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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 studnet 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 motivation 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 :Pierre Nyquist . homepage and contact information
Course literature.:
-
(1) Gut Allan Gut: An Intermediate Course in Probability. Second Edition. Springer-Verlag, 2009.
The book can be found at the Kårbokhandel at the adress Osquars Backe 21 at the campus
(2) LN =Supplemental Lecture Notes (2012 Edition, forthcoming) .
This compendium can be bought at the student expedition of the mathematics department.
Examination:
There will be a written examination on Wednesday 17th of October, 2012, 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 respective Chapters in Gut and
from LN
(TK=Timo Koski, PN =Pierre Nyquist )
The addresses of the lecture halls and guiding instructions are found by clicking on the Hall links below
| Day |
Date |
Time |
Hall |
Topic |
Lecturer |
| Tue |
30/08 |
15-17 |
E2
|
Lecture 1:Sigma-fields, Probability space,
Axioms of probability calculus, Some Theorems of Probability calculus. Distribution functions. Multivariate random
variables. Chapter 1 in Gut, Chapter 1 in LN.
|
TK |
| Wed |
31/08
|
10-12 |
V2 |
Lecture 2:Multivariate random
variables. Marginal density, Independence, Density of a transformed
random vector, Conditional density, Conditional Expectation.
Chapter 2.1-2.2 Gut,
Chapter 2 in LN
|
TK
|
Thu
|
01/09
|
08-10 |
E2 |
Exercises 1: Gut Chapter 1: 13, 20, 23,
Additional recommended: Gut Chapter 1. 9, 29,39, 41
|
PN
|
Fri
|
02/09 |
08-10 |
V3
|
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 2.2-2.4 in Gut, Chapter 2 in LN .
|
TK
|
Mo
|
05/09 |
10-12 |
M3 |
Lecture 4:Probability generating function, examples and properties. Chapter 3.2 Gut.
Moment generating function, examples and properties Chapter 3.3 Gut
|
TK
|
Wed
|
07/09 |
10-12 |
H1 |
Lecture 5: Characteristic function, examples and properties, table of formulas, characteristic function of a normal distribution.
Sums of a random number of random variables Chapter 3.4, 3.6 in Gut. |
TK
|
Thu
|
08/09 |
08-10 |
E2 |
Exercises 2: Gut Chapter 1: 14,
Gut Chapter 2:1,4, 36
Additional recommended Gut chapter 2:9, 30 |
PN
|
Fri
|
09/09 |
15-17 |
E3 |
Exercises3: Gut Chapter 2:5, 7
Gut Chapter 3:1,2,3
LN section 2.7.2: 1,2,3,4 |
PN
|
Mo
|
12/09 |
13-15 |
V2 |
Lecture 6: Concepts of convergence in probability theory, Chapter 6.1- 4 in Gut. The proof of theorem 6.2.1 of uniqueness of limits is skipped. Proof of property IV (as indexed in Gut) is included. Chapter 3 in LN.
|
TK
|
Tue
|
13/09 |
08-10 |
E3 |
Exercises4: Gut Chapter 3:6, 13, 21, 32
LN section 3.6.3: 1,2,3
Additional recommended Gut chapter 3:8, 18, 22, 23, 24
|
PN
|
Fri
|
14/09 |
10-12 |
B1 |
Lecture 7: Concepts of convergence in probability theory: convergence via transforms
(Chapter 6.4-6 in Gut.). Convergence of sums and functions of
random variables. Borel Cantelli, strong law of large numbers.
LN Chapter 3
|
TK
|
Thu
|
15/09 |
08-10 |
E2 |
Exercises 5: Gut Chapter 6: 2, 16, 23, 25
recommended: Gut Chapter 6: 5, 8
| PN
|
Mon
|
19/09 |
13-15 |
V3 |
Lecture 8: Multivariate Gaussian variables, Gut Chapter 5,
LN Chapter 4
|
TK
|
Tu
|
20/09 |
08-10 |
E2
|
Exercises 6: Gut Chapter 6:6, 9,10, 11, 17
recommended: Gut Chapter 6:6, 8, 13, 19, 49 |
PN
|
| We |
21/09 |
10-12 |
H1
|
Exercises 7 : Gut Chapter 5:4,5, 20, 37
|
PN
|
Mon
|
26/09 |
13-15 |
V3 |
Lecture 9: Gaussian process, covariance properties, Wiener process, independent increments. LN Chapter 5 .
|
TK
|
Th
|
29/09 |
10-12 |
V3
|
Exercises 8: Gut Chapter 5: 2, 10, 9
recommended: Gut Chapter 5: 31
|
PN
|
Fri
|
30/09 |
10-12 |
V3
|
Lecture 10: More on Gaussian Processes. LN Chapter 5
| TK
|
| Mon |
03/10 |
13-15 |
V2
|
Lecture 11: Poisson process, definition (def.I in Gut), independent Exp(1/lambda) duration times (tau) , occurrence times T, start from scratch. Gut Chapter 8: 8.1.8.2
|
TK
|
Thu
|
06/10 |
10-12 |
M3 |
Exercises 9: LN section 4.5.2: 7, 9
LN section 5.7.2: 2
LN section 5.7.4: 3, 4
LN 5 section 5.7.5: 1,2
LN section 5.7.6: 7 |
PN
|
Fri
|
07/10 |
15-17 |
E3
|
Exercises 10: LN section 6.7.2: 1, 2,3, 5, 6 (a), 8, 9 |
PN
|
| Mon |
10/10 |
13-15 |
V2
|
Exercises 11: Wiener integrals, LN section 6.7.3: 1,2
Poisson process Gut Chapter 8: 1, 3, 8
|
PN
|
Thu
|
13/10 |
10-12 |
V3 |
Lecture 12: Reserve, repetition, summary |
TK
|
Fr
|
14/10 |
10-12 |
V3
|
Exercises 12: Repetition and old exams
|
PN
|
Welcome, we hope you will enjoy the course!
Timo and Pierre
To course
web page
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