## Course data

Lectures: Tue & Thu 12:15–14:00 MaD380 (8.1.2013 – 28.2.2013)

Exercises: Tue 10:15–12:00 MaD380 (15.1.2013 – 5.3.2013).

Instructors: Lasse Leskelä and Mikko Kuronen

Course data in Korppi: https://korppi.jyu.fi/kotka/r.jsp?course=134915

### Contents

• Basic concepts of probability: sample space, event space, probability measure
• Independent and almost sure events
• Random variable and its distribution on a general sample space
• Distribution function of a real random variable
• Expectation of a random variable and its basic properties
• Independent random variables and product measure
• Joint distribution and marginal distributions of a random vector

### Learning outcomes

After completing the course the participant:

• Can discuss probabilities in exact mathematical terms
• Can construct a mathematical model for a simple random phenomenon
• Can compute the expectation of a random variable on a general sample space
• Can analyze independent random variables using a product measure

### Literature

The course contents approximately correspond to Chapters 1–12 in the book:

Jean Jacod & Philip Protter.
Probability Essentials.
Springer 2005.
or Chapters 1–8 in the book:
David Williams.
Probability with Martingales.
Cambridge University Press 1991.

### Handouts and solutions

There will be no separate lectures notes distributed, except possibly for selected topics which are not covered in the aforementioned excellent textbooks. The students are encouraged to write down and distribute their own notes with each other. The same goes for exercise solutions. Because written solutions to the exercises shall not be systematically delivered, it is a good idea to compare and share written solutions with other students.

### Language

The classroom language will be Finnish by default. The language may also be switched to English if a sufficiently great majority of the participants so wishes.

### Evaluation

The course is completed by passing a written exam. To pass the course, you must obtain at least 12 points from the final exam (max 24). In this case the course grade is obtained from the formula

```  g = min(floor(max(x+b-9,0)/3),5)
```
where x is the number of points from the final exam (max 24), and the exercise bonus
```  b = min(floor(k/8),4)
```
is computed from the total number of solved exercises k (max 40). Eight completed exercises thus correspond to one bonus point, up to four bonus points.

The weekly exercises are not compulsory, but they are warmly recommended. You can mark an exercise as solved if you think that you understand the exercise well enough to present it to the others — the solution does not need to be correct. You are recommended to solve the exercise problem together with your fellow students.

To be announced later.

The major part of the workload consists of independent study (solving the exercises, finding out how things work, etc.). Passing the course successfully requires allocating sufficient weekly time for independent study.

```Attending lectures       (8 x 4 h) 32 h
Attending exercises      (8 x 2 h) 16 h
Solving exercises        (8 x 6 h) 48 h
Weekly independent study (8 x 3 h) 24 h
Preparing for the exam             11 h
Attending the exam                  4 h
---------------------------------------
Total                             135 h
```
1 cr amounts to 27 h total work -> 5 cr equals 135 h of work.

### Prerequisites

As a prerequisite, the participants are assumed familiarity with the elementary calculus on the level of MATA111 Calculus 1 and MATA112 Calculus 2. Useful, but not mandatory, additional background includes elements of probability such as TILA120 Probability A and TILA120 Probability B.