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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 : Timo Koski, Prof. homepage and contact information
The course web page. http://www.math.kth.se/matstat/gru/sf2940/
Teaching assistants :
- Gaultier Lambert
email
- Davit Karagulyan email
- Harald Lang
- The teaching assistants will each have an office hour open for consultation (1h per week). The hours will be announced later.
Exercise groups
- Gaultier Lambert --- The participants with first letter of last name
- Davit Karagulyan --- The participants with first letter of last name
- Harald Lang ---
Workshop There will be a 2-hour workshop (räknestuga) on the 16th of October The date is preliminary (see the Plan below)
Course literature:
- T.Koski Lecture Notes: Probability and Random Processes Edition 2014 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.
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 sante15-2.
Examination:
There will be a written examination on Monday 28th of October, 2015, 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 and the Collection of Formulas.
Each student must bring her/his own calculator to the examination. The department will distribute the "Formulas and survey" and it is not allowed to use your own copy.
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.
Homework:
There will be two sets of elective homework assignments that will give bonus points (scale forthcoming) in the written exam on the 28th of October. The deadllines of submission will be announced later.
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.
(TK=Timo Koski, GL= Gaultier Lambert, DK=Davit Karagulyan, HL=Harald Lang )
The addresses of the lecture halls and guiding instructions are found by clicking on the Hall links below
| Day |
Date |
Time |
Hall |
Topic |
Lecturer |
| Mon |
31/08 |
15-17 |
K1
|
Lecture 1:Sigma-fields, Probability space,
Axioms of probability calculus, Some Theorems of Probability calculus. Distribution functions. Chapter 1 in LN.
|
TK |
| Tue |
01/09
|
13-15 |
K1 |
Lecture 2:Multivariate random
variables. Marginal density, Independence, Density of a transformed
random vector, Conditional density, Conditional Expectation.
Chapters 2-3.5 in LN
|
TK
|
Thu
|
03/09
|
10-12 |
E31
K1 |
Exercises 1: Sect 1.12.2: 1,12,Sect 1.12.3: 6, 9
Recommended: Sect 1.12.2: 6,7,9
|
GL
DK
|
Fri
|
04/09 |
10-12 |
K1 |
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 .
|
TK
|
Mo
|
07/09 |
15-17 |
K1 |
Lecture 4: Characteristic fuctions Chapter 4.1. - 4.4 LN .
|
TK
|
Tue
|
08/09 |
13-15 |
K1
K51
L22
|
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 |
GL
DK
HL
|
Thu
|
10/09 |
10-12 |
E36
E52
K1
|
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
|
GL
DK
HL
|
Fri
|
11/09 |
10-12 |
B2 |
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. |
TK
|
Mon
|
14/09 |
15-17 |
Q2
Q34
V3
|
Exercises 4: Sect 3.8.5: 1,3,4, 6(a), 7
Recommended Sect 3.8.5: 2,5,8
|
GL
DK
HL
|
Tue
|
15/09 |
13-15 |
M2 |
Lecture 6: Concepts of convergence in probability 6.2 - 6.5 LN
|
TK
|
Thu
|
17/09 |
10-12 |
E3
Q17
Q2
|
Exercises 5: Sect 4.7.1: 3,6, 7, 12 Sect 4.7.2: 1
Recommended: Sect 4.7.1: 2,5,8
| GL
DK
HL
|
Fri
| 18/09 |
10-12 |
K1 |
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
|
TK
|
Tue
|
22/09 |
13-15 |
K1
V01
V35
|
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 |
GL
DK
HL
|
Thu
|
24/09 |
10-12 |
K1 |
Lecture 8: Multivariate Gaussian variables,
LN Chapter 8
|
TK
|
Fri |
25/09 |
10-12 |
E3
E31
Q2
|
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: 7,8,9,12
|
GL
DK
HL
|
Mon
|
28/09 |
15-17 |
F2 |
Lecture 9: Gaussian process, covariance properties. Chapter 9.1-9.4.
|
TK
|
Tue
|
29/09 |
13-15 |
B3
E3
E53
|
Exercises 8: Sect 6.8.1: 13, Sect 8.5.1:
8,10, 13, 15, 17
Recommended: Sect 8.5.1: 6,14,16
|
GL
DK
HL
|
Thu
|
01/10 |
10-12 |
F2 |
Lecture 10: Wiener process chapter 10.2-10.4, Wiener integral 10.5.1-10.5.2 LN
| TK
|
Mon
|
05/10 |
15-17 |
B1
V21
V3
|
Exercises 9: Sect 9.7.2: 2, Sect 9.7.4: 4,
Sect 9.7.5: 1,2 Sect 9.7.6: 7 |
GL
DK
HL
|
Tue
|
06/10 |
13-15 |
F2 |
Lecture 11: Ornstein Uhlenbeck process, chapter 11.2 LN Poisson process 12.2 - 12.3 LN
|
TK
|
Fri
|
09/10 |
10-12 |
D2
FB53
K51
|
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 |
GL
DK
HL
|
Tue
|
13/10 |
13-15 |
M2 |
Lecture 12: Reserve, repetition, summary |
TK
|
Thu |
15/10 |
10-12 |
D3
E3
E31
|
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
|
GL
DK
HL
|
Fri
|
16/10 |
10-12 |
M2
M31
M33
|
Exercises 12: Repetition and old exams
|
GL
DK
HL
|
Thu
|
22/10 |
10-12 |
L51
L52
|
Workshop (Räknestuga) in Probability Theory
|
GL
DK
|
Wed
|
28/10 |
08-13 |
D1, D3, D32, D33, D34, D35, D41, D42, E1, E3, E31, E32, E33, E34, E35, E36, E51, E52, E53, K53, L21, L22, L31, L41, L42, L43, L44, L51, L52, M37 |
Exam
|
TK
|
Welcome, we hope you will enjoy the course (and learn a lot)!
Timo and Gaultier
To course
web page
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