Last modified: 2 April 2013
MASC01: Probability Theory
Course scheme:
See below. The course will be given in English.
Department:
Mathematical Statistics, Centre for Mathematical Sciences
Credits:
7.5 ECTS credits (5 poäng)
Requirements:
45 ECTS credits in Mathematics and a course corresponding to MASA01
Mathematical
Statistics, General Course.
Time period:
The first part of the spring semester. The first lecture is on Monday
2013-1-21, 10:15-12:00 in MH:362D.
Course literature:
A. Gut, "An
Intermediate Course in Probability Theory", 2nd edition,
Springer 2009.
(The book is also available as e-book
through the Lund University Libraries.)
Lecturer Spring 2013:
Stanislav
Volkov
PRELIMINARY COURSE CONTENT (and old detailed
version)
1. The transformation theorem.
2. Conditional density function. Conditional expectation.
Conditional variance.
3. The moment generating function, the characteristic function:
definitions, properties. Uniqueness theorem.
4. Sums of a random number of random variables: expected value,
variance, moment generating function.
5. Branching process: definition, moment generating function
(iteration formula), expected value, variance, probability of ultimate
extinction.
6. Order statistics: joint distribution.
7. Multivariate normal distribution: characteristic function,
density.
8. Multivariate normal distribution: conditional distribution, incl. general case (not in the
textbook).
9. Multivariate normal distribution: independence, Daly's theorem.
10. Four types of convergence for the random variables. Theorems
of
uniqueness of the limits.
11. Relations between different types of convergence. Proof of all relations (incl. a.s.
implies in probability)
12. Borel-Cantelli lemma.
13. The Weak Law of Large Numbers.
14. The Central Limit Theorem.
15. Convergence of the sums of the sequences of random variables.
16. The Poisson process: two definitions and their equivalence.
17. The Poisson process: superposition and thinning.
18. The distribution of the consecutive moments of occurences of
the Poisson process.
19. The joint distribution of the consecutive moments of
occurences of the Poisson process given the value of the process at
time 1.
ACTUAL INFORMATION
We will have 3 lectures during the first week (also on Tuesday 22
January instead of exercises)
Exam: (1) theoretical questions WILL BE on the written exam (e.g.
proofs of theorems we had in class); (2) you WILL get a list of
distributions (densities/pmf) relevant to the exam.
Problems for exercises session on
January 29
Problems for exercises session on February
5
Problems for exercises session on February
12
Problems for exercises session on February
19
Problems for exercises session on February
26
Problems for exercises session on March 5
Sample old exam and its solution.
Course schedule from 21 January till 7 March 2013:
Monday
|
10:15-12:00
|
Lecture
|
MH:362D
|
Tuesday
|
10:15-12:00 |
Exercises
|
E:3316
|
Thursday
|
10:15-12:00 |
Lecture
|
MH:B
|
EXAMS (will be writtten only)
Exam: 2013-03-14. Time: 8-13. Place: Sparta B
Resit exam (omtenta): 2013-04-06. Time: 8-13. Place:
VIC 3D
Resit exam 2 (omtenta): 2013-08-23. Time: 8-13 (?).
Place: Sparta B(?)