Mathematics for statisticians

Subject description

Analysis and linear algebra

Sequences and number series.

Functions (domain, range, continuity, limit).

Derivatives (differentiation rules, geometric interpretation, applications of derivatives).

Integrals (antiderivative, definite integral, applications if integrals).

Function series (Taylor series).

Functions of more variables (domain, range, partial derivatives and their applications, multiple integrals).

Vectors (basic operations, scalar product, vector product, basis of the vector space).

Matrices (basic operations, multiplication, rank, determinant, special kinds of matrices, eigenvalues, eigenvectors, linear transformations, similarity of matrices, square form).

Systems of linear equations (Gauss method).

Probability

Sample spaces, events, probability.

Conditional probability and independence.

Random variables, discrete and continuous distributions.

Expectation, variance, moments.

Joint distributions; distributions of functions of random variables and random vectors.

Conditional distributions, conditional expectations.

Convergence of random variables.

Laws of large numbers.

Convergence in distribution, central limit theorem.

The subject is taught in programs

Objectives and competences

The first part of the course aims to present to the students the fundamental mathematical concepts, methods and principles, which are necessary tools needed for the study of statistics, and to unify the mathematical background of students coming from different first level university programmes.

In the second part of the course the students learn the fundamental probability concepts which are the basis of the statistics and are therefore indispensable for the study of statistics.

Provide the students with computer skills to do mathematical calculations and to make graphical presentations of the obtained results. The development of analytical thinking and careful and precise inference.

Teaching and learning methods

Lectures, laboratory work, consultations, homework, Group analysis, interpretation and solving of statistical problems.

Part of the pedagogical process will be carried out with the help of ICT technologies and the opportunities they offer.

Expected study results

Knowledge and understanding:

Knowledge and understanding of basic concepts of mathematical analysis (including sequences, functions, derivatives, integrals, function series) and linear algebra (including vectors, determinant, matrices, systems of linear equations).

Acquaintance with the basic concepts of probability calculus (including random variables, expectation, variance, moments, joint distributions, conditional distributions, convergence of random variables, laws of large numbers, central limit theorem, estimation of parameters).

The ability to analyze and give mathematical interpretation of fundamental statistical problems.

The ability to use computers to do mathematical calculations and to make graphical presentations of the obtained results.

The ability to apply mathematical concepts in real world problems.

Basic sources and literature

J. A. Rice: Mathematical Statistics and Data Analysis, Thomson Learning, 2006.
S. Ross: A first course in probability, Pearson education, 2006.
G. Dolinar, Matematika 1, Založba FE in FRI, 2010.
P. Šemrl, Osnove višje matematike 1, DMFA-založništvo, 2009.
R. Jamnik: Verjetnostni račun, DMFA, 1987.
G. B. Tomas, M. D. Weir, J. Hass, F. R. Giordano: Thomas' Calculus, Pearson Education, 2005.
R. W. Hamming: Methods of Mathematics Applied to Calculus, Probability, and Statistics, Dover Publications, 2004.
D. C. Lay: Linear algebra and its applications, Pearson education, 2003.