Discrete and Stochastic Mathematical Models

Entry requirements: Basic knowledge in probability theory and mathematical statistics

Credits: 4

Course: Core

Language of the course: Russian

Objectives

The course mainly aims at understanding the key aspects of discrete and stochastic modelling, with a focus on examples and application in such areas as production, service and computations. Students will be able to deepen their knowledge of probability theory, and develop their skills in multivariate statistical analysis. The course covers the description of key modelling principles, popular statistical models, ways of input data generation and analysis of modelling process input data.

Contents

Discrete modelling:
The course is not restricted to a specific programming language, although most of the examples will be in Python. Students will learn how to select appropriate modelling means, will understand the advantages and disadvantages of certain models, estimate models' productivity, and know the approaches to realizing scientific projects connected to modelling.

  • Understanding basic principles of discrete modelling
  • Modelling random-in systems
  • Developing discrete-event models
  • Using statistical methods to generate input data for modelling
  • Realizing queue models, models of in storage stocks and supply systems
  • Analyzing modelling results
  • Estimating models' productivity

Stochastic modelling:

  • Stochastic models for one-dimensional random variable. Basic concepts of distribution law, distribution function, distribution density (and its properties). Estimation methods of distribution parameters. Probabilistic interval, confidence interval, tolerance interval.
  • Stochastic models for multidimensional random variable. Regression analysis. Correlation analysis. Principal component analysis. Multidimensional interval estimations of distribution parameters, and regression.
  • Stochastic models for one-dimensional and multidimensional random processes, and fields. Random function and its connection to time processes and fields. Stationary state definition in narrow and road sense. Ergodic processes. Periodically correlated random processes. Gaussian processes. Markov processes. Dynamic system model. Regression models for random processes. Correlation analysis of random processes. Autoregression model. Wold's model. Rice model. Autoregressive-moving average model. Trend modelling. Auto- and cross-spectral
    analysis. Fourier transform Wiener-Khinchin theorem.

Format

Lectures and labs

Assessment

Attendance is mandatory. Students cannot miss more than one class. The final grade is 40% individual work and 60% examination.