NATIONAL UNIVERSITYOF LIFE AND ENVIRONMENTAL SCIENCES OF UKRAINE

Economic Faculty

Department of Statistics and Economic Analysis

EDUCATIONALLY-METHODICAL COMPLEX

ldquoTheory of Probabilityrdquo

for preparation specialists of bachelorrsquos degree specialty 072 Finance Bank Activity and Insurance

Prepared by Lesia VoliakPhD in Economics Senior Lecture

Kyiv ndash 2019

NATIONAL UNIVERSITYOF LIFE AND ENVIRONMENTAL SCIENCES OF UKRAINE

Department of Statistics and Economic Analysis

ldquoAPPROVEDrdquoDean of the Economic faculty

________ AD Dibrova______ _______________ 2019 y

CONSIDERED AND APPROVEDat the meeting of the Department

of Statistics and Economic AnalysisProtocol 11 from ldquo10rdquo June 2019 y

Head of the Department__________ I D Lazaryshyna

WORKING PROGRAM OF EDUCATIONAL DISCIPLINE

ldquo Theory of Probability rdquo

for preparation specialists of bachelorrsquos degree specialty 072 ldquoFinance Bank Activity and Insurancerdquo Economic FacultyDevelopers Lesia Voliak Senior Lecture of the Department of Statistics and Economic Analysis PhD in Economics

Kyiv ndash 2019

INTRODUCTION

The Theory of Probability is a major tool that can be used to explain

and understand the various phenomena in different natural physical and social

sciences This book provides a systematic exposition of the theory in a setting

which contains a balanced mixture of the classical approach and the modern day

axiomatic approach After reviewing the basis of the theory the book considers

univariate distributions bivariate normal distribution multinomial distribution

convergence of random variables and elements of stochastic process Difficult

ideas have been explained lucidly and augmented with explanatory notes

examples and exercises The basic requirement for reading the book is the

knowledge of mathematics at graduate level

The aim of the notes is to combine the mathematical and theoretical

underpinning of statistics and statistical data analysis with computational

methodology and practical applications Hopefully the notes pave the way for an

understanding of the foundation of data analysis with a focus on the probabilistic

model and the methodology that we can develop from this point of view In a

single course there is no hope that we can present all models and all relevant

methods that the students will need in the future and for this reason we develop

general ideas so that new models and methods can be more easily approached by

students after the course We can on the other hand not develop the theory

without a number of good examples to illustrate its use Due to the history of the

course most examples in the notes are biological of nature but span a range of

different areas from molecular biology and biological sequence analysis over

molecular evolution and genetics to toxicology and various assay procedures

1 Description of the discipline STATISTICS

Sphere of knowledge training direction specialty education and qualification level

Sphere of knowledge 07 Management and administration (code and name)

Training direction ____________________ (code and name)

Specialty 072 ldquoFinance Bank Activity and Insurancerdquo (code and name)

Education and qualification level bachelor (Bachelor Specialist Master)

Characterization of disciplineType Normative (selective)Total number of hours 150Number of ECTS credits ____5____The number of structural modules ___2___Course project (work) (if your curriculum)

_______________ (name)

Form of control Test exam

Indicators of discipline for full-time and distance learning

full-time education distance educationYear of training _____1____Semester _____2____Lectures ____30___ hoursPractical seminars ____30___ hoursLaboratory studies _______ hoursIndependent work 90 hoursIndividual tasks _______ hoursThe number of weekly hours for full-time study classroom ___4__ hours

2 Purposes of the CourseThe working program for the study of regulatory disciplines ldquoTheory of

probability and mathematical statisticsrdquo prepared in accordance with the

educational-professional programs of preparation of bachelors of the direction

ldquoEconomics of enterpriserdquo

The subject of study of the discipline is the regularities of random

phenomena and their use for building stochastic economic models

The purpose of study of discipline ldquoTheory of probability and mathematical

statisticsrdquo is forming the base of students knowledge and practical skills on basic

probability and statistical apparatus the main methods of quantitative

measurement of the randomness of the action of factors affecting any processes

principles of mathematical statistics which is used during planning organization

and production management quality evaluation of products systems analysis of

economic structures and technological processes application of mathematical

methods in Economics

Objectives of the discipline which needs to be solved in the process of

study theoretical and practical training of students aimed at mastery of modern

mathematics probability theory and statistical methods which should help for

future economists to solve applied economic problems

Discipline program consists of the following content modules

1 Probability theory

2 Mathematical statistics

As a result of studying of discipline the student should

know basic concepts of probability theory such as random phenomenos

definition and properties of probability random variable basic distributions of

discrete and continuous random variables numerical characteristics the statistical

distribution of the sample and their numerical characteristics statistical estimation

of distribution parameters statistical hypotheses and their verification

to be able to select mathematical methods and probabilistic models

methods of statistical analysis for the study of applied problems to apply modern

statistical methods to solve practical economic problems and to acquire skills of

independent application of mathematical literature and application of software

packages for statistical analysis of economic data

Program material of the course ldquoTheory of Probabilityrdquo

Theoretical and practical course subject ldquoTheory of Probabilityrdquo is taught

using a large number of examples calculations tables graphs solving common

tasks and multimedia equipment computer programs

3 The Program of educational disciplineThe structure of the discipline

STATISTICSThe names of the modules and themes

Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theoryTheme 2 Fundamental theorems of probability theoryTheme 3 The sequence of independent trials Bernoulli schemeTheme 4 Random variables and distribution functionsTheme 5 The law of large numbers and central limit theorem

Module 2 Mathematical statisticsTheme 6 Basics of Mathematical StatisticsTheme 7 Statistical estimation of population parametersTheme 8 Verification of statistical hypothesesTheme 9 Elements of dispersion analysisTheme 10 Basic theory of correlation analysis

Theme 1 Basic concepts and theorems of probability theory

Basic concepts of probability theory The subject and tasks of probability

theory Stochastic experiment elementary space Operations over the elementary

events Elements of combinatorics their use in probability theory Different types

of connections host permutations and combinations

Classical statistical and geometrical definition of probability

Theorems of addition and multiplication of probabilities The amount of

events addition theorem of probabilities Conditional probability and the theorem

Kyiv ndash 2019

NATIONAL UNIVERSITYOF LIFE AND ENVIRONMENTAL SCIENCES OF UKRAINE

Department of Statistics and Economic Analysis

ldquoAPPROVEDrdquoDean of the Economic faculty

________ AD Dibrova______ _______________ 2019 y

CONSIDERED AND APPROVEDat the meeting of the Department

of Statistics and Economic AnalysisProtocol 11 from ldquo10rdquo June 2019 y

Head of the Department__________ I D Lazaryshyna

WORKING PROGRAM OF EDUCATIONAL DISCIPLINE

ldquo Theory of Probability rdquo

for preparation specialists of bachelorrsquos degree specialty 072 ldquoFinance Bank Activity and Insurancerdquo Economic FacultyDevelopers Lesia Voliak Senior Lecture of the Department of Statistics and Economic Analysis PhD in Economics

Kyiv ndash 2019

INTRODUCTION

The Theory of Probability is a major tool that can be used to explain

and understand the various phenomena in different natural physical and social

sciences This book provides a systematic exposition of the theory in a setting

which contains a balanced mixture of the classical approach and the modern day

axiomatic approach After reviewing the basis of the theory the book considers

univariate distributions bivariate normal distribution multinomial distribution

convergence of random variables and elements of stochastic process Difficult

ideas have been explained lucidly and augmented with explanatory notes

examples and exercises The basic requirement for reading the book is the

knowledge of mathematics at graduate level

The aim of the notes is to combine the mathematical and theoretical

underpinning of statistics and statistical data analysis with computational

methodology and practical applications Hopefully the notes pave the way for an

understanding of the foundation of data analysis with a focus on the probabilistic

model and the methodology that we can develop from this point of view In a

single course there is no hope that we can present all models and all relevant

methods that the students will need in the future and for this reason we develop

general ideas so that new models and methods can be more easily approached by

students after the course We can on the other hand not develop the theory

without a number of good examples to illustrate its use Due to the history of the

course most examples in the notes are biological of nature but span a range of

different areas from molecular biology and biological sequence analysis over

molecular evolution and genetics to toxicology and various assay procedures

1 Description of the discipline STATISTICS

Sphere of knowledge training direction specialty education and qualification level

Sphere of knowledge 07 Management and administration (code and name)

Training direction ____________________ (code and name)

Specialty 072 ldquoFinance Bank Activity and Insurancerdquo (code and name)

Education and qualification level bachelor (Bachelor Specialist Master)

Characterization of disciplineType Normative (selective)Total number of hours 150Number of ECTS credits ____5____The number of structural modules ___2___Course project (work) (if your curriculum)

_______________ (name)

Form of control Test exam

Indicators of discipline for full-time and distance learning

full-time education distance educationYear of training _____1____Semester _____2____Lectures ____30___ hoursPractical seminars ____30___ hoursLaboratory studies _______ hoursIndependent work 90 hoursIndividual tasks _______ hoursThe number of weekly hours for full-time study classroom ___4__ hours

2 Purposes of the CourseThe working program for the study of regulatory disciplines ldquoTheory of

probability and mathematical statisticsrdquo prepared in accordance with the

educational-professional programs of preparation of bachelors of the direction

ldquoEconomics of enterpriserdquo

The subject of study of the discipline is the regularities of random

phenomena and their use for building stochastic economic models

The purpose of study of discipline ldquoTheory of probability and mathematical

statisticsrdquo is forming the base of students knowledge and practical skills on basic

probability and statistical apparatus the main methods of quantitative

measurement of the randomness of the action of factors affecting any processes

principles of mathematical statistics which is used during planning organization

and production management quality evaluation of products systems analysis of

economic structures and technological processes application of mathematical

methods in Economics

Objectives of the discipline which needs to be solved in the process of

study theoretical and practical training of students aimed at mastery of modern

mathematics probability theory and statistical methods which should help for

future economists to solve applied economic problems

Discipline program consists of the following content modules

1 Probability theory

2 Mathematical statistics

As a result of studying of discipline the student should

know basic concepts of probability theory such as random phenomenos

definition and properties of probability random variable basic distributions of

discrete and continuous random variables numerical characteristics the statistical

distribution of the sample and their numerical characteristics statistical estimation

of distribution parameters statistical hypotheses and their verification

to be able to select mathematical methods and probabilistic models

methods of statistical analysis for the study of applied problems to apply modern

statistical methods to solve practical economic problems and to acquire skills of

independent application of mathematical literature and application of software

packages for statistical analysis of economic data

Program material of the course ldquoTheory of Probabilityrdquo

Theoretical and practical course subject ldquoTheory of Probabilityrdquo is taught

using a large number of examples calculations tables graphs solving common

tasks and multimedia equipment computer programs

3 The Program of educational disciplineThe structure of the discipline

STATISTICSThe names of the modules and themes

Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theoryTheme 2 Fundamental theorems of probability theoryTheme 3 The sequence of independent trials Bernoulli schemeTheme 4 Random variables and distribution functionsTheme 5 The law of large numbers and central limit theorem

Module 2 Mathematical statisticsTheme 6 Basics of Mathematical StatisticsTheme 7 Statistical estimation of population parametersTheme 8 Verification of statistical hypothesesTheme 9 Elements of dispersion analysisTheme 10 Basic theory of correlation analysis

Theme 1 Basic concepts and theorems of probability theory

Basic concepts of probability theory The subject and tasks of probability

theory Stochastic experiment elementary space Operations over the elementary

events Elements of combinatorics their use in probability theory Different types

of connections host permutations and combinations

Classical statistical and geometrical definition of probability

Theorems of addition and multiplication of probabilities The amount of

events addition theorem of probabilities Conditional probability and the theorem

NATIONAL UNIVERSITYOF LIFE AND ENVIRONMENTAL SCIENCES OF UKRAINE

Department of Statistics and Economic Analysis

ldquoAPPROVEDrdquoDean of the Economic faculty

________ AD Dibrova______ _______________ 2019 y

CONSIDERED AND APPROVEDat the meeting of the Department

of Statistics and Economic AnalysisProtocol 11 from ldquo10rdquo June 2019 y

Head of the Department__________ I D Lazaryshyna

WORKING PROGRAM OF EDUCATIONAL DISCIPLINE

ldquo Theory of Probability rdquo

for preparation specialists of bachelorrsquos degree specialty 072 ldquoFinance Bank Activity and Insurancerdquo Economic FacultyDevelopers Lesia Voliak Senior Lecture of the Department of Statistics and Economic Analysis PhD in Economics

Kyiv ndash 2019

INTRODUCTION

The Theory of Probability is a major tool that can be used to explain

and understand the various phenomena in different natural physical and social

sciences This book provides a systematic exposition of the theory in a setting

which contains a balanced mixture of the classical approach and the modern day

axiomatic approach After reviewing the basis of the theory the book considers

univariate distributions bivariate normal distribution multinomial distribution

convergence of random variables and elements of stochastic process Difficult

ideas have been explained lucidly and augmented with explanatory notes

examples and exercises The basic requirement for reading the book is the

knowledge of mathematics at graduate level

The aim of the notes is to combine the mathematical and theoretical

underpinning of statistics and statistical data analysis with computational

methodology and practical applications Hopefully the notes pave the way for an

understanding of the foundation of data analysis with a focus on the probabilistic

model and the methodology that we can develop from this point of view In a

single course there is no hope that we can present all models and all relevant

methods that the students will need in the future and for this reason we develop

general ideas so that new models and methods can be more easily approached by

students after the course We can on the other hand not develop the theory

without a number of good examples to illustrate its use Due to the history of the

course most examples in the notes are biological of nature but span a range of

different areas from molecular biology and biological sequence analysis over

molecular evolution and genetics to toxicology and various assay procedures

1 Description of the discipline STATISTICS

Sphere of knowledge training direction specialty education and qualification level

Sphere of knowledge 07 Management and administration (code and name)

Training direction ____________________ (code and name)

Specialty 072 ldquoFinance Bank Activity and Insurancerdquo (code and name)

Education and qualification level bachelor (Bachelor Specialist Master)

Characterization of disciplineType Normative (selective)Total number of hours 150Number of ECTS credits ____5____The number of structural modules ___2___Course project (work) (if your curriculum)

_______________ (name)

Form of control Test exam

Indicators of discipline for full-time and distance learning

full-time education distance educationYear of training _____1____Semester _____2____Lectures ____30___ hoursPractical seminars ____30___ hoursLaboratory studies _______ hoursIndependent work 90 hoursIndividual tasks _______ hoursThe number of weekly hours for full-time study classroom ___4__ hours

2 Purposes of the CourseThe working program for the study of regulatory disciplines ldquoTheory of

probability and mathematical statisticsrdquo prepared in accordance with the

educational-professional programs of preparation of bachelors of the direction

ldquoEconomics of enterpriserdquo

The subject of study of the discipline is the regularities of random

phenomena and their use for building stochastic economic models

The purpose of study of discipline ldquoTheory of probability and mathematical

statisticsrdquo is forming the base of students knowledge and practical skills on basic

probability and statistical apparatus the main methods of quantitative

measurement of the randomness of the action of factors affecting any processes

principles of mathematical statistics which is used during planning organization

and production management quality evaluation of products systems analysis of

economic structures and technological processes application of mathematical

methods in Economics

Objectives of the discipline which needs to be solved in the process of

study theoretical and practical training of students aimed at mastery of modern

mathematics probability theory and statistical methods which should help for

future economists to solve applied economic problems

Discipline program consists of the following content modules

1 Probability theory

2 Mathematical statistics

As a result of studying of discipline the student should

know basic concepts of probability theory such as random phenomenos

definition and properties of probability random variable basic distributions of

discrete and continuous random variables numerical characteristics the statistical

distribution of the sample and their numerical characteristics statistical estimation

of distribution parameters statistical hypotheses and their verification

to be able to select mathematical methods and probabilistic models

methods of statistical analysis for the study of applied problems to apply modern

statistical methods to solve practical economic problems and to acquire skills of

independent application of mathematical literature and application of software

packages for statistical analysis of economic data

Program material of the course ldquoTheory of Probabilityrdquo

Theoretical and practical course subject ldquoTheory of Probabilityrdquo is taught

using a large number of examples calculations tables graphs solving common

tasks and multimedia equipment computer programs

3 The Program of educational disciplineThe structure of the discipline

STATISTICSThe names of the modules and themes

Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theoryTheme 2 Fundamental theorems of probability theoryTheme 3 The sequence of independent trials Bernoulli schemeTheme 4 Random variables and distribution functionsTheme 5 The law of large numbers and central limit theorem

Module 2 Mathematical statisticsTheme 6 Basics of Mathematical StatisticsTheme 7 Statistical estimation of population parametersTheme 8 Verification of statistical hypothesesTheme 9 Elements of dispersion analysisTheme 10 Basic theory of correlation analysis

Theme 1 Basic concepts and theorems of probability theory

Basic concepts of probability theory The subject and tasks of probability

theory Stochastic experiment elementary space Operations over the elementary

events Elements of combinatorics their use in probability theory Different types

of connections host permutations and combinations

Classical statistical and geometrical definition of probability

Theorems of addition and multiplication of probabilities The amount of

events addition theorem of probabilities Conditional probability and the theorem

Kyiv ndash 2019

INTRODUCTION

The Theory of Probability is a major tool that can be used to explain

and understand the various phenomena in different natural physical and social

sciences This book provides a systematic exposition of the theory in a setting

which contains a balanced mixture of the classical approach and the modern day

axiomatic approach After reviewing the basis of the theory the book considers

univariate distributions bivariate normal distribution multinomial distribution

convergence of random variables and elements of stochastic process Difficult

ideas have been explained lucidly and augmented with explanatory notes

examples and exercises The basic requirement for reading the book is the

knowledge of mathematics at graduate level

The aim of the notes is to combine the mathematical and theoretical

underpinning of statistics and statistical data analysis with computational

methodology and practical applications Hopefully the notes pave the way for an

understanding of the foundation of data analysis with a focus on the probabilistic

model and the methodology that we can develop from this point of view In a

single course there is no hope that we can present all models and all relevant

methods that the students will need in the future and for this reason we develop

general ideas so that new models and methods can be more easily approached by

students after the course We can on the other hand not develop the theory

without a number of good examples to illustrate its use Due to the history of the

course most examples in the notes are biological of nature but span a range of

different areas from molecular biology and biological sequence analysis over

molecular evolution and genetics to toxicology and various assay procedures

1 Description of the discipline STATISTICS

Sphere of knowledge training direction specialty education and qualification level

Sphere of knowledge 07 Management and administration (code and name)

Training direction ____________________ (code and name)

Specialty 072 ldquoFinance Bank Activity and Insurancerdquo (code and name)

Education and qualification level bachelor (Bachelor Specialist Master)

Characterization of disciplineType Normative (selective)Total number of hours 150Number of ECTS credits ____5____The number of structural modules ___2___Course project (work) (if your curriculum)

_______________ (name)

Form of control Test exam

Indicators of discipline for full-time and distance learning

full-time education distance educationYear of training _____1____Semester _____2____Lectures ____30___ hoursPractical seminars ____30___ hoursLaboratory studies _______ hoursIndependent work 90 hoursIndividual tasks _______ hoursThe number of weekly hours for full-time study classroom ___4__ hours

2 Purposes of the CourseThe working program for the study of regulatory disciplines ldquoTheory of

probability and mathematical statisticsrdquo prepared in accordance with the

educational-professional programs of preparation of bachelors of the direction

ldquoEconomics of enterpriserdquo

The subject of study of the discipline is the regularities of random

phenomena and their use for building stochastic economic models

The purpose of study of discipline ldquoTheory of probability and mathematical

statisticsrdquo is forming the base of students knowledge and practical skills on basic

probability and statistical apparatus the main methods of quantitative

measurement of the randomness of the action of factors affecting any processes

principles of mathematical statistics which is used during planning organization

and production management quality evaluation of products systems analysis of

economic structures and technological processes application of mathematical

methods in Economics

Objectives of the discipline which needs to be solved in the process of

study theoretical and practical training of students aimed at mastery of modern

mathematics probability theory and statistical methods which should help for

future economists to solve applied economic problems

Discipline program consists of the following content modules

1 Probability theory

2 Mathematical statistics

As a result of studying of discipline the student should

know basic concepts of probability theory such as random phenomenos

definition and properties of probability random variable basic distributions of

discrete and continuous random variables numerical characteristics the statistical

distribution of the sample and their numerical characteristics statistical estimation

of distribution parameters statistical hypotheses and their verification

to be able to select mathematical methods and probabilistic models

methods of statistical analysis for the study of applied problems to apply modern

statistical methods to solve practical economic problems and to acquire skills of

independent application of mathematical literature and application of software

packages for statistical analysis of economic data

Program material of the course ldquoTheory of Probabilityrdquo

Theoretical and practical course subject ldquoTheory of Probabilityrdquo is taught

using a large number of examples calculations tables graphs solving common

tasks and multimedia equipment computer programs

3 The Program of educational disciplineThe structure of the discipline

STATISTICSThe names of the modules and themes

Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theoryTheme 2 Fundamental theorems of probability theoryTheme 3 The sequence of independent trials Bernoulli schemeTheme 4 Random variables and distribution functionsTheme 5 The law of large numbers and central limit theorem

Module 2 Mathematical statisticsTheme 6 Basics of Mathematical StatisticsTheme 7 Statistical estimation of population parametersTheme 8 Verification of statistical hypothesesTheme 9 Elements of dispersion analysisTheme 10 Basic theory of correlation analysis

Theme 1 Basic concepts and theorems of probability theory

Basic concepts of probability theory The subject and tasks of probability

theory Stochastic experiment elementary space Operations over the elementary

events Elements of combinatorics their use in probability theory Different types

of connections host permutations and combinations

Classical statistical and geometrical definition of probability

Theorems of addition and multiplication of probabilities The amount of

events addition theorem of probabilities Conditional probability and the theorem

INTRODUCTION

The Theory of Probability is a major tool that can be used to explain

and understand the various phenomena in different natural physical and social

sciences This book provides a systematic exposition of the theory in a setting

which contains a balanced mixture of the classical approach and the modern day

axiomatic approach After reviewing the basis of the theory the book considers

univariate distributions bivariate normal distribution multinomial distribution

convergence of random variables and elements of stochastic process Difficult

ideas have been explained lucidly and augmented with explanatory notes

examples and exercises The basic requirement for reading the book is the

knowledge of mathematics at graduate level

The aim of the notes is to combine the mathematical and theoretical

underpinning of statistics and statistical data analysis with computational

methodology and practical applications Hopefully the notes pave the way for an

understanding of the foundation of data analysis with a focus on the probabilistic

model and the methodology that we can develop from this point of view In a

single course there is no hope that we can present all models and all relevant

methods that the students will need in the future and for this reason we develop

general ideas so that new models and methods can be more easily approached by

students after the course We can on the other hand not develop the theory

without a number of good examples to illustrate its use Due to the history of the

course most examples in the notes are biological of nature but span a range of

different areas from molecular biology and biological sequence analysis over

molecular evolution and genetics to toxicology and various assay procedures

1 Description of the discipline STATISTICS

Sphere of knowledge training direction specialty education and qualification level

Sphere of knowledge 07 Management and administration (code and name)

Training direction ____________________ (code and name)

Specialty 072 ldquoFinance Bank Activity and Insurancerdquo (code and name)

Education and qualification level bachelor (Bachelor Specialist Master)

Characterization of disciplineType Normative (selective)Total number of hours 150Number of ECTS credits ____5____The number of structural modules ___2___Course project (work) (if your curriculum)

_______________ (name)

Form of control Test exam

Indicators of discipline for full-time and distance learning

full-time education distance educationYear of training _____1____Semester _____2____Lectures ____30___ hoursPractical seminars ____30___ hoursLaboratory studies _______ hoursIndependent work 90 hoursIndividual tasks _______ hoursThe number of weekly hours for full-time study classroom ___4__ hours

2 Purposes of the CourseThe working program for the study of regulatory disciplines ldquoTheory of

probability and mathematical statisticsrdquo prepared in accordance with the

educational-professional programs of preparation of bachelors of the direction

ldquoEconomics of enterpriserdquo

The subject of study of the discipline is the regularities of random

phenomena and their use for building stochastic economic models

The purpose of study of discipline ldquoTheory of probability and mathematical

statisticsrdquo is forming the base of students knowledge and practical skills on basic

probability and statistical apparatus the main methods of quantitative

measurement of the randomness of the action of factors affecting any processes

principles of mathematical statistics which is used during planning organization

and production management quality evaluation of products systems analysis of

economic structures and technological processes application of mathematical

methods in Economics

Objectives of the discipline which needs to be solved in the process of

study theoretical and practical training of students aimed at mastery of modern

mathematics probability theory and statistical methods which should help for

future economists to solve applied economic problems

Discipline program consists of the following content modules

1 Probability theory

2 Mathematical statistics

As a result of studying of discipline the student should

know basic concepts of probability theory such as random phenomenos

definition and properties of probability random variable basic distributions of

discrete and continuous random variables numerical characteristics the statistical

distribution of the sample and their numerical characteristics statistical estimation

of distribution parameters statistical hypotheses and their verification

to be able to select mathematical methods and probabilistic models

methods of statistical analysis for the study of applied problems to apply modern

statistical methods to solve practical economic problems and to acquire skills of

independent application of mathematical literature and application of software

packages for statistical analysis of economic data

Program material of the course ldquoTheory of Probabilityrdquo

Theoretical and practical course subject ldquoTheory of Probabilityrdquo is taught

using a large number of examples calculations tables graphs solving common

tasks and multimedia equipment computer programs

3 The Program of educational disciplineThe structure of the discipline

STATISTICSThe names of the modules and themes

Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theoryTheme 2 Fundamental theorems of probability theoryTheme 3 The sequence of independent trials Bernoulli schemeTheme 4 Random variables and distribution functionsTheme 5 The law of large numbers and central limit theorem

Module 2 Mathematical statisticsTheme 6 Basics of Mathematical StatisticsTheme 7 Statistical estimation of population parametersTheme 8 Verification of statistical hypothesesTheme 9 Elements of dispersion analysisTheme 10 Basic theory of correlation analysis

Theme 1 Basic concepts and theorems of probability theory

Basic concepts of probability theory The subject and tasks of probability

theory Stochastic experiment elementary space Operations over the elementary

events Elements of combinatorics their use in probability theory Different types

of connections host permutations and combinations

Classical statistical and geometrical definition of probability

Theorems of addition and multiplication of probabilities The amount of

events addition theorem of probabilities Conditional probability and the theorem

1 Description of the discipline STATISTICS

Sphere of knowledge training direction specialty education and qualification level

Sphere of knowledge 07 Management and administration (code and name)

Training direction ____________________ (code and name)

Specialty 072 ldquoFinance Bank Activity and Insurancerdquo (code and name)

Education and qualification level bachelor (Bachelor Specialist Master)

Characterization of disciplineType Normative (selective)Total number of hours 150Number of ECTS credits ____5____The number of structural modules ___2___Course project (work) (if your curriculum)

_______________ (name)

Form of control Test exam

Indicators of discipline for full-time and distance learning

full-time education distance educationYear of training _____1____Semester _____2____Lectures ____30___ hoursPractical seminars ____30___ hoursLaboratory studies _______ hoursIndependent work 90 hoursIndividual tasks _______ hoursThe number of weekly hours for full-time study classroom ___4__ hours

2 Purposes of the CourseThe working program for the study of regulatory disciplines ldquoTheory of

probability and mathematical statisticsrdquo prepared in accordance with the

educational-professional programs of preparation of bachelors of the direction

ldquoEconomics of enterpriserdquo

The subject of study of the discipline is the regularities of random

phenomena and their use for building stochastic economic models

The purpose of study of discipline ldquoTheory of probability and mathematical

statisticsrdquo is forming the base of students knowledge and practical skills on basic

probability and statistical apparatus the main methods of quantitative

measurement of the randomness of the action of factors affecting any processes

principles of mathematical statistics which is used during planning organization

and production management quality evaluation of products systems analysis of

economic structures and technological processes application of mathematical

methods in Economics

Objectives of the discipline which needs to be solved in the process of

study theoretical and practical training of students aimed at mastery of modern

mathematics probability theory and statistical methods which should help for

future economists to solve applied economic problems

Discipline program consists of the following content modules

1 Probability theory

2 Mathematical statistics

As a result of studying of discipline the student should

know basic concepts of probability theory such as random phenomenos

definition and properties of probability random variable basic distributions of

discrete and continuous random variables numerical characteristics the statistical

distribution of the sample and their numerical characteristics statistical estimation

of distribution parameters statistical hypotheses and their verification

to be able to select mathematical methods and probabilistic models

methods of statistical analysis for the study of applied problems to apply modern

statistical methods to solve practical economic problems and to acquire skills of

independent application of mathematical literature and application of software

packages for statistical analysis of economic data

Program material of the course ldquoTheory of Probabilityrdquo

Theoretical and practical course subject ldquoTheory of Probabilityrdquo is taught

using a large number of examples calculations tables graphs solving common

tasks and multimedia equipment computer programs

3 The Program of educational disciplineThe structure of the discipline

STATISTICSThe names of the modules and themes

Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theoryTheme 2 Fundamental theorems of probability theoryTheme 3 The sequence of independent trials Bernoulli schemeTheme 4 Random variables and distribution functionsTheme 5 The law of large numbers and central limit theorem

Module 2 Mathematical statisticsTheme 6 Basics of Mathematical StatisticsTheme 7 Statistical estimation of population parametersTheme 8 Verification of statistical hypothesesTheme 9 Elements of dispersion analysisTheme 10 Basic theory of correlation analysis

Theme 1 Basic concepts and theorems of probability theory

Basic concepts of probability theory The subject and tasks of probability

theory Stochastic experiment elementary space Operations over the elementary

events Elements of combinatorics their use in probability theory Different types

of connections host permutations and combinations

Classical statistical and geometrical definition of probability

Theorems of addition and multiplication of probabilities The amount of

events addition theorem of probabilities Conditional probability and the theorem

2 Purposes of the CourseThe working program for the study of regulatory disciplines ldquoTheory of

probability and mathematical statisticsrdquo prepared in accordance with the

educational-professional programs of preparation of bachelors of the direction

ldquoEconomics of enterpriserdquo

The subject of study of the discipline is the regularities of random

phenomena and their use for building stochastic economic models

The purpose of study of discipline ldquoTheory of probability and mathematical

statisticsrdquo is forming the base of students knowledge and practical skills on basic

probability and statistical apparatus the main methods of quantitative

measurement of the randomness of the action of factors affecting any processes

principles of mathematical statistics which is used during planning organization

and production management quality evaluation of products systems analysis of

economic structures and technological processes application of mathematical

methods in Economics

Objectives of the discipline which needs to be solved in the process of

study theoretical and practical training of students aimed at mastery of modern

mathematics probability theory and statistical methods which should help for

future economists to solve applied economic problems

Discipline program consists of the following content modules

1 Probability theory

2 Mathematical statistics

As a result of studying of discipline the student should

know basic concepts of probability theory such as random phenomenos

definition and properties of probability random variable basic distributions of

discrete and continuous random variables numerical characteristics the statistical

distribution of the sample and their numerical characteristics statistical estimation

of distribution parameters statistical hypotheses and their verification

to be able to select mathematical methods and probabilistic models

methods of statistical analysis for the study of applied problems to apply modern

statistical methods to solve practical economic problems and to acquire skills of

independent application of mathematical literature and application of software

packages for statistical analysis of economic data

Program material of the course ldquoTheory of Probabilityrdquo

Theoretical and practical course subject ldquoTheory of Probabilityrdquo is taught

using a large number of examples calculations tables graphs solving common

tasks and multimedia equipment computer programs

3 The Program of educational disciplineThe structure of the discipline

STATISTICSThe names of the modules and themes

Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theoryTheme 2 Fundamental theorems of probability theoryTheme 3 The sequence of independent trials Bernoulli schemeTheme 4 Random variables and distribution functionsTheme 5 The law of large numbers and central limit theorem

Module 2 Mathematical statisticsTheme 6 Basics of Mathematical StatisticsTheme 7 Statistical estimation of population parametersTheme 8 Verification of statistical hypothesesTheme 9 Elements of dispersion analysisTheme 10 Basic theory of correlation analysis

Theme 1 Basic concepts and theorems of probability theory

Basic concepts of probability theory The subject and tasks of probability

theory Stochastic experiment elementary space Operations over the elementary

events Elements of combinatorics their use in probability theory Different types

of connections host permutations and combinations

Classical statistical and geometrical definition of probability

Theorems of addition and multiplication of probabilities The amount of

events addition theorem of probabilities Conditional probability and the theorem

statistical methods to solve practical economic problems and to acquire skills of

independent application of mathematical literature and application of software

packages for statistical analysis of economic data

Program material of the course ldquoTheory of Probabilityrdquo

Theoretical and practical course subject ldquoTheory of Probabilityrdquo is taught

using a large number of examples calculations tables graphs solving common

tasks and multimedia equipment computer programs

3 The Program of educational disciplineThe structure of the discipline

STATISTICSThe names of the modules and themes

Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theoryTheme 2 Fundamental theorems of probability theoryTheme 3 The sequence of independent trials Bernoulli schemeTheme 4 Random variables and distribution functionsTheme 5 The law of large numbers and central limit theorem

Module 2 Mathematical statisticsTheme 6 Basics of Mathematical StatisticsTheme 7 Statistical estimation of population parametersTheme 8 Verification of statistical hypothesesTheme 9 Elements of dispersion analysisTheme 10 Basic theory of correlation analysis

Theme 1 Basic concepts and theorems of probability theory

Basic concepts of probability theory The subject and tasks of probability

theory Stochastic experiment elementary space Operations over the elementary

events Elements of combinatorics their use in probability theory Different types

of connections host permutations and combinations

Classical statistical and geometrical definition of probability

Theorems of addition and multiplication of probabilities The amount of

events addition theorem of probabilities Conditional probability and the theorem

of multiplication of probabilities The probability of the opposite event The

probability of occurrence of at least one event

The formula of total probability and Bayes formula The formula of total

probability The probability of the hypothesis and Bayes formula

Theme 2 Fundamental theorems of probability theory

Sample space sigma algebra Axioms of probability theory Conditional

probability independent events Bayersquos theorem

Definitions and Examples Discrete Random Variables Probability Mass

Function and Distribution Function Special kinds of Discrete Random Variables

Continuous Random Variables Probability Density Function and istribution

Function

Theme 3 A sequence of independent trials The Scheme of Bernoulli

The probability of complex events and the formula of Bernoulli Limit

theorems in Bernoulli scheme the Poisson theorem the local and integral Laplace

theorem The probability of deviation of relative frequency from the probability

The formula of Bernoulli and limit theorems in Bernoulli scheme

Theme 4 Random variables and distribution of functions

One-dimensional random variables and distribution functions The concept

of discrete random variable and its distribution laws The concept of continuous

random variables and the function and density of distribution of probabilities

Multidimensional random variable Functions of random variables System

of two random variables its law of distribution and numerical characteristics of

components The conditional probability distribution of components of the system

of two continuous random variables Stochastic dependence between the random

variables

The concept of a function of a random variable The law of distribution and

numerical characteristics of functions of random argument

Numerical characteristics of random variables Numerical characteristics of

discrete random variables mathematical expectation variance and standard

deviation Properties of numerical characteristics of discrete random variable

Numerical characteristics of continuous random variables and their properties

The main kinds distributions of random variables Benny geometric

distribution a discrete random variable Uniform normal exponential distribution

continuous random variable

Theme 5 The law of large numbers and central limit theorem

The concept of the law of large numbers and central limit theorem

The Chebyshev Inequality Theorem Chebyshevs law of large numbers The

law of large numbers in the model re-tests

The Central limit theorem of Lyapunov

Theme 6 Basics of Mathematical Statistics

The basic goal of statistics draw conclusions based on data There are

various aspects of statistics ranging from formulating the question designing

experiments to address the question collecting the data and analyzing the data but

well be stressing the role of probability and probability distributions in this

process Well often begin with a random sample drawn from a parameterized

family of distributions and our job is to make conclusions about the parameter

Theme 7 Statistical estimation of the parameters of the population

Statistical point estimation of distribution parameters of a random variable

The statistical evaluation Requirements for statistical estimations Point

estimates for the mathematical expectation and dispersion of the population

Assessment of the portion of the characteristic Methods of constructing statistical

estimates

Interval estimation of distribution parameters of a random variable

The distribution of the point estimates of the parameters of normally

distributed random variables Accuracy confidence intervals and boundaries

Interval estimates of mathematical expectation and variance of the normally

distributed random variable The task of interval estimation of the parameter of a

binomial distribution

Theme 8 Verification of statistical hypothesis

Define statistical hypothesis The General concept of inference The main

hypothesis and alternative Errors of the first and second kind The power of

criteria for testing hypotheses Case parameter estimates

Test hypotheses about the distribution using χ2 - criterion

Theme 9 Elements of dispersion analysis

One-way of analysis of variance as procedure for testing the hypothesis

about absence of influence of factor on the investigated value

The concept of multivariate analysis of variance

Theme 10 Basic theory of correlation analysis

Functional and statistical dependence

Correlation table for finding connections between random variables in the

form of the regression line

A linear correlation Direct regression and finding their parameters by the

method of least squares The coefficient of regression The coefficient of

correlation

The simplest cases of curvilinear correlation Correlation ratio The concept

of multiple correlation and multivariate regression model

3 Structure of educational Discipline

Names of Modules and Themes

Quantity of HoursFull-time

Total

includingL Sem

Pr Lab Indw

1 2 3 4 5 6 7Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theory 12 2 2 8

Theme 2 Fundamental theorems of probability theory 18 4 4 10

Theme 3 The sequence of independent trials Bernoulli scheme 18 4 4 10

Theme 4 Random variables and distribution functions 16 4 2 10

Theme 5 The law of large numbers and central limit theorem 14 2 2 10

Total for module 1 78 16 0 14 0 48Module 2 Mathematical statistics

Theme 6 Basics of Mathematical Statistics 12 2 2 8Theme 7 Statistical estimation of population parameters 12 2 2 8

Theme 8 Verification of statistical hypotheses 14 2 4 8Theme 9 Elements of dispersion analysis 18 4 4 10Theme 10 Basic theory of correlation analysis 16 4 4 8Total for module 2 72 14 0 16 0 42Total sum 150 30 0 30 0 90

4 Themes of seminars Name of Themes Quantity of

hours12

Total

5 Topics of practical classes Name of Themes Quantity of

hours1 Basic concepts and theorems of probability theory 22 Fundamental theorems of probability theory 43 The sequence of independent trials Bernoulli scheme 44 Random variables and distribution functions 25 The law of large numbers and central limit theorem 26 Basics of Mathematical Statistics 27 Statistical estimation of population parameters 28 Verification of statistical hypotheses 49 Elements of dispersion analysis 410 Basic theory of correlation analysis 4

Total 30

deviation Properties of numerical characteristics of discrete random variable

Numerical characteristics of continuous random variables and their properties

The main kinds distributions of random variables Benny geometric

distribution a discrete random variable Uniform normal exponential distribution

continuous random variable

Theme 5 The law of large numbers and central limit theorem

The concept of the law of large numbers and central limit theorem

The Chebyshev Inequality Theorem Chebyshevs law of large numbers The

law of large numbers in the model re-tests

The Central limit theorem of Lyapunov

Theme 6 Basics of Mathematical Statistics

The basic goal of statistics draw conclusions based on data There are

various aspects of statistics ranging from formulating the question designing

experiments to address the question collecting the data and analyzing the data but

well be stressing the role of probability and probability distributions in this

process Well often begin with a random sample drawn from a parameterized

family of distributions and our job is to make conclusions about the parameter

Theme 7 Statistical estimation of the parameters of the population

Statistical point estimation of distribution parameters of a random variable

The statistical evaluation Requirements for statistical estimations Point

estimates for the mathematical expectation and dispersion of the population

Assessment of the portion of the characteristic Methods of constructing statistical

estimates

Interval estimation of distribution parameters of a random variable

The distribution of the point estimates of the parameters of normally

distributed random variables Accuracy confidence intervals and boundaries

Interval estimates of mathematical expectation and variance of the normally

distributed random variable The task of interval estimation of the parameter of a

binomial distribution

Theme 8 Verification of statistical hypothesis

Define statistical hypothesis The General concept of inference The main

hypothesis and alternative Errors of the first and second kind The power of

criteria for testing hypotheses Case parameter estimates

Test hypotheses about the distribution using χ2 - criterion

Theme 9 Elements of dispersion analysis

One-way of analysis of variance as procedure for testing the hypothesis

about absence of influence of factor on the investigated value

The concept of multivariate analysis of variance

Theme 10 Basic theory of correlation analysis

Functional and statistical dependence

Correlation table for finding connections between random variables in the

form of the regression line

A linear correlation Direct regression and finding their parameters by the

method of least squares The coefficient of regression The coefficient of

correlation

The simplest cases of curvilinear correlation Correlation ratio The concept

of multiple correlation and multivariate regression model

3 Structure of educational Discipline

Names of Modules and Themes

Quantity of HoursFull-time

Total

includingL Sem

Pr Lab Indw

1 2 3 4 5 6 7Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theory 12 2 2 8

Theme 2 Fundamental theorems of probability theory 18 4 4 10

Theme 3 The sequence of independent trials Bernoulli scheme 18 4 4 10

Theme 4 Random variables and distribution functions 16 4 2 10

Theme 5 The law of large numbers and central limit theorem 14 2 2 10

Total for module 1 78 16 0 14 0 48Module 2 Mathematical statistics

Theme 6 Basics of Mathematical Statistics 12 2 2 8Theme 7 Statistical estimation of population parameters 12 2 2 8

Theme 8 Verification of statistical hypotheses 14 2 4 8Theme 9 Elements of dispersion analysis 18 4 4 10Theme 10 Basic theory of correlation analysis 16 4 4 8Total for module 2 72 14 0 16 0 42Total sum 150 30 0 30 0 90

4 Themes of seminars Name of Themes Quantity of

hours12

Total

5 Topics of practical classes Name of Themes Quantity of

hours1 Basic concepts and theorems of probability theory 22 Fundamental theorems of probability theory 43 The sequence of independent trials Bernoulli scheme 44 Random variables and distribution functions 25 The law of large numbers and central limit theorem 26 Basics of Mathematical Statistics 27 Statistical estimation of population parameters 28 Verification of statistical hypotheses 49 Elements of dispersion analysis 410 Basic theory of correlation analysis 4

Total 30

Theme 8 Verification of statistical hypothesis

Define statistical hypothesis The General concept of inference The main

hypothesis and alternative Errors of the first and second kind The power of

criteria for testing hypotheses Case parameter estimates

Test hypotheses about the distribution using χ2 - criterion

Theme 9 Elements of dispersion analysis

One-way of analysis of variance as procedure for testing the hypothesis

about absence of influence of factor on the investigated value

The concept of multivariate analysis of variance

Theme 10 Basic theory of correlation analysis

Functional and statistical dependence

Correlation table for finding connections between random variables in the

form of the regression line

A linear correlation Direct regression and finding their parameters by the

method of least squares The coefficient of regression The coefficient of

correlation

The simplest cases of curvilinear correlation Correlation ratio The concept

of multiple correlation and multivariate regression model

3 Structure of educational Discipline

Names of Modules and Themes

Quantity of HoursFull-time

Total

includingL Sem

Pr Lab Indw

1 2 3 4 5 6 7Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theory 12 2 2 8

Theme 2 Fundamental theorems of probability theory 18 4 4 10

Theme 3 The sequence of independent trials Bernoulli scheme 18 4 4 10

Theme 4 Random variables and distribution functions 16 4 2 10

Theme 5 The law of large numbers and central limit theorem 14 2 2 10

Total for module 1 78 16 0 14 0 48Module 2 Mathematical statistics

Theme 6 Basics of Mathematical Statistics 12 2 2 8Theme 7 Statistical estimation of population parameters 12 2 2 8

Theme 8 Verification of statistical hypotheses 14 2 4 8Theme 9 Elements of dispersion analysis 18 4 4 10Theme 10 Basic theory of correlation analysis 16 4 4 8Total for module 2 72 14 0 16 0 42Total sum 150 30 0 30 0 90

4 Themes of seminars Name of Themes Quantity of

hours12

Total

5 Topics of practical classes Name of Themes Quantity of

hours1 Basic concepts and theorems of probability theory 22 Fundamental theorems of probability theory 43 The sequence of independent trials Bernoulli scheme 44 Random variables and distribution functions 25 The law of large numbers and central limit theorem 26 Basics of Mathematical Statistics 27 Statistical estimation of population parameters 28 Verification of statistical hypotheses 49 Elements of dispersion analysis 410 Basic theory of correlation analysis 4

Total 30

3 Structure of educational Discipline

Names of Modules and Themes

Quantity of HoursFull-time

Total

includingL Sem

Pr Lab Indw

1 2 3 4 5 6 7Module 1 Probability theory

Theme 1 Basic concepts and theorems of probability theory 12 2 2 8

Theme 2 Fundamental theorems of probability theory 18 4 4 10

Theme 3 The sequence of independent trials Bernoulli scheme 18 4 4 10

Theme 4 Random variables and distribution functions 16 4 2 10

Theme 5 The law of large numbers and central limit theorem 14 2 2 10

Total for module 1 78 16 0 14 0 48Module 2 Mathematical statistics

Theme 6 Basics of Mathematical Statistics 12 2 2 8Theme 7 Statistical estimation of population parameters 12 2 2 8

Theme 8 Verification of statistical hypotheses 14 2 4 8Theme 9 Elements of dispersion analysis 18 4 4 10Theme 10 Basic theory of correlation analysis 16 4 4 8Total for module 2 72 14 0 16 0 42Total sum 150 30 0 30 0 90

4 Themes of seminars Name of Themes Quantity of

hours12

Total

5 Topics of practical classes Name of Themes Quantity of

hours1 Basic concepts and theorems of probability theory 22 Fundamental theorems of probability theory 43 The sequence of independent trials Bernoulli scheme 44 Random variables and distribution functions 25 The law of large numbers and central limit theorem 26 Basics of Mathematical Statistics 27 Statistical estimation of population parameters 28 Verification of statistical hypotheses 49 Elements of dispersion analysis 410 Basic theory of correlation analysis 4

Total 30

6 Topics of lab classes

Name of Themes Quantity of hours

1 -

7 Individual tasksParticipation in scientific conferences settlement graphics solving situational

problems

8 Methods of education

Practical Visual Verbal Working with book

Video- method

Experiments exercises

training and productive work

Illustrations demonstration observation of

students

Explanation explanation

narration conversation instruction

lecture discussion

debate

Reading Viewing Training

Exercises under the supervision of

electronic teacher control

9 Control forms

Control measures include current and final evaluation of student knowledge

Current control is carried out during practice and in the process of self-study in the

following areas rapid surveys tests tasks right-wrong problem

10 Distribution of points that students receive

Current control Rating of Academic

AffairsR НР

Rating of additional

work R ДР

Fine ratingR ШТР

Final certification

(exam or test)

Total quantity of marksModule 1 Module 2 Module 3 Module 4

0-100 0-100 0-100 0-100 0-70 0-20 0-5 0-30 0-100

Remark1 According to the Regulations on credit-modular system of education in NUBiP Ukraine approved by the university rector 27022019 ranking students for Academic R НР

study concerning certain discipline is given by the formula

07 (R(1)ЗМ К(1)

ЗМ + + R(n)ЗМ К(n)

ЗМ )

RНР = -------------------------------------------------------- + RДР - RШТР

КДИС

where R(1)ЗМ hellip R(n)

ЗМ minus ratings of content modules on a 100-point scale

n - number of structural modules

К(1)ЗМ hellip К(n)

ЗМ minus number of ECTS credits provided work curriculum for the

corresponding semantic module

КДИС = К(1)ЗМ + hellip + К(n)

ЗМ minus number of ECTS credits provided work curriculum for

courses in the current semester

R ДР minus rating with additional work

R ШТР minus fine rating

The formula can be simplified if we take К(1)ЗМ = hellip= К(n)

ЗМ Then it will look like

07 (R(1)ЗМ + + R(n)

ЗМ )

RНР = ------------------------------------ + RДР - RШТР

n

Rating of additional work R ДР added to the R НР and can not exceed 20 points It is determined by the lecturer and the students given the decision of the Department for the execution of work not provided for by the curriculum but enhance students knowledge in the discipline Fine rating R ШТР less than 5 points and subtracted from R НР It is determined by the lecturer and the decision of the Department administered to students who have learned the material semantic module on time do not comply with work schedules missed classes and more

2 According to the principle the preparation and defense of a course project (work) measured at 100 point scale and then translated into a score for the national scale and scale ECTS

Grading scale

Evaluation on the scale Total points for all activities

excellent 90 ndash 100good 74 ndash 89

satisfactorily 60-73unsatisfactorily 0-59

13 Methodological support 1 Regulations 2 Complex teaching of the discipline 3 Methodological guidelines for independent study courses 4 Methodological guidelines for writing a term paper5 Elearn httpselearnnubipeduuacourseviewphpid=1880

where R(1)ЗМ hellip R(n)

ЗМ minus ratings of content modules on a 100-point scale

n - number of structural modules

К(1)ЗМ hellip К(n)

ЗМ minus number of ECTS credits provided work curriculum for the

corresponding semantic module

КДИС = К(1)ЗМ + hellip + К(n)

ЗМ minus number of ECTS credits provided work curriculum for

courses in the current semester

R ДР minus rating with additional work

R ШТР minus fine rating

The formula can be simplified if we take К(1)ЗМ = hellip= К(n)

ЗМ Then it will look like

07 (R(1)ЗМ + + R(n)

ЗМ )

RНР = ------------------------------------ + RДР - RШТР

n

Rating of additional work R ДР added to the R НР and can not exceed 20 points It is determined by the lecturer and the students given the decision of the Department for the execution of work not provided for by the curriculum but enhance students knowledge in the discipline Fine rating R ШТР less than 5 points and subtracted from R НР It is determined by the lecturer and the decision of the Department administered to students who have learned the material semantic module on time do not comply with work schedules missed classes and more

2 According to the principle the preparation and defense of a course project (work) measured at 100 point scale and then translated into a score for the national scale and scale ECTS

Grading scale

Evaluation on the scale Total points for all activities

excellent 90 ndash 100good 74 ndash 89

satisfactorily 60-73unsatisfactorily 0-59

13 Methodological support 1 Regulations 2 Complex teaching of the discipline 3 Methodological guidelines for independent study courses 4 Methodological guidelines for writing a term paper5 Elearn httpselearnnubipeduuacourseviewphpid=1880

14 Recommended literature

Tutorials1 Жильцов ОБ Теорія ймовірностей та математична статистика у прикладах і

задачах навч посіб для студ вищ навч закл ОБ Жильцов за ред ГО Михаліна mdash К Київ ун-т ім Б Грінченка 2015 mdash 336 с

2 ІА Рудоміно-Дусятська ЛМ Козубцова ОЮ Пояркова ТВ Соловйова ВЄ Сновида ЛМ Цитрицька Теорія ймовірностей теорія випадкових процесів та математична статистика (частина І) ndash К ВІТІ 2018 ndash 187 с

3 Кушлик-Дивульська ОІ Теорія ймовірностей та математична статистика навч посіб frasl О І Кушлик-Дивульська Н В Поліщук Б П Орел ПІ Штабалюкminus К НТУУ laquoКПІraquo 2014 ndash 212 с

4 Огірко О І Галайко Н В Теорія ймовірностей та математична статистика навчальний посібник О І Огірко Н В Галайко ndash Львів ЛьвДУВС 2017 ndash 292 с

Supplementary1 Lo GS(2016) Measure Theory and Integration By and For the Learner SPAS

Books Series Saint-Louis Senegal - Calgary Canada Doi httpdxdoiorg1016929sbs20160005 ISBN 978-2-9559183-5-7 [Lo (2016)] Lo GS(2016) Mathematical Foundation Probability Theory SPAS Books Series Saint-Louis Senegal - Calgary Canada To be posted soon

2 Турчин ВМ Теорія ймовірностей і математична статистика Основні поняття приклади задачі ndash Д ІМАпрес 2014

3 Булаєнко М В Теорія ймовірностей Конспект лекцій з дисципліни ldquoТеорія ймовірностей і математична статистикаrdquo ndash Х ХНАМГ 2011

15 Information Resources1 Верховна Рада України httpzakonradagovua2 Кабінет Міністрів України httpwwwkmugovuacontrol3 Державна Служба статистики України httpukrstatgovua4 Продовольча та сільськогосподарська організація ООН (ФАО) httpwwwfaoorg5 Світовий банк httpwwwworldbankorg6 Євростат httpeppeurostateceuropaeuportalpageportaleurostathome7 Міністерство сільського господарства США httpswwwusdagov 8 Положення про екзамени та заліки у Національному університеті

біоресурсів і природокористування України [Електронний ресурс] httpsnubipeduuasitesdefaultfilesu284polozh_ekzameni_zaliki_02_2019_ostatochniy_dlya_nakazupdf ndash 2019

TESTSTask 1 Name the type of grouping that is used for studying the availability and direction of connection between signs from which one is effective and another - factor that affects the result (answer-word)

Task 2 On 20 (n) plots of agricultural lands the average yield is 30 centners ha ( ) N1=10 n2= 5 n3=5 =25 =31) Calculate the average yield of third groupTask 3 QuizzesQuestion 1 If all frequencies of the row of distribution increase or decrease in the constant number of times k the arithmetic middling will increase or decrease in k times

Answer 1Correct2Incorrect

Question 2 Net profit during the first year increased on 10 during the second - on 5 third - 15 On what percentage did the profit increase over three yearsQuestion 3 Choose the correct sequence of growth of different types of middling calculated for one and the same variation row (majorization)

1 geometric2 quadratic3 harmonic

4 arithmeticQuestion 4 System of measures that turn to increase crop yield in agriculture is called (answer ndash in a word)Question 17 Square of average error is directly proportional to the dispersion of the sign in the general aggregate and is inversely proportional to size of sampling

1 Yes 2 NoQuestion 5 Write linear form of regressionQuestion 6 What is region of acceptability Choose one correct answer

1 number of acceptability position of sample point in sample space that lead to acceptance of null hypothesis

2 sample space of sample variable3 critical region4 statistical criterion

Question 7 What means statistical hypothesis Choose one or several correct answers 1 scientifical assumption that demands control confirmation2 scientifical assumption about size of statistical characteristic3 scientifical assumption that needs to be controlledQuestion 8 Value of varying sign which is in the middle of ranged distribution row Choose one correct answer1 dispersion2 mode3 median4 arithmetic

Question 9 Match names of average values with the methods of their calculation

A

1 quadratic middling2 arithmetic middling3 harmonic middling4 geometric middling

B

C

D Question 10 Determine correspondence of interconnection between phenomenonsA particular significance of factor respons only one concrete significance of result

1correlation relationship2function relationship

B particular significance of factor respons multiple significance of result

• Program material of the course ldquoTheory of Probabilityrdquo
• Theoretical and practical course subject ldquoTheory of Probabilityrdquo is taught using a large number of examples calculations tables graphs solving common tasks and multimedia equipment computer programs
• • Module 2 Mathematical statistics
  • • Total sum

Question 9 Match names of average values with the methods of their calculation

A

1 quadratic middling2 arithmetic middling3 harmonic middling4 geometric middling

B

C

D Question 10 Determine correspondence of interconnection between phenomenonsA particular significance of factor respons only one concrete significance of result

1correlation relationship2function relationship

B particular significance of factor respons multiple significance of result

• Program material of the course ldquoTheory of Probabilityrdquo
• Theoretical and practical course subject ldquoTheory of Probabilityrdquo is taught using a large number of examples calculations tables graphs solving common tasks and multimedia equipment computer programs
• • Module 2 Mathematical statistics
  • • Total sum