Theory of probability and mathematical statistics
Description: The course explores the fundamentals of probability theory and statistics for application in data analysis and decision-making under uncertainty. It covers probabilistic models, random variables, probability distributions (including normal and binomial), as well as statistical analysis methods such as parameter estimation, hypothesis testing, correlation, and regression. Students will learn to apply probabilistic and statistical techniques to interpret data, estimate parameters, and test hypotheses, enabling them to make informed decisions based on the analysis of uncertain information.
Amount of credits: 5
Пререквизиты:
- Mathematical Analysis 2
Course Workload:
| Types of classes | hours |
|---|---|
| Lectures | 15 |
| Practical works | 30 |
| Laboratory works | |
| SAWTG (Student Autonomous Work under Teacher Guidance) | 30 |
| SAW (Student autonomous work) | 75 |
| Form of final control | Exam |
| Final assessment method |
Component: University component
Cycle: Base disciplines
Goal
- Study of basic concepts and mathematical methods for solving practical problems in probability theory and mathematical statistics
Objective
- -formation of concepts about the theory of probability and mathematical statistics, features of application in the process of solving professional problems;
- -mastering the skills of formulating mathematical statements of problems, finding a suitable method for solving them
Learning outcome: knowledge and understanding
- Knows the method of collecting and processing information; relevant sources of information in the field of professional activity; system analysis method
- Point estimation of parameters and determination of the confidence interval, the main methods of statistical processing
Learning outcome: applying knowledge and understanding
- knows how to use mathematical methods in technical applications, calculate the basic numerical characteristics of random variables, solve the main problems of mathematical statistics; solve typical calculation problems
- owns methods of mathematical analysis and modeling; methods for solving problems of analysis and calculation of the characteristics of physical systems, basic methods for processing experimental data, methods for working with applied software products
Learning outcome: formation of judgments
- Analyzes the effectiveness of the obtained model, applying mathematical methods and has an idea about mathematical models and methods for solving applied problems from various fields of natural science.
Learning outcome: communicative abilities
- Able to solve applied problems in a team using mathematical methods, to correctly defend his point of view, to propose new solutions
Learning outcome: learning skills or learning abilities
- Strive for professional and personal growth by mastering the techniques and skills for solving specific problems from different areas of the discipline, helping to further solve engineering, production and scientific problems
Teaching methods
Information and communication technology;
Technology for the development of critical thinking;
Integrated learning technology;
Traditional technologies (lectures, practical classes)
Assessment of the student's knowledge
Teacher oversees various tasks related to ongoing assessment and determines students' current performance twice during each academic period. Ratings 1 and 2 are formulated based on the outcomes of this ongoing assessment. The student's learning achievements are assessed using a 100-point scale, and the final grades P1 and P2 are calculated as the average of their ongoing performance evaluations. The teacher evaluates the student's work throughout the academic period in alignment with the assignment submission schedule for the discipline. The assessment system may incorporate a mix of written and oral, group and individual formats.
| Period | Type of task | Total |
|---|---|---|
| 1 rating | IH 1 "Random Events" | 0-100 |
| Independent work #1 on the topic "Random events" | ||
| test 1 | ||
| Border control 1 | ||
| 2 rating | IH 2 "Random variables" | 0-100 |
| Independent work #2 on the topic "Random variables" | ||
| test 2 | ||
| Border control 2 | ||
| Total control | Exam | 0-100 |
The evaluating policy of learning outcomes by work type
| Type of task | 90-100 | 70-89 | 50-69 | 0-49 |
|---|---|---|---|---|
| Excellent | Good | Satisfactory | Unsatisfactory | |
| Interview on control questions (colloquium) | demonstrates systematic theoretical knowledge, has a command of terminology, logically and consistently explains the essence of phenomena and processes, makes reasoned conclusions and generalizations, gives examples, demonstrates fluency in monologue speech and the ability to quickly respond to clarifying questions | demonstrates solid theoretical knowledge, has a command of terminology, logically and consistently explains the essence of phenomena and processes, makes reasoned conclusions and generalizations, gives examples, demonstrates fluency in monologue speech, but at the same time makes minor mistakes that he corrects independently or with minor correction from the teacher | demonstrates shallow theoretical knowledge, displays poorly developed skills in analyzing phenomena and processes, insufficient ability to make reasoned conclusions and give examples, demonstrates insufficiently fluent command of monologue speech, terminology, logic and consistency of presentation, makes mistakes that can only be corrected with correction by the teacher. | demonstrates systematic theoretical knowledge, has a command of terminology, logically and consistently explains the essence of phenomena and processes, makes reasoned conclusions and generalizations, gives examples, demonstrates fluency in monologue speech and the ability to quickly respond to clarifying questions |
| IH (individual homework) or written work/exam | completed the practical work in full, observing the required sequence of actions; in the answer, correctly and accurately completes all entries, tables, drawings, diagrams, graphs, calculations; correctly performs error analysis.When answering questions, correctly understands the essence of the question, gives an accurate definition and interpretation of the main concepts; accompanies the answer with new examples, is able to apply knowledge in a new situation; can establish a connection between the material being studied and previously studied material, as well as with material learned in studying other disciplines. | выполнил требования к оценке «5», но допущены 2-3 недочета. Ответ обучающегося на вопросы удовлетворяет основным требованиям к ответу на 5, но дан без применения знаний в новой ситуации, без использования связей с ранее изученным материалом и материалом, усвоенным при изучении других дисциплин; допущены одна ошибка или не более двух недочетов, обучающийся может их исправить самостоятельно или с небольшой помощью преподавателя. | выполнил работу не полностью, но не менее 50% объема практической работы, что позволяет получить правильные результаты и выводы; в ходе проведения работы были допущены ошибки. При ответе на вопросы обучающийся правильно понимает сущность вопроса, но в ответе имеются отдельные проблемы в усвоении вопросов курса, не препятствующие дальнейшему усвоению программного материала; допущено не более одной грубой ошибки и двух недочетов. | completed the practical work in full, observing the required sequence of actions; in the answer, correctly and accurately completes all entries, tables, drawings, diagrams, graphs, calculations; correctly performs error analysis.When answering questions, correctly understands the essence of the question, gives an accurate definition and interpretation of the main concepts; accompanies the answer with new examples, is able to apply knowledge in a new situation; can establish a connection between the material being studied and previously studied material, as well as with material learned in studying other disciplines. |
Evaluation form
The student's final grade in the course is calculated on a 100 point grading scale, it includes:
- 40% of the examination result;
- 60% of current control result.
The final grade is calculated by the formula:
| FG = 0,6 | MT1+MT2 | +0,4E |
| 2 |
Where Midterm 1, Midterm 2are digital equivalents of the grades of Midterm 1 and 2;
E is a digital equivalent of the exam grade.
Final alphabetical grade and its equivalent in points:
The letter grading system for students' academic achievements, corresponding to the numerical equivalent on a four-point scale:
| Alphabetical grade | Numerical value | Points (%) | Traditional grade |
|---|---|---|---|
| A | 4.0 | 95-100 | Excellent |
| A- | 3.67 | 90-94 | |
| B+ | 3.33 | 85-89 | Good |
| B | 3.0 | 80-84 | |
| B- | 2.67 | 75-79 | |
| C+ | 2.33 | 70-74 | |
| C | 2.0 | 65-69 | Satisfactory |
| C- | 1.67 | 60-64 | |
| D+ | 1.33 | 55-59 | |
| D | 1.0 | 50-54 | |
| FX | 0.5 | 25-49 | Unsatisfactory |
| F | 0 | 0-24 |
Topics of lectures
- Algebra of events
- Elements of Combinatorics
- Probability addition theorem
- Formula of total probability
- Repetition tests
- Random variables
- Numerical characteristics of a discrete random variable
- Density distribution of continuous random variables
- Uniform, normal, exponential distribution of continuous random variables and their numerical characteristics
- Initial and central theoretical moments of random variables
- Multivariate random variables
- Elements of mathematical statistics
- Interval estimates
- Criteria and its application to various proposed tests
- Determination of parameters of linear and nonlinear regression using the least squares method
Key reading
- Севастьянов Б. А. Курс теории вероятностей и математической статистики. — Москва-Ижевск: Институт компьютерных исследований, 2019
- Гмурман В.Е. Руководство к решению задач по теории вероятностей и математической статистике. – М.: Высшая школа, 2008.
- Горбиков С.П., Филатов Л.В. Лекции по теории вероятностей и математической статистике. [Текст]: учебное пособие для вузов./ Горбиков С.П., Филатов Л.В.; Нижегор. Гос. Архитектур.- строит. ун-т – Н.Новгород: ННГАСУ, 2011
- Вентцель Е.С. Теория вероятностей. – М.: Физматгиз, 2002.
- Гмурман В.Е. Введение в теорию вероятностей и математическую статистику. – М.: Высшая школа, 2008.
- Письменный Д.Т. Конспект лекций по теории вероятностей и математической статистике. – М.: Айрис-пресс, 2004.
- Кибзун А.И. и др. Теория вероятностей и математическая статистика. Базовый курс с примерами и задачами. – М.: Физматлит, 2002.
- Тыныбекова С.Д., Рахметуллина Ж.Т., Конырханова А.А.Теория вероятностей и математическая статистика в вопросах и задачах. – Усть-Каменогорск: ВКГТУ, 2011.
- Рябушко А.П., Бархатов В.В. и др. Индивидуальные задания по высшей математике. – Минск: Высшая школа, 2009. – Т. 4.
