Course detail

# Statistics and Probability

FIT-MSPAcad. year: 2019/2020

Summary

of elementary concepts from probability theory and mathematical statistics.

Limit theorems and their applications. Parameter estimate methods and their

properties. Scattering analysis including post hoc analysis. Distribution

tests, tests of good compliance, regression analysis, regression model

diagnostics, non-parametric methods, categorical data analysis. Markov

decision-making processes and their analysis, randomized algorithms.

Supervisor

Learning outcomes of the course unit

Students

will extend their knowledge of probability and statistics, especially in the

following areas:

- Parameter

estimates for a specific distribution - simultaneous

testing of multiple parameters - hypothesis

testing on distributions - regression

analysis including regression modeling - nonparametric

methods - Markov

processes

Prerequisites

Foundations of differential and integral calculus.

Foundations
of descriptive statistics, probability theory and mathematical statistics.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Anděl, Jiří. Základy matematické statistiky. 3., opr. vyd. Praha: Matfyzpress, 2011. ISBN 978-80-7378-001-2.

Meloun M., Militký J.: Statistické zpracování experimentálních dat (nakladatelství PLUS, 1994).

FELLER, W.: An Introduction to Probability Theory and its Applications. J. Wiley, New York 1957. ISBN 99-00-00147-X

Hogg, V.R., McKean J.W. and Craig A.T. Introduction to Mathematical Statistics. Seventh Edition, 2012. Macmillan Publishing Co., INC. New York. ISBN-13: 978-0321795434 2013

Zvára K.. Regresní analýza, Academia, Praha, 1989

D. P. Bertsekas, J. N. Tsitsiklis. Introduction to Probability, Athena, 2008. Scientific

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Three tests will be written during the semester - 3rd, 6th and 11th week. The exact term will be specified by the lecturer. The test duration is 60 minutes. The evaluation of each test is 0-10 points.

Projected evaluated 0-10 points.

**Final written exam - 60 points**

Exam prerequisites:

The credit will be awarded to the one who meets the attendance conditions and whose total test scores will reach at least 15 points and project score at least 5 points. The points earned in the exercise are transferred to the exam.

Language of instruction

Czech

Work placements

Not applicable.

Aims

Introduction

of further concepts, methods and algorithms of probability theory, descriptive

and mathematical statistics. Development of probability and statistical topics

from previous courses. Formation of a stochastic way of thinking leading to

formulation of mathematical models with emphasis on information fields.

Specification of controlled education, way of implementation and compensation for absences

Participation

in lectures in this subject is not controlled

Participation

in the exercises is compulsory. During the semester two abstentions

are tolerated. Replacement of missed lessons is determined by the leading exercises.

Classification of course in study plans

- Programme MITAI Master's
specialization NADE , 1. year of study, winter semester, 6 credits, compulsory

specialization NBIO , 1. year of study, winter semester, 6 credits, compulsory

specialization NGRI , 1. year of study, winter semester, 6 credits, compulsory

specialization NNET , 1. year of study, winter semester, 6 credits, compulsory

specialization NVIZ , 1. year of study, winter semester, 6 credits, compulsory

specialization NCPS , 1. year of study, winter semester, 6 credits, compulsory

specialization NSEC , 1. year of study, winter semester, 6 credits, compulsory

specialization NEMB , 1. year of study, winter semester, 6 credits, compulsory

specialization NHPC , 1. year of study, winter semester, 6 credits, compulsory

specialization NISD , 1. year of study, winter semester, 6 credits, compulsory

specialization NIDE , 1. year of study, winter semester, 6 credits, compulsory

specialization NISY , 1. year of study, winter semester, 6 credits, compulsory

specialization NMAL , 1. year of study, winter semester, 6 credits, compulsory

specialization NMAT , 1. year of study, winter semester, 6 credits, compulsory

specialization NSEN , 1. year of study, winter semester, 6 credits, compulsory

specialization NVER , 1. year of study, winter semester, 6 credits, compulsory

specialization NSPE , 1. year of study, winter semester, 6 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

- Summary

of basic theory of probability: axiomatic definition of probability,

conditioned probability,

dependent and independent events, Bayes formula. - Summary

of discrete and continuous random variables: probability, probability

distribution density, distribution function and their properties, functional

and numerical characteristics of random variable, basic discrete and continuous distributions. - Discrete

and continuous random vector (distribution functions, characteristics,

multidimensional distribution). Transformation of random variables.

Multidimensional normal distribution. - Limit

theorems and their use (Markov and Chebyshev Inequalities, Convergence, Law of

Large Numbers, Central Limit Theorem) - Parameter

estimation. Unbiased and consistent estimates. Method of moments, Maximum

likelihood method, Bayesian approach - parameter estimates. - Analysis

of variance (simple sorting, ANOVA). Multiple comparison (Scheffy and Tukey

methods). - Testing

statistical hypotheses on distributions. Goodness of fit tests. - Regression

analysis. Creating a regression model. Test hypotheses on regression model

parameters. Comparison of regression models. Diagnostics. - Project assignment, demonstration of programs and tools for solving statistical problems.
- Nonparametric

methods for testing statistical hypotheses. - Analysis

of categorical data: contingency table, chi-square test, Fisher test. - Markov processes, Markov decision processes, and their analysis and applications.
- Introduction

to randomized algorithms and their use (Monte Carlo, Las Vegas, applications).

Fundamentals seminar

34 hours, compulsory

Teacher / Lecturer

Syllabus

- Sets, relations, and their basic properties.
- Propositional calculus and its formal system.
- Repetition of the basic probability theory and statistics.
- Important distribution and their use in Limit theorems.
- Parameter estimate: properties, methods
- Analysis of variance (simple sorting, ANOVA), post hos analysis.
- Testing statistical hypotheses on distributions. Goodness of fit tests.
- Regression analysis. Creating a regression model. Test hypotheses on regression model parameters.
- Regression analysis. Test hypotheses on regression model parameters. Diagnostics.
- Nonparametric methods for testing statistical hypotheses.
- Analysis of categorical data: contingency table, chi-square test.
- Application and analysis of Markov processes and Markov decision processes.
- Introduction to randomized algorithms

**Demo exercise focusing on algebra and logic (only the first two weeks -- 4-times 2 hours):**

- Sets, Cartesian product, relations, and functions. Properties and types of relations and functions. Congruence.
- Basic algebraic structures (group, Boolean algebra, lattice, field). Homomorfism.
- Propositional calculus. Syntax and semantics. Formal system for propositional calculus. Posts completeness theorem.
- Predicate logic. Syntax and semantics. Formal system for predicate logic. Gödels completeness theorem. Gödels incompleteness theorem.

Project

5 hours, compulsory

Teacher / Lecturer

Syllabus

- Usage of tools for solving statistical problems (data processing and interpretation).