KTH Mathematics  


Probability Theory SF2940

The aim of the course is to introduce basic theories and methods of pure probability theory at an intermediate level. For example, the student will learn how to compute limits of sequences of stochastic variables by transform techniques. No knowledge of measure and integration theory is required, and only bare first statements of that will be included in the course. Techniques developed in this course are important in statistical inference, statistical physics, time series analysis, financial analysis, signal processing, statistical mechanics, econometrics, and other branches of engineering and science. The course gives also a background and tools required for studies of advanced courses in probability and statistics. The course is lectured and examined in English.

Prerequisites:

  • SF 1901 or equivalent course a la 'a first course in probability and statistics (for engineers)'
  • Basic differential and integral calculus, basic linear algebra.
  • Previous knowledge of transform theory (e.g., Fourier transforms) and generating functions is helpful, but not a necessary piece of prerequisites.
  • The concept of Hilbert space will make an appearance, but is not actively required.

Lecturer and Examiner : Boualem Djehiche homepage and contact information

The course web page. http://www.math.kth.se/matstat/gru/sf2940/

Teaching assistants :

  • Martina Favero email. Office hours: Wednesdays at 11:00 - 12:00, Room 3738.
  • Boris Petkovic email.
  • Lukas Schoug email Office hours: Tuesdays at 10.30-11.30, Room 3732.
  • Johan Westerborn email. Office hours: Tuesdays at 14.00-15.00, Room 3747.

  • The teaching assistants will each have an office hour open for consultation (1h per week). The hours will be announced later.

Exercise groups

  • Martina Favero
  • Boris Petkovic
  • Lukas Schoug
  • Johan Westerborn

Workshop There will be a 2-hour workshop (räknestuga) on a date to be announced later on

Course literature:

  • T. Koski Lecture Notes: Probability and Random Processes Edition 2017 LN pdf
A hardcopy of this text can be bought at THS kårbokhandel (i.e., the bookstore at Campus Valhallavägen), address: Drottning Kristinas väg 19.
  • The book by A. Gut An Intermediate Course in Probability, Springer-Verlag 1995 or later editions may be used for a secondary reading reference.


    Important: Students, who are admitted to a course and who intend to attend it, need to activate themselves in Rapp . Log in there using your KTH-id and click on "activate" (aktivera). The codename for sf2940 in Rapp is SF2940:sante16.


    Examination:
    There will be a written examination on Wednesday 25th of October, 2017, 08.00- 13.00. Allowed means of assistance for the exam are a calculator (but not the manual for it!) and the Appendix B of Gut, the Collection of Formulas and L. Råde & B. Westergren: Mathematics Handbook for Science and Engineering. Each student must bring her/his own calculator, Appendix B of Gut and the Collection of Formulas (that should be downloaded from this homepage) as well as the book by Råde & Westergren to the examination. The department will NOT distribute the "Formulas and survey". Grades are set according to the quality of the written examination. Grades are given in the range A-F, where A is the best and F means failed. Fx means that you have the right to a complementary examination (to reach the grade E). The criteria for Fx is a grade F on the exam, and that an isolated part of the course can be identified where you have shown a particular lack of knowledge and that the examination after a complementary examination on this part can be given the grade E.

    The Re-exam is scheduled to take place on Tuesday December 19, 2017, 08.00-13.00.

    Homework:
    There will be two sets of elective homework assignments that will give bonus points in the written exam on the 25th of October 2017, AND in the Re-exam 19th of December 2017. If you have received 5 bonus points you may skip Problem 1(a). If you have received 10 bonus points you may skip the whole Problem 1.

    Preliminary plan Exercises are from the Sections of Problems of LN. For example: Section 1.12.2 1 is the first exercise in section 1.12.2 in LN.
    (BD=Boualem Djehiche, MF= Martina Favero, BP=Boris Petkovic, LS=Lukas Schoug, JW=Johan Westerborn) 
    The addresses of the lecture halls and guiding instructions are found at KTH website.



    Day Date Time Hall Topic Lecturer
    Mon 28/08 08-10 FR4 (Albanova) Lecture 1: Sigma-fields, Probability space, Axioms of probability calculus, Some Theorems of Probability calculus. Distribution functions. Chapter 1 in LN.
    BD
    Tue
    29/08
    15-17 D34, D41,E31, E35
    Exercises 1: Sect 1.12.2: 1,12,Sect 1.12.3: 6, 9
    Recommended: Sect 1.12.2: 6,7,9
    MF
    LS
    JW
    BP
    Wed 30/08
    13-15 D2 Lecture 2: Multivariate random variables. Marginal density, Independence, Density of a transformed random vector, Conditional density, Conditional Expectation.
    Chapters 2-3.5 in LN

    BD
    Fri
    01/09 10-12 M35,M36, Q21, Q36
    Exercises 2: Sect 2.6.2: 4, Sect 2.6.3: 13,15, 17, 20, 21

    Recommended Sect 2.6.2: 4,8,5,8; Sect 2.6.3.: 1,4,5,10, 25
    MF
    LS
    JW
    BP
    Mon
    04/09 08-10 FR4 (Albanova) Lecture 3: The Rule of Double Expectation E(Y) = E(E(Y|X)|X), Conditional variance, The Formula Var(Y) = E (Var(Y|X)) + Var( E(Y | X)) and its applications, Conditional expectation w.r.t. a sigma-field. Chapter 3 in LN .
    BD
    Tue
    05/09 15-17 D41,
    E31,
    E35, E51
    Exercises 3: Sect 2.6.5: 2, Sect 3.8.3: 5,10,12,14,
    Recommended: Sect 3.8.3: 11, Sect 3.8.4: 8,11
    MF
    LS
    JW
    BP
    Wed
    06/09 13-15 D2 Lecture 4: Characteristic fuctions Chapter 4.1. - 4.4 LN .

    BD
    Fri
    08/09 10-12 M32,
    M35, M36,
    V34
    Exercises 4: Sect 3.8.5: 1,3,4, 6(a), 7
    Recommended Sect 3.8.5: 2,5,8
    MF
    LS
    JW
    BP
    Mon
    11/09 08-10 M1 Lecture 5: More on characteristic functions chapter 4.4 LN
    Generating functions, Sums of a random number of random variables Chapter 5.2- 5.5, 5.7 in LN.
    BD
    Tue
    12/09 15-17 D41
    E31
    E35, E51
    Exercises 5: Sect 4.7.1: 3,6, 7, 12 Sect 4.7.2: 1
    Recommended: Sect 4.7.1: 2,5,8
    MF
    LS
    JW
    BP
    Wed
    13/09 13-15 FR4 (Albanova) Lecture 6: Concepts of convergence in probability 6.2-6.5 LN
    BD
    Fri
    15/09 10-12 Q21, Q22, Q26, Q34
    Exercises 6: Sect 5.8.1: 4,5 Sect 5.8.2: 5,6,7 Sect: 5.8.3 12,13
    Recommended: Sect 5.8.2 3, Sect 5.8.3: 3
    MF
    LS
    JW
    BP
    Tue
    19/09 15-17 B2 Lecture 7: Concepts of convergence in probability theory: convergence by transforms Convergence of sums and functions of random variables. Almost sure convergence, strong law of large numbers.
    Chapter 6.6 6.7 LN
    BD
    Wed 20/09 13-15 V23
    V33, V35, D33
    Exercises 7: Sect 6.8.1: 15, 16, 17, Sect 6.8.2: 1,7, Sect 6.8.4: 1,2,3
    Recommended: sect 6.8.1: 7,8,12
    MF
    LS
    JW
    BP
    Fri
    22/09 10-12 FR4 (Albanova) Lecture 8: Multivariate Gaussian variables,
    LN Chapter 8
    BD
    Tue
    26/09 15-17 E31,
    E32,
    E35, E51
    Exercises 8: Sect 6.8.1: 13, Sect 8.5.1: 8,10, 13, 15, 17
    Recommended: Sect 8.5.1: 6,14,16
    MF
    LS
    JW
    BP
    Wed
    27/09 13-15 FR4 (Albanova) Lecture 9: Gaussian process, covariance properties. Chapter 9.1-9.4. BD
    Fri
    29/09 10-12 M32
    M35
    M36, V3
    Exercises 9: Sect 9.7.2: 2, Sect 9.7.4: 4,
    Sect 9.7.5: 1,2 Sect 9.7.6: 7
    MF
    LS
    JW
    BP
    Tue
    03/10 15-17 E1 Lecture 10: Wiener process chapter 10.2-10.4, Wiener integral 10.5.1-10.5.2 LN
    BD
    Wed
    04/10 13-15 E35
    E51
    E52, Q26
    Exercises 10: Sect 10.7.2: 1,2,3,4, 6 (d), 8, 9, Sect 10.7.3: 1
    Recommended Sect 10.7.2: 6(a), 6(c) Sect 10.7.3: 6
    MF
    LS
    JW
    BP
    Fri
    06/10 10-12 B2 Lecture 11: Ornstein Uhlenbeck process, chapter 11.2 LN Poisson process 12.2 - 12.3 LN
    BD
    Tue 10/10 15-17 E31, E35,
    E51,
    E52
    Exercises 11: Sect 11.5: 2
    Sect 12.6.1: : 1,2, 3, 4 Sect 12.6.2: 4
    Recommended Sect 12.6.1: Sect 12.6.2: 4
    MF
    LS
    JW
    BP
    Wed
    11/10 13-15 F1 Lecture 12: Reserve, repetition, summary BD
    Fri
    13/10 10-12 Q22
    Q26
    Q34, M31
    Exercises 12: Repetition and old exams
    Johan will have workshop.
    MF
    LS
    JW
    BP
    Thu
    19/10 10-12 To be announced later on

    Workshop (Räknestuga) in Probability Theory
    MF
    LS
    JW
    Wed
    25/10 08-13 See the relevant web page for further information or this web page Exam
    BD

    Welcome, we hope you will enjoy the course (and learn a lot)!

    Boualem, Martina, Boris, Johan and Lukas


    To course web page

Published by: Boualem Djehiche
Updated:2017-08-23