Теория вероятностей и математическая статистика для применения в анализе данных

ФЕДЕРАЛЬНОЕ ГОСУДАРСТВЕННОЕ ОБРАЗОВАТЕЛЬНОЕ  
БЮДЖЕТНОЕ УЧРЕЖДЕНИЕ ВЫСШЕГО ОБРАЗОВАНИЯ  
«ФИНАНСОВЫЙ УНИВЕРСИТЕТ ПРИ ПРАВИТЕЛЬСТВЕ  
РОССИЙСКОЙ ФЕДЕРАЦИИ»  
(ФИНАНСОВЫЙ УНИВЕРСИТЕТ)

Департамент математики

О.Е. Пыркина

PROBABILITY THEORY  
AND MATHEMATICAL STATISTIC FOR 
APPLICATIONS IN DATA ANALYSIS
 
Учебное пособие
по дисциплине «Анализ данных» (на английском языке)

Направление подготовки: 38.03.01 Экономика 

профили: 
Международные финансы
Международная торговля и налогообложение,
Мировая экономика и международный бизнес,
Мировые финансы,
Международный бизнес энергетических компаний,

Программы подготовки бакалавра
С реализацией или частичной реализацией образовательных 
программ на английском языке

МОСКВА
2023

FEDERAL STATE-FUNDED EDUCATIONAL INSTITUTION  
OF HIGHER EDUCATION  
«FINANCIAL INIVERSITY UNDER THE GOVERNMENT  
OF RUSSIAN FEDERATION»

Department of Mathematics

O.E. Pyrkina

PROBABILITY THEORY AND 
MATHEMATICAL STATISTIC FOR 
APPLICATIONS IN DATA ANALYSIS
 
Textbook 
(in English)

BSc in Economics (38.03.01) 

Concentration: 
International Finance 
International Trade and Taxation
Accounting and Financial Analysis
World Economy and International Business
World Finance
Economics and Finance of the Fuel and Energy Sector

МОСКВА
2023

ISBN 978-5-00172-475-9

УДК 519.2                                                  
ББК 22.17
 
П95
Рецензенты:
С.А. Зададаев, кандидат физико-математических наук, 
доцент, руководитель департамента математики;
В.В. Булатов, доктор физико-математических наук, доктор 
экономических наук, профессор, действительный член РАЕН, 
ведущий научный сотрудник Института проблем механики им. 
А.Ю. Ишлинского РАН

П95
Пыркина О.Е.
Теория вероятностей и математическая статистика 
для применения в анализе данных: Учебное пособие / 
О.Е. Пыркина. — М.: Прометей, 2023. — 582 с.

ISBN 978-5-00172-475-9
Учебное пособие "Probability Theory and Mathematical Statistic 
for Applications in Data Analysis" на английском языке (русский 
вариант названия —  «Теория вероятностей и математическая статистика для применения в анализе данных») готовит читателей 
к успешной работе с информацией в рамках современной науки 
о данных (Data Science). Продуктивное развитие цифровой экономики невозможно без умения специалистов грамотно и эффективно 
оперировать непрерывно поступающим потоком цифровых данных 
статистического характера. Для обработки таких данных и принятия управленческих решений на основе данных необходимы умения 
и навыки как технического, так и теоретического уровня, позволяющие проводить обобщения и делать выводы на основе поступившей 
информации.
В пособии последовательно рассматриваются традиционные 
темы курсов теории вероятностей и математической статистики 
как теоретической основы анализа данных. Все вопросы курса рассматриваются с применением статистических функций и пакета 
анализа данных Excel. Курс дополнен примерами, задачами 
и тестовыми вопросами для самопроверки. Пособие состоит из 20 
глав, введения и заключения.
Учебное пособие может быть использовано студентами и преподавателями университетов (в частности, Финансового университета при Правительстве РФ) в курсе дисциплины «Анализ данных» 
(дисциплины базовой части математического цикла дисциплин 
по направлению подготовки 38.03.01 «Экономика», профили: 
«Международные финансы» (на английском языке), «Международная торговля и налогообложение» (на английском языке), «Мировая экономика и международный бизнес» (с частичной реализацией 
на английском языке), «Мировые финансы» (с частичной реализацией на английском языке), «Международный бизнес энергетических компаний» (с частичной реализацией на английском языке), 
программы подготовки бакалавра.
Одобрено советом департамента математики, протокол 
№03 от 23 сентября 2022 года.

©  Пыркина О.Е., 2023
© Издательство «Прометей», 2023

ISBN 978-5-00172-475-9

UDC 519.2                                                                      
BBC 22.17
 
P95
Readers:
S.A. Zadadaev, Ph.D Physics & Mathematics, associate 
professor, Head of Mathematics Department 
V.V. Bulatov, Doctor of Science Physics & Mathematics, 
Doctor of Science Economics, professor, Member of Russian 
Academy of Natural Science, Leading Research Scientist in the 
Ishlinsky Institute for Problems in Mechanics of the Russian 
Academy of Sciences

P95
Pyrkina O.E.
Probability Theory and Mathematical Statistic 
for Applications in Data Analysis: Textbook / 
O.E. Pyrkina. — M.: Prometej, 2023. — 582 pages.

ISBN 978-5-00172-475-9
The textbook "Probability Theory and Mathematical Statistic 
for Applications in Data Analysis" prepares readers for successful 
operation with information as a part of contemporary data science. 
The productive formation and development of the digital economy is 
impossible without the ability of specialists to operate competently 
and effectively with a continuously incoming stream of digital 
statistical data. To process such data and to make management 
decisions based on the data, skills and abilities of both technical and 
theoretical levels are required, that allows to carry out generalizations 
and make conclusions based on the information received.
The textbook discusses step by step traditional topics of courses 
in probability theory and mathematical statistics as a theoretical 
foundation for data analysis. All course questions are considered 
with application of statistical functions and the Excel data analysis 
package. The course is supplemented with examples, tasks and test 
questions for self-examination. The textbook includes 20 chapters, 
an introduction and conclusion.
The textbook can be used by students and lecturers of universities 
(in particular, the Financial University under the Government of the 
Russian Federation) in the course of "Data Analysis" (disciplines of 
the basic part of the mathematical cycle of disciplines, for a field 
of study 38.03.01 "Economics", study programs (concentrations): 
"International Finance" ( in English), "International Trade and 
Taxation" (in English), "World Economy and International Business" 
(with partial implementation in English), "World Finance" (with 
partial implementation in English), "International Business of 
Energy companies" (with partial implementation in English), level 
of study: bachelor's degree programs.

©  Pyrkina O.E., 2023
© Prometheus publishing house, 2023

ОГЛАВЛЕНИЕ

Chapter 1. Event Algebra. Basic Concepts ...................................12
1.1. Introduction: What is Probability? ............................12
1.2. Random experiment ...............................................13
1.3. Events .................................................................15
Self-testing questions ...................................................27

Chapter 2. Probability and Its Postulates. Probability Rules ......29
2.1. How Could We Define Probability? ............................29
2.2. Formalism: Postulates and Consequences ...................29
2.3. Introduction into Combinatorial Calculus:  
Permutation and Combinations ................................35
Self-testing questions ...................................................42
Self-testing questions: answers .......................................47

Chapter 3. Conditional Probability. Statistical Independence. ...48
3.1. The notion of conditional probability .........................48
3.2. The multiplication rule of probabilities  
and statistical independence ....................................50
Self-testing questions ...................................................58
Self-testing questions: answers .......................................63

Chapter 4. Bayes’ Theorem and Total Probability Formula. 
Bivariate probabilities ..................................................................64
4.1. Bayes’ Theorem .....................................................64
4.2. Total Probability Formula .......................................66
4.3. Bivariate probabilities: general setup .........................70
Self-testing questions ...................................................77
Self-testing questions: answers .......................................79

Chapter 5. Random variables. Probability distributions for 
discrete random variables ............................................................80
5.1. Random variables ..................................................80
5.2. Probability distributions for discrete random  
variables ..............................................................83
5.3. Expectations for Discrete Random Variables ...............89

5.4. Variances for Discrete Random Variables ....................93
5.5. The Linear Function of Discrete Random Variable. .......95
Self-testing questions ...................................................99
Self-testing questions: answers ..................................... 103

Chapter 6. Jointly Distributed Discrete Random Variables. .....105
6.1. The Joint probability Function and Marginal  
Probability Functions ........................................... 105
6.2. The Conditional Probabilities and Independence of 
Discrete Random Variables .................................... 108
6.3. The Joint Cumulative Probability Function. .............. 111
6.4. The Covariance and Correlation Coefficient ............... 112
Self-testing questions ................................................. 120
Self-testing questions: answers ..................................... 122

Chapter 7. Bernoulli Trials and Binomial Distribution. The 
Hypergeometric Distribution. The Geometric Distribution. 
The Poisson Distribution. ...........................................................125
7.1. Bernoulli Trials ................................................... 125
7.2. The Binomial Distribution ..................................... 128
7.3. The Hypergeometric Distribution ........................... 138
7.4. The Geometric Distribution ................................... 142
7.5. The Poisson Distribution ....................................... 145
Self-testing questions ................................................. 153
Self-testing questions: answers ..................................... 155

Chapter 8. Continuous Random Variables .................................157
8.1. Continuous random variables: Statement of a Problem ... 157
8.2. Probability Distributions for continuous random 
variables. ........................................................... 158
8.3. Numerical characteristics for continuous random 
variables ............................................................ 167
8.4. Jointly Distributed Continuous Random Variables ..... 171
8.5. Uniform Distribution: general view. ........................ 176
8.6. Normal Distribution: general view .......................... 180
8.7. The central limit theorem. ..................................... 197
8.8. The normal distribution as an approximation to the 
binomial and Poisson distributions .......................... 202

8.9. The Exponential Distribution. ................................ 208
8.10. The Lognormal distribution ................................. 212
Self-testing questions ................................................. 214
Self-testing questions . Answers ................................... 220

Chapter 9. Laws of large numbers ..............................................221
9.1. Chebyshev inequality ............................................ 221
9.2. Laws of large numbers. ......................................... 224
9.3. Bernoulli’s theorem ............................................. 230
Self-testing questions ................................................. 232
Self-testing questions. Answers .................................... 235

Chapter 10. Moments of a single random variable and 
jointly distributed continuous random variables ......................236
10.1. Moments and higher-order moments of probability 
distribution ........................................................ 236
10.2. Moments of two or more random variables .............. 243
10.3. Conditional distributions ..................................... 244
10.4. Moment generating functions ............................... 250
Self-testing questions ................................................. 256
Self-testing questions. Answers .................................... 258

Chapter 11. Jointly Distributed Continuous Random 
Variables .....................................................................................260
11.1. Joint density functions ....................................... 260
11.1. Function of two random variables .......................... 267
11.2. Bivariate normal distribution ............................... 277
Self-testing questions ................................................. 285
Self-testing questions. Answers .................................... 287

Chapter 12. Introduction in the theory of Markov Chains ........288
12.1. The main notions ............................................... 288
12.2. Specifying a Markov Chain ................................... 290
12.3. Long-term behavior of a Markov chain ................... 294
12.4. Absorbing Markov Chains .................................... 297
Self-testing questions ................................................. 305
Self-testing questions. Answers .................................... 306

Chapter 13. Summarizing Numerical Information ...................307
13.1. Population and samples ....................................... 307
13.2. Distinction between two types of data sets .............. 308
13.3. Numerical Summary: Measures of Central Tendency ...309
13.4. Numerical Summary: Measures of Dispersions ........ 314
Self-testing questions ................................................. 323
Self-testing questions . Answers ................................... 324

Chapter 14. Summarizing Numerical Information  
for Grouped Data ........................................................................326
14.1. Grouping the observations ................................... 326
14.2. Numerical summary of grouped data ..................... 335
Self-testing questions ................................................. 348
Self-testing questions. Answers .................................... 350

Chapter 15. Sampling and Sampling Distributions ..................351
15.1. Sampling from a population ................................. 351
15.2. Sampling distribution of the sample mean. ............. 355
15.3. Sampling distribution of the sample proportion ....... 362
15.4. Sampling distribution of the sample variance .......... 366
Self-testing questions ................................................. 375
Self-testing questions. Answers .................................... 377

Chapter 16. Point Estimations and Methods of its’ Creation ....379
16.1. Introduction: main definitions .............................. 379
16.2. Unbiased estimators, their efficiency  
and consistency ................................................... 383
16.3. Method of moments ............................................ 392
16.4. Method of Maximum Likelihood Estimation ............ 394
Self-testing questions ................................................. 403
Self-testing questions. Answers .................................... 404

Chapter 17. Confidence intervals ...............................................405
17.1. Interval estimation: introduction .......................... 405
17.2. Interval estimation: the center and boundaries ........ 407
17.3. Confidence intervals for the mean of a normal 
distribution: population variance known .................. 410

17.4. Confidence intervals for the mean of a normal 
distribution: population variance unknown, large 
sample size ......................................................... 419
17.5. The Student’s t Distribution ................................ 421
17.6. Confidence intervals for the mean of a normal 
distribution: population variance unknown, small 
sample size ......................................................... 425
17.7. Confidence intervals for the population proportion  
(large samples) .................................................... 428
17.8. Confidence intervals for the variance of a normal 
population .......................................................... 431
17.9. Estimating the sample size................................... 435
Self-testing questions ................................................. 440
Self-testing questions. Answers .................................... 442

Chapter 18. Hypothesis Testing .................................................443
18.1. The concept of statistical hypothesis testing............ 443
18.2. Tests of the mean of a normal distribution: simple 
null, population variance known ............................. 454
18.3. What is meant by the rejection of a null 
hypothesis? P-value of the test ............................... 457
18.4. Tests of the mean of a normal distribution:  
population variance known. Composite null  
and alternative hypothesis ..................................... 459
18.5. Test of the mean of a normal distribution, 
population variance unknown: large sample sizes. ...... 464
18.6. Test of the mean of a normal distribution, 
population variance unknown ................................. 467
18.7. Test of the variance of a normal distribution ........... 470
18.8. Test of the population proportions (large samples). ... 474
18.9. Tests for the differences between two means. Test 
based on matched pairs. Test based on independent 
samples .............................................................. 477
18.10. Tests for the differences between two population 
proportions (large samples) .................................... 487
18.11. Testing the equality of the variances of two 
normal populations. F-distribution ......................... 491

18.12. Measuring the power of a test ............................. 495
18.13. Some comments of hypothesis testing .................. 503
18.14. Test of normality .............................................. 505
18.15. Goodness-of-fit tests ......................................... 508
18.16. A test of association in contingency tables ............. 517
Self-testing questions ................................................. 521
Self-testing questions. Answers .................................... 528

Chapter 19. Some nonparametric tests ......................................531
19.1. Introduction. The sigh test .................................. 531
19.2. The Wilcoxon test .............................................. 537
19.3. The Mann-Whitney test ....................................... 543
19.4. Discussion ........................................................ 549
Self-testing questions ................................................. 551
Self-testing questions. Answers .................................... 554

Chapter 20. ANOVA (Analysis of variance) ...............................554
20.1. Comparison of several population means ................. 554
20.2. One-way analysis of variance ................................ 561
20.3. The Kruskal-Wallis test ....................................... 573
Self-testing questions ................................................. 577
Self-testing questions. Answers .................................... 579
Bibliography ...............................................................................580