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 10.3011.30, Room 3732.
 Johan Westerborn email. Office hours: Tuesdays at 14.0015.00, 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 2hour 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, SpringerVerlag 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 KTHid 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 AF, 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 Reexam is scheduled to take place on Tuesday December 19, 2017, 08.0013.00.
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 Reexam 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 
0810 
FR4 (Albanova)

Lecture 1: Sigmafields, Probability space,
Axioms of probability calculus, Some Theorems of Probability calculus. Distribution functions. Chapter 1 in LN.

BD 
Tue

29/08

1517 
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

1315 
D2 
Lecture 2: Multivariate random
variables. Marginal density, Independence, Density of a transformed
random vector, Conditional density, Conditional Expectation.
Chapters 23.5 in LN

BD

Fri

01/09 
1012 
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 
0810 
FR4 (Albanova) 
Lecture 3: The Rule of Double Expectation E(Y) =
E(E(YX)X), Conditional
variance, The Formula Var(Y) = E (Var(YX)) + Var( E(Y  X)) and its applications, Conditional expectation w.r.t. a sigmafield. Chapter 3 in LN .

BD

Tue

05/09 
1517 
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 
1315 
D2 
Lecture 4: Characteristic fuctions Chapter 4.1.  4.4 LN . 
BD

Fri

08/09 
1012 
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 
0810 
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 
1517 
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 
1315 
FR4 (Albanova) 
Lecture 6: Concepts of convergence in probability 6.26.5 LN

BD

Fri

15/09 
1012 
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 
1517 
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 
1315 
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 
1012 
FR4 (Albanova) 
Lecture 8: Multivariate Gaussian variables, LN Chapter 8

BD

Tue

26/09 
1517 
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 
1315 
FR4 (Albanova) 
Lecture 9: Gaussian process, covariance properties. Chapter 9.19.4.

BD

Fri

29/09 
1012 
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 
1517 
E1 
Lecture 10: Wiener process chapter 10.210.4, Wiener integral 10.5.110.5.2 LN
 BD

Wed

04/10 
1315 
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 
1012 
B2 
Lecture 11: Ornstein Uhlenbeck process, chapter 11.2 LN Poisson process 12.2  12.3 LN

BD

Tue 
10/10 
1517 
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 
1315 
F1 
Lecture 12: Reserve, repetition, summary 
BD

Fri

13/10 
1012 
Q22
Q26
Q34, M31

Exercises 12: Repetition and old exams Johan will have workshop.

MF LS JW BP

Thu

19/10 
1012 
To be announced later on

Workshop (Räknestuga) in Probability Theory

MF LS JW

Wed

25/10 
0813 
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
