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PřF:M3121 Probability and Statistics I - Course Information

## M3121 Probability and Statistics I

**Faculty of Science**

autumn 2021

**Extent and Intensity**- 2/2/0. 4 credit(s) (fasci plus compl plus > 4). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).

Taught in person. **Teacher(s)**- doc. Mgr. Jan Koláček, Ph.D. (lecturer)

Mgr. Ondřej Pokora, Ph.D. (seminar tutor)

Mgr. Jan Ševčík (seminar tutor) **Guaranteed by**- doc. Mgr. Jan Koláček, Ph.D.

Department of Mathematics and Statistics - Departments - Faculty of Science

Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science **Timetable**- Thu 14:00–15:50 A,01026
- Timetable of Seminar Groups:

*O. Pokora*

M3121/02: Fri 8:00–9:50 M5,01013,*J. Ševčík*

M3121/03: Mon 16:00–17:50 M2,01021,*J. Ševčík* **Prerequisites**-
**M2100**Mathematical Analysis II ||**FI:MB001**Calculus II ||**FI:MB102**Calculus ||**M2B02**Calculus II ||**FI:MB202**Calculus B || NOW (**MIN301**Mathematics III ) ||**MIN301**Mathematics III ||**FI:MB152**Calculus

Differential and integral calculus of functions of n real variables. Basic knowledge of linear algebra. **Course Enrolment Limitations**- The course is also offered to the students of the fields other than those the course is directly associated with.
**fields of study / plans the course is directly associated with**- Applied Mathematics for Multi-Branches Study (programme PřF, B-MA)
- Financial and Insurance Mathematics (programme PřF, B-MA)
- Mathematical Biology (programme PřF, B-EXB)
- Mathematics (programme PřF, B-MA)
- Statistics and Data Analysis (programme PřF, B-MA)

**Course objectives**- The basic course of probability and mathematical statistics and introductory course for other theoretically oriented and applied stochastic subjects. The content of the course is axiomatical approach to probability theory, random variables and random vectors, probability distributions and characteristics of the distribution. The last part of the course is devoted to the laws of large numbers and to the central limit theorem. On the completion of this course, the student is expected to obtain sufficient mastery of basic probability theory to be able to study topics on statistical inference.
**Learning outcomes**- On the completion of this course, the student is expected to obtain sufficient mastery of basic probability theory; to define a random variable and a random vector; to characterize basic types of probability distribution; to model probability distribution in practical examples; be able to study topics on statistical inference.
**Syllabus**- Elements of probability: axiomatic definition of probability, probability space, conditional probability, independence. Random variables: borel functions, definition of random variable, distribution function, discrete and continuous probability distributions, probability and density function, examples of discrete and continuous random variables, distribution of transformed random variables. Random vectors: joint distributions, independence, examples of multivariate distributions (multivariate normal and multinomial distributions), distribution of the sum and ratio of random variables, distributions derived from normal distribution, marginal distributions. Characteristics: expectation, variance, covariance, moments and their properties, covariance and correlation matrices, characteristic function of random vector. Limit theorems: Borel and Cantelli theorem, Cebyshev's inequality, Laws of large numbers, central limit theorem.

**Literature**- Ash, R.B. and Doléans-Dade C.A. Probability and measure theory. Academic Press. San Diego.2000
- MICHÁLEK, Jaroslav.
*Úvod do teorie pravděpodobnosti a matematické statistiky*. 1. vyd. Praha: Státní pedagogické nakladatelství, 1984. 204 s. info - Karr, A.F. Probability. Springer. 1992
- Dupač, V. a Hušková, M.: Pravděpodobnost a matematická statistika. Karolinum. Praha 1999.

**Teaching methods**- Lectures: theoretical explanation with practical examples Excercises: solving problems for acquirement of basic concepts, solving theoretical problems, solving simple tasks and also complicated problems
**Assessment methods**- Lecture with exercises. Active work in exercises. 2 written tests. Each test consists of 4-5 examples and is for 20 points. 50% of points is needed to pass fulfilling requirements. Examination consists of two parts: written and oral. Written part consists of 4 theoretical questions, each for 10 points. The final result is corrected by the oral part. Final grade: A: 37 - 40 points B: 32 - 36 points C: 27 - 31 points D: 22 - 26 points E: 18 - 21 points F: 0 - 17 points
**Language of instruction**- Czech
**Follow-Up Courses****Further comments (probably available only in Czech)**- Study Materials

Information on completion of the course: Ukončení předmětu zápočtem je možné pouze pro studenty Matematické biologie.

The course is taught annually. **Listed among pre-requisites of other courses**

- Enrolment Statistics (recent)

- Permalink: https://is.muni.cz/course/sci/autumn2021/M3121