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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 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.
Prerequisities:
- SF 1901 or equivalent course of the type '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) 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
Teaching assistant : Gaultier Lambert
Course literature.:
- T.Koski Lecture Notes: Probability and Random Processes Edition 2013 LN pdf
This compendium can be bought at the student expedition of the mathematics department.
Examination:
There will be a written examination on Tuesday 29th of October, 2013, 08.00-
13.00.
Registration for the written examination via "mina sidor"/"my pages"
is required.
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.
Homeworks:
There will no be homework assignments.
Preliminary plan Exercises are from the Sections of Problems of LN. For example: Section 1.12.2 1 is the first exerecise in section 1.12.2 in LN.
(TK=Timo Koski, GL= Gaultier Lambert )
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 |
02/09 |
08-10 |
V2
|
Lecture 1:Sigma-fields, Probability space,
Axioms of probability calculus, Some Theorems of Probability calculus. Distribution functions. Chapter 1 in LN.
|
TK |
| Wed |
04/09
|
08-10 |
K1 |
Lecture 2:Multivariate random
variables. Marginal density, Independence, Density of a transformed
random vector, Conditional density, Conditional Expectation.
Chapters 2-3.5
|
TK
|
Thu
|
05/09
|
08-10 |
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, Random parameters, Conditional expectation w.r.t. a sigma-field. Chapter 3 in LN .
|
TK
|
Fri
|
06/09 |
08-10 |
K1 |
Lecture 4: Characteristic fuctions Chapter 4.1. - 4.4 LN .
|
TK
|
Mon
|
09/09 |
08-10 |
D2
| Exercises 1: Sect 1.12.2: 1,12,Sect 1.12.3: 6, 9
Recommended: Sect 1.12.2: 6,7,9
|
GL
|
Wed
|
11/09 |
15-17 |
D2 |
Lecture 5: More on characteristic function chapter 4.4
Generating functions, Sums of a random number of random variables Chapter 5.2- 5.5, 5.7 . |
TK
|
Thu
|
12/09 |
08-10 |
K1 |
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
|
Fri
|
13/09 |
15-17 |
D2 |
Lecture 6: Concepts of convergence in probability 6.2 - 6.5 LN
|
TK
|
Mon
|
16/09 |
08-10 |
V2 |
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
|
Wed
|
18/09 |
13-15 |
K1 |
Exercises 4: Sect 3.8.5: 1,3,4, 6(a), 7
Recommended Sect 3.8.5: 2,5,8
|
GL
|
Thu
|
19/09 |
13-15 |
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
|
Fri
|
20/09 |
15-17 |
K1 |
Exercises 5: Sect 4.7.1: 3,6, 7, 12 Sect 4.7.2: 1
Recommended: Sect 4.7.1: 2,5,8
| GL
|
Mon
|
23/09 |
08-10 |
V2 |
Lecture 8: Multivariate Gaussian variables,
LN Chapter 8
|
TK
|
Tue
|
24/09 |
13-15 |
V1
|
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
|
| Wed |
25/09 |
15-17 |
D2
|
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
|
Wed
|
02/10 |
14-16 |
M2 |
Lecture 9: Gaussian process, covariance properties. Chapter 9.1-9.4.
|
TK
|
Fri
|
04/10 |
10-12 |
D2
|
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
|
Tue
|
08/10 |
15-17 |
K1
|
Lecture 10: Wiener process chapter 10.2-10.4, Wiener integral 10.5.1-10.5.2 LN
| TK
|
| Wed |
09/10 |
15-17 |
D2
|
Lecture 11: Ornstein Uhlenbeck process, chapter 11.2 LN Poisson process 12.2 - 12.3 LN
|
TK
|
Thu
|
10/10 |
15-17 |
V1 |
Exercises 9: Sect 9.7.2: 2, Sect 9.7.4: 4,
Sect 9.7.5: 1,2 Sect 9.7.6: 7 |
GL
|
Mon
|
14/10 |
08-10 |
V2
|
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
|
| Tue |
15/10 |
15-17 |
D2
|
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
|
Wed
|
16/10 |
15-17 |
D2 |
Lecture 12: Reserve, repetition, summary |
TK
|
Fri
|
18/10 |
10-12 |
D2
|
Exercises 12: Repetition and old exams
|
GL
|
Tue
|
29/10 |
08-13 |
Rooms |
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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