Probability Theory and Mathematical Statistics in Tver State University

Project MetaMath Probability theory and mathematical statistics in Tver State University

Probability theory and mathematical statistics in TSU

Duration: 2 semesters (37 weeks)Preferred year/semester: 2, 3 / 4, 5 Prerequisite Courses: Linear algebra, Mathematical analysis, Differential EquationsAmount of credits – 9 (1 ≡36 hours)

Forms of control: modules, course project (each semester), exam (each semester)


Classroom load: •lectures – 74 hours,•laboratory work / seminars – 74 hours,•exam preparation – 27 hours.

Homework – necessary.Assessment – 2-5Methodological materials, lectures, guidelines for implementation of course work in electronic form in the public domain.


Discipline = 100 marks: work in the semester – 60, exam – 40

The exam includes:

1. two questions (20)

2. list of basic knowledge (theorems, definitions, formulas) (10)

3. solution of task (10)

Exam preparation – 27 hours.

Assessment (2-5): < 50 – 2

< 69 – 3

< 84 – 4

≥ 85 – 5


The goal of the course

The purpose of this course is to present basic information about the construction and analysis of models that take into

account random factors.

Output results

During this course, students should: master the basic concepts of probability theory and mathematical

statistics, master the basic techniques of formulating and solving probabilistic statistical problems.

SEFI. Level 0CORE 0 TSU

interpret data presented in the form of line diagrams, bar charts, pie charts

interpret data presented in the form of stem and leaf diagrams, box plots, histogramsconstruct line diagrams, bar charts, pie charts, setm and leaf diagrams, box plots, histograms for suitable data sets

2/4 semester

calculate the mode, median and mean for a set of data items 2/4 semester

define the terms 'outcome','event' and 'probability' 2/ 4 semester

calculate the probability of an event by counting outcomes 2/ 4 semester

calculate the probability of the complement of an event 2/ 4 semester

calculate the probability of the union of two mutually-exclusive events 2/ 4 semester

calculate the probability of the union of two events 2/ 4 semester

calculate the probability of the intersection of two independent events 2/ 4 semester

SEFI. Level 1LEVEL 1 TSU

calculate the range, inter-quartile range, variance and standard deviation for a set of data items

2/ 4 semester

distinguish between a population and a sample 3/5 semesterknow the difference between the characteristic values (moments) of a population and of a sample

3/5 semester

construct a suitable frequency distribution from a data set 3/5 semester

calculate realtive frequencies 3/5 semestercalculate measures of average and dispersion for a grouped set of data 3/5 semester

understand the effect of grouping on these measures 3/5 semester

Combinatorics 2/4 semesteruse the multiplication principle for combinations 2/4 semesterinterpret probability as a degree of belief 2/4 semesterunderstand the distinction between a priori and a posteriori probabilities 2/4 semester

use a tree diagram to calculate probabilities 2/4 semesterknow what conditional probability is and be able to use it (Bayes' theorem) 2/4 semester

SEFI. Level 1calculate probabilities for series and parallel connections 2/4 semester

define a random variable and a discrete probability distribution 2/4 semester

state the criteria for a binomial model and define its parameters 2/4 semestercalculate probabilities for a binomial model 2/4 semester

state the criteria for a Poisson model and define its parameters 2/4 semester

calculate probabilities for a Poisson model 2/4 semester

state the expected value and variance for each of these models 2/4 semester

understand what a random variable is continuous 2/4 semester

explain the way in which probability calculations are carried out in the continuous case

2/4 semester

Normal distribution 2/4 semester

relate the general normal distribution to the standardised normal distribution 2/4 semesterdefine a random sample 3/5 semester

know what a sampling distribution is 3/5 semester

SEFI. Level 1understand the term 'mean squared error' of an estimate 3/5 semesterunderstand the term 'unbiasedness' of an estimate 3/5 semester

apply confidence intervals to sample estimates 3/5 semester

follow the main steps in a test of hypothesis 3/5 semester

understand the difference between a test of hypothesis and a significance test (pvalue)

3/5 semester

define the level of a test (error of the first kind) 3/5 semester

define the power of a test (error of the second kind) 3/5 semester

state the link between the distribution of a normal variable and that of means of samples

3/5 semester

place confidence intervals around the sample estimate of a population mean 3/5 semester

test claims about the population mean using results from sampling 3/5 semester

recognise whether an alternative hypothesis leads to a one-tail or a two-tail test

3/5 semester

compare the approaches of using confidence intervals and hypothesis tests 3/5 semester

SEFI. Level 2LEVEL 2 TSU

compare empirical and theoretical distributions 3/5 semesterapply the exponential distribution to simple problems 2/4 semester

apply the normal distribution to simple problems 2/4 semesterapply the gamma distribution to simple problems 4, 5 semesterunderstand the concept of a joint distribution 2/4 semesterunderstand the terms 'joint density function', 'marginal distribution functions'

2/4 semester

define independence of two random variables 2/4 semestersolve problems involving linear combinations of random variables 2/4 semester

determine the covariance of two random variables 2/4 semester

SEFI. Level 2determine the correlation of two random variables 2/4 semester

realise what the normal distribution is not reliable when used with small samples

3/5 semester

use tables of the t-distribution 2/4 semesteruse tables of the F-distribution 3/5 semesteruse the method of pairing where appropriate 3/5 semesteruse tables for chi-squared distributions 3/5 semesterdecide on the number of degrees of freedom appropriate to a particular problem

3/5 semester

use the chi-square distribution in tests of independence (contigency tables)

3/5 semester

use the chi-square distribution in tests of goodness of fit. 3/5 semester

set up the information for a one-way analysis of variance 3/5 semesterAnalysis of variance in another courseSimple linear regression, Multiple linear regression in another course

SEFI. Level 3

LEVEL 3 TSU

Stochastic processes 3 year (Stochastic processes)Statistical quality controlReliability 4 year Experimental designQueueing theory and discrete simulation 4 year (Simulation modeling), Master (Queueing theory)Filtering and control 4 year Markov processes and renewal theory 3 year (Stochastic processes), Master (Queueing theory)Statistical inferenceMultivariate analysis elective courses

in SEFI offline

ADD:

•Conditional distribution and conditional mean (3/5 semester)•characteristic functions (3/5 semester)•effective estimates (3/5 semester)•information Fischer, inequality of Rao-Cramer (3/5 semester)•central statistics, the construction of confidence intervals using central statistics (3/5 semester)•methods of construction estimates (3/5 semester)•likelihood function (3/5 semester)


Modernization course:

• pre-test;

• homework in Math Bridge;

• tests on the topics in Math Bridge;

• bridging courses in Math Bridge.

Probability theory and mathematical statistics in TSU: Modernization course

Pre-test (content):

• Theory of sets;

• combinatorics;

• graphs of functions;

• area of the figure;

• geometric progression;

• image of a set of points on the plane;

• diagrams.


Pre-test (example):1. Тридцать человек разбиты на три группы по 10 человек в каждой. Сколько может различных составов групп?

2. Сколько четырехзначных чисел, составленных из цифр 0,1,2,3,4,5 содержат цифру 3?

3.Садовник должен в течение 3-х дней посадить 10 деревьев. Сколькими способами он может распределить по дням работу,

если будет сажать не менее 1 дерева в день?

4. Расписание одного дня содержит 4 семинара. Определить количество таких расписаний из 9 дисциплин.

5. Какова вероятность того, что при двух подбрасываниях монеты герб появится один раз?

6. В группе 35 студентов. Каждый из них пользуется хотя бы одним из видов городского транспорта: метро, автобусом и

троллейбусом. Всеми тремя видами транспорта пользуются 6 студентов, метро и автобусом – 15 студентов, метро и

троллейбусом – 13 студентов, троллейбусом и автобусом – 9 студентов. Сколько студентов используют только один вид

транспорта?


7. Первую или вторую контрольные работы по математике успешно написали 33 студента, первую или третью – 31 студент, вторую

или третью – 32 студента. Не менее двух контрольных работ выполнили 20 студентов. Сколько студентов успешно решили только

одну контрольную работу?

8. Найти суммы бесконечно убывающих прогрессий: a).

9. Построить график функции:

10. Изобразить с помощью линейной диаграммы тенденцию перевозок пассажиров общественным транспортом в России, если

имеются следующие данные:Годы

2006

2007

2008

2009

2010

2011

Млн.чел.

47885

48114

46283

45037

45412

45817


Homework in Math Bridge:

At the present time for homework developed methodological guidelines containing a brief theoretical material,

examples of problem solving and task list for homework.

It makes sense to use the platform Math Bridge for homework, use bank of available tasks.


Bridging courses in Math Bridge (for example):

1. theory of sets;

2. combinatorics;

3. graphs of functions.

Since in our department to first-year students entered the leveling course of elementary mathematics, it is very convenient

in this course with the help of Math Bridge consider necessary topics.

Project MetaMath

Thank you for your attention!

[email protected]

Variants of the course project

1. Write a program that calculates the value of the function

• Random value has a t - distribution (Student distribution).

2. Write a program that calculates the value of the integral

• - function of distribution of betta – law, if - Euler gamma

function.

3. Write a program that calculates the area of the figure given lines

Education

Probability Theory and Mathematical Statistics in Tver State University