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:
M3121/01: Thu 10:00–11:50 M2,01021, 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
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
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020.
  • Enrolment Statistics (recent)
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