Statistics and Data Analysis" by John Rice.

Lecture time Room
Tuesday 10:15-11:55 CAB G 61
Thursday 13:15-15:00 HG E 5

The first lecture will be on Tuesday 19.02.2019 and the last lecture will be on Tuesday 28.05.2019. Note that there will be no lectures or classes during Easter break (Fri 19.04.2019 - Sun 28.04.2019) and on Ascension day (Thu 30.05.2019).

Future changes and announcements will be posted here.

Updated 18.03.19 The program will be updated during the semester. Please make sure you have the latest version.

Note 18.03.19 Chapter 5 (generating functions) will not be discussed in lectures, but it will be studied through exercises.

• AD: the book of Anirban DasGupta
• JR: the book of John Rice
• *: Starred section from the book
• +: Additional topic, not in the book
• x: Topic from the book not done in the course
Lecture Date Topics
1 Tue 19.02 Chapter 1 of AD, Introduction, counting methods
2 Thu 21.02 Chapter 1 of AD, Introduction, counting methods
3 Tue 26.02 Chapter 2 of AD, Birthday and matching problem
(*) Stirling's approximation
4 Thu 28.02 Chapters 2 and 3 of AD, Conditional probability, independence
5 Tue 05.03 Chapters 3 and 4 of AD, discrete random variables
(*) Other fundamental moment inequalities
6 Thu 07.03 Chapter 4 of AD, discrete random variables
(+) Overview of measurability
7 Tue 12.03 Chapter 4 of AD, discrete random variables
8 Thu 14.03 Chapter 4 of AD, discrete random variables
(+) Jensen's inequality
9 Tue 19.03 Chapters 4 and 6 of AD, standard discrete distributions
10 Thu 21.03 Chapter 6 of AD
11 Tue 26.03 Chapter 6 of AD
(+) Overview of Kolmogorov's extension theorem
12 Thu 28.03 Chapter 6 of AD
(+) Overview of Poisson processes
13 Tue 02.04 Chapter 6 of AD
(+) Domination of measures, Radon-Nikodym derivative
14 Thu 04.04 Chapters 7 and 8 of AD
(+) Absolute continuity with respect to Lebesgue measure
15 Tue 09.04 Chapters 7 and 8 of AD
(+) Review of integrals with respect to a measure
16 Thu 11.04 Chapters 7 and 8 of AD
17 Tue 16.04 Chapters 7&8, chapters 9&10 of AD
18 Thu 18.04 Chapters 9 and 10 of AD
A 15min movie about Carl Friedrich Gauss
Tue 23.04  EASTER BREAK
Thu 25.04  EASTER BREAK
19 Tue 30.04 Chapters 10 and 11 of AD
20 Thu 02.05 Chapter 11 of AD
(x) Continuity correction
21 Tue 07.05 Chapters 11 and 12 of AD
(x) 11.5 on the Multivariate case (joint MGF, multinomial distribution, etc.)
22 Thu 09.05 Chapter 12 of AD
(x) 12.5 on Bivariate normal conditional distributions
(x) 12.6 on Order statistics
23 Tue 14.05 Chapter 12 of AD
Chapter 8 of JR: estimators, moment estimators, MLE
24 Thu 16.05 Chapter 8 of JR: Large sample theory for the MLE
25 Tue 21.05 Chapter 8 of JR: Efficiency, sufficiency
26 Thu 23.05 Chapter 9 of JR (Hypothesis testing): Introduction, Neyman-Pearson Lemma.
(x) 8.6 on the Bayesian approach
27 Tue 28.05 Neyman-Pearson Lemma (continued), examples of testing simple hypotheses
(x) Sections 9.3 to 9.10

UPDATE 01 Mar 2019: Class registrations through the echo system, available on echo.ethz.ch, have now closed. In case of questions, please contact the coordinator.

Solutions should be submitted to your assistant's folder in the box dedicated to this course, next to HG G 53.2. The deadline is 2pm of the Friday before the corresponding class in order to guarantee that they are marked in time. The marked exercise sheets will be returned in the next class or otherwise returned to the box for collection.

The exercises will be uploaded here one week in advance, together with the solutions for the previous sheet. Submitting solutions is not mandatory, but attempting to solve the sheets is very helpful with practicing the contents of the course and preparing for the exam.

Exercise sheet Due by Solution
Sheet 1 01 March 2019 Solutions 1
Sheet 2 08 March 2019 Solutions 2
Sheet 3 15 March 2019 Solutions 3
Sheet 4 22 March 2019 Solutions 4
Sheet 5 29 March 2019 Solutions 5
Sheet 6 4 April 2019 Solutions 6
Sheet 7 11 April 2019 Solutions 7
Sheet 8 29 April 2019 by 12:00 Solutions 8
Sheet 9 3 May 2019 Solutions 9
Sheet 10 10 May 2019 Solutions 10
Sheet 11 17 May 2019 Solutions 11
Sheet 12 24 May 2019 Solutions 12
Sheet 13 31 May 2019 Solutions 13

## Schedule

TimeRoomAssistantContactTeaching language
Tuesday 13-15CAB G 56Marcello Longolongom@student.ethz.chen
Tuesday 13-15CHN E 42Raphael Oberlioberlir@ethz.chen
Tuesday 13-15HG E 22 David Pires Tavares Martinsdavid.martins@math.ethz.chen
Tuesday 13-15HG E 33.5Angelo Abaecherliangelo.abaecherli@math.ethz.chen
Tuesday 13-15HG G 26.3Chong Liuchong.liu@math.ethz.chen

The first classes will be held on the second week of the semester (26 February 2019).

Grades for this course will be based solely on the written final exam, to be held during the August examination session. The exam will cover all the material taught in the lectures, and will be broadly similar in content to the problems appearing on the exercise sheets.

Students may not bring any paper or calculators to the exam, and should only bring writing materials (pens etc.) along with their student cards (Legi). Paper for drafts and for writing your answers will be provided. You will also be provided with a printed copy of the following formulae sheet:

Formulae sheet

Some old exams can be found here on the homepage of Group 3. It is not advisable to prepare from old exams only, since the topics covered in previous years may not match exactly the contents from the current course.

For questions before the exam, please attend one of the Präsenz sessions during the semester break. During the second and the third week of the semester after the exam, you have the possibility to look at your exam during the regular Präsenz hours.

Other references include: