*Last modified: 2017-01-12*

# FMSF05/MASC01: Probability Theory

**The course schedule can be found at this** link

The course will be given in English.

**Department:**

Mathematical Statistics, Centre for Mathematical Sciences

**Credits:**

7.5 ECTS credits

**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.

**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 2019:** Tatyana Turova

**
Teacher assistant Spring 2019: Vassiliy Goryachkin
**

**
**

### COURSE CONTENT

1. Probability space and random variables.
2. Multivariate random variables. The transformation theorem.
3. Conditional density function. Conditional expectation. Conditional variance.
4. The moment generating function, the characteristic function: definitions, properties. Uniqueness theorem.
5. Sums of a random number of random variables: expected value, variance, moment generating function.
6. Branching process: definition, moment generating function (iteration formula), expected value, variance, probability of ultimate extinction.
7. Order statistics.
8. Multivariate normal distribution. Quadratic forms.
9. Four types of convergence for random variables. Theorems of uniqueness of the limits.
10. Relations between different types of convergence.
11. Borel-Cantelli lemma.
12. The Weak Law of Large Numbers.
13. The Central Limit Theorem.
14. Convergence of the sums of the sequences of random variables.
15. The Poisson process.

### ACTUAL INFORMATION

*
*

NEW: THE EXAMS ARE GRADED. THE RESULTS WILL BE WITH THE ADMINISTRATOR. MEETING FOR QUESTIONS and CHECKING YOUR GRADES:
27/3 12:15-13:00 in MH:227

For 2018:
Exam: (1) Written exam. The student will be asked to solve some problems; (2) the student WILL get a list of
distributions (densities/pmf) relevant to the exam, but no other material is permitted.

#### Reading :

Ch. 1, Ch. 2: § 1,2,3, Ch. 3, Ch. 4: § 1, 3, Ch. 5: § 1, 2, 3, 4, 5,
Ch. 6: § 1, 2, 3, 4, 5,6.

For future remianing lectures:
Ch. 8: § 1, 3, 4, 5,6.

#### Assigned problems

The problems are chosen from the textbook. Please, note that we use the 2nd edition of the book for the numbering of problems but one can use the 1st edition as well!

Exercises class week 4: Ch. 1, no. 1,2,6,7, 12, 13

Exercises class week 5: Ch. 2, no. 1,3,4,11,35, 39

Exercises class week 6: Ch. 3, no. 3, 4, 10 , 13, 18,22, 26, 30, 40, 45, 46

Exercises class week 7: Ch. 4, no. 2, 3, 4, 10, 13, 19

Exercises class week 8: Ch. 5, no. 2, 3, 4, 6, 7, 11

Exercises class week 9: Ch. 5, no. 15, 20, 21, 29.
Ch. 6, no. 2, 5, 7, 11, 15, 20, 28, 30

Exercises class week 10: Ch. 8, no. 1, 5, 6, 9, 11, 12, 15, 18

####

#### Course schedule is to be found in the separate file with the time-table.

The OLD written exams are found
here and
here.

Sample old exam and its solution.

#### EXAMS (written only)

**Exam:** Date: 18 March 2019.
Time: 8-13. Place: Vic 2B-C.

**Resit exam (***omtenta*): Date: ......
Time: 8-13. Place: MA9

**Resit exam 2 (***omtenta*): Date: .....
Time: 08-13. Place: VIC: 2A