- •Introduction
- •Basic concepts of probability theory
- •Classical definition of probability
- •Relative frequency
- •Geometric probabilities
- •Glossary
- •Exercises for Seminar 1
- •Exercises for Homework 1
- •Basic formulas of combinatorial analysis
- •Operations over events
- •Glossary
- •Exercises for Seminar 2
- •Exercises for Homework 2
- •Theorem of addition of probabilities of incompatible events
- •Complete group of events
- •Opposite events
- •Conditional probability
- •Theorem of multiplication of probabilities
- •Glossary
- •Exercises for Seminar 3
- •Exercises for Homework 3
- •Independent events
- •Where a is the appearance of at least one of the events a1, a2, …, An; .
- •Glossary
- •Exercises for Seminar 4
- •Exercises for Homework 4
- •Theorem of addition of probabilities of compatible events
- •Formula of total probability
- •Probability of hypotheses. Bayes’s formulas.
- •Glossary
- •Exercises for Seminar 5
- •Exercises for Homework 5
- •Repetition (recurrence) of trials. The Bernoulli formula
- •Local theorem of Laplace
- •Integral theorem of Laplace
- •Glossary
- •Exercises for Seminar 6
- •Exercises for Homework 6
- •Random variables. The law of distribution of a discrete random variable
- •A random variable is understood as a variable which as result of a trial takes one of the possible set of its values (which namely – it is not beforehand known).
- •Mathematical operations over random variables
- •(Mathematical) expectation of a discrete random variable
- •Dispersion of a discrete random variable
- •Glossary
- •Exercises for Seminar 7
- •Exercises for Homework 7
- •Distribution function of a random variable
- •Properties of a distribution function
- •Continuous random variables. Probability density
- •Properties of probability density
- •Glossary
- •Exercises for Seminar 8
- •Exercises for Homework 8
- •Basic laws of distribution of discrete random variables
- •1. Binomial law of distribution
- •2. The law of distribution of Poisson
- •3. Geometric distribution
- •4. Hypergeometric distribution
- •Glossary
- •Exercises for Seminar 9
- •Exercises for Homework 9
- •Basic laws of distribution of continuous random variables
- •1. The uniform law of distribution
- •2. Exponential law of distribution
- •3. Normal law of distribution
- •Glossary
- •Exercises for Seminar 10
- •Exercises for Homework 10
- •The law of large numbers and limit theorems
- •The central limit theorem
- •Glossary
- •Exercises for Seminar 11
- •Exercises for Homework 11
- •Mathematical statistics. Variation series and their characteristics
- •Numerical characteristics of variation series
- •Glossary
- •Exercises for Seminar 12
- •Exercises for Homework 12
- •Bases of the mathematical theory of sampling
- •Glossary
- •Exercises for Seminar 13
- •Exercises for Homework 13
- •Methods of finding of estimations
- •Notion of interval estimation
- •Glossary
- •Exercises for Seminar 14
- •Exercises for Homework 14
- •Testing of statistical hypotheses
- •Glossary
- •Exercises for Seminar 15
- •Exercises for Homework 15
- •Individual homeworks
- •Variant 1
- •Variant 2
- •Variant 3
- •Variant 4
- •Variant 5
- •Variant 6
- •Variant 7
- •Variant 8
- •Variant 9
- •Variant 10
- •Variant 11
- •Variant 12
- •Variant 13
- •Variant 14
- •Variant 15
- •Variant 16
- •Variant 17
- •Variant 18
- •Variant 19
- •Variant 20
- •Variant 21
- •Variant 22
- •Variant 23
- •Variant 24
- •Variant 25
- •Final exam trial tests (for self-checking)
- •Appendix
- •Values the functions and
- •List of the used books
- •Contents
List of the used books
Гмурман В.Е., Теория вероятностей и математическая статистика. Издание седьмое, стереотипное, Москва: «Высшая школа», 2000.
Сборник задач. Математика для экономистов. Под редакцией профессора Кельтеновой Р.Т., Алматы: «Экономика», 2000.
Горелова Г.В., Кацко И.А., Теория вероятностей и математическая статистика в примерах и задачах с применением Excel. – Издание четвертое, Ростов на Дону: «Феникс», 2006.
Коршунов Д.А., Фосс С.Г., Сборник задач и упражнений по теории вероятностей, Новосибирск, 1997.
Коршунов Д.А., Чернова Н.И., Сборник задач и упражнений по математической статистике, Новосибирск. 2001.
Кремер Н.Ш., Теория вероятностей и математическая статистика: Учебник для ВУЗов. — 2-е изд., перераб. и доп. М: ЮНИТИ-ДАНА, 2004. — 573 с.
Данко П.Е., Попов А.Г., Кожевникова Т.Я., Высшая математика в упражнениях и задачах. Издание пятое, исправленное. – Москва: «Высшая школа», 1999, часть 2.
Манита А.Д., Теория вероятностей и математическая статистика: Учебное пособие. – М.: Издательский отдел УНЦ ДО, 2001, 120 с.
Гмурман В.Е., Руководство к решению задач по теории вероятностей и математической статистике: Учебное пособие — 11-е изд., перераб. — М.: Высшее образование, 2006, 404 с.
Olav Kallenberg, Foundations of Modern Probability, second edition, Springer Series in Statistics, 2002, 650 pages.
Olav Kallenberg, Probabilistic Symmetries and Invariance Principles, Springer-Verlag, New York, 2005, 510 pages.
Hopkins W.G., A New View of Statistics, 2000, Internet Society for Sport Science: http://www.sportsci.org/resource/stats/.
Sheldon Ross, A First Course in Probability, seventh edition, University of Southern California, Pearson Education International, 2006, 561 pages.
Contents
INTRODUCTION ………………………………………………………..
|
3 |
LECTURE 1. Basic concepts of probability theory……………………….. |
5 |
Classical definition of probability ………………………………………….. |
6 |
Relative frequency …………………………………………………………. |
7 |
Geometric probabilities …………………………………………………….. |
8 |
Exercises ……………………………………………………………...........
|
10 |
LECTURE 2. Basic formulas of combinatorial analysis ……………..…… |
12 |
Operations over events …………………………………..………………..… |
15 |
Exercises ……………………………………………………………………
|
16 |
LECTURE 3. Theorem of addition of probabilities of incompatible events |
19 |
Complete group of events …………………………………………..……….. |
20 |
Opposite events………………………………………………………………. |
20 |
Conditional probability ……………………………………………………… |
21 |
Theorem of multiplication of probabilities…………………………………… |
22 |
Exercises ……………………………………………………………………
|
23 |
LECTURE 4. Independent events …………………………………………. |
26 |
Probability of appearance of at least one event …………………..………… |
28 |
Exercises ………………………………………………………..……………
|
30 |
LECTURE 5. Theorem of addition of probabilities of compatible events … |
32 |
Formula of total probability………………………………………………… |
33 |
Probability of hypotheses. Bayes’s formulas……………………………….. |
34 |
Exercises ……………………………………………………………………
|
36 |
LECTURE 6. Repetition (recurrence) of trials. The Bernoulli formula…… |
38 |
Local theorem of Laplace…………………………………..………… |
40 |
Integral theorem of Laplace ………………………………………………… |
41 |
Exercises ……………………………………………………………..………
|
43 |
LECTURE 7. Random variables. The law of distribution of a discrete random variable …………………………………………………………….. |
45 |
Mathematical operations over random variables……………………………. |
48 |
(Mathematical) expectation of a discrete random variable………………….. |
49 |
Dispersion of a discrete random variable…………………………………… |
51 |
Exercises ………………………………………………………………..……
|
53 |
LECTURE 8. Distribution function of a random variable………………….. |
56 |
Properties of a distribution function…………………..………………..…… |
57 |
Continuous random variables. Probability density………………………….. |
58 |
Properties of probability density…………………………………………….. |
59 |
Exercises …………………………………………………………………...
|
61 |
LECTURE 9. Basic laws of distribution of discrete random variables…….. |
65 |
Binomial law of distribution…………………………………………………. |
65 |
The law of distribution of Poisson……………………………………………. |
66 |
Geometric distribution ………………………………………..……………… |
68 |
Hypergeometric distribution………………………………………………….. |
69 |
Exercises …………………………………………………………..…….…...
|
70 |
LECTURE 10. Basic laws of distribution of continuous random variables…. |
73 |
Uniform law of distribution ……………………………………………......... |
73 |
Exponential law of distribution………………………………………………. |
74 |
Normal law of distribution……………………………………………………. |
75 |
Exercises …………………………………………………………………….
|
78 |
LECTURE 11. The law of large numbers and limit theorems………………. |
81 |
The central limit theorem…………………………………………………….. |
85 |
Exercises ……………………………………………………………............
|
87 |
LECTURE 12. Mathematical statistics. Variation series and their characteristics…………………………………………………………………. |
91 |
Numerical characteristics of variation series…………………………………. |
94 |
Exercises ………………………………………………………………….....
|
96 |
LECTURE 13. Bases of the mathematical theory of sampling…………….. |
99 |
Exercises ……………………………………………………………………..
|
102 |
LECTURE 14. Methods of finding of estimations…………………………. |
104 |
Method of moments …………………………………………………………. |
104 |
Method of maximal (the greatest) plausibility………………………………. |
105 |
Method of the least squares…………………………………………………. |
107 |
Notion of interval estimation……………………………………………….. |
107 |
Exercises …………………………………………………………………….
|
108 |
LECTURE 15. Testing of statistical hypotheses…………………………… |
110 |
Exercises ……………………………………………………………….........
INDIVIDUAL HOME WORKS …………………………………………
FINAL EXAM TRIAL TESTS (FOR SELF-CHECKING)…………….
|
113
116
152 |
APPENDIX………………………………………………………………….
|
162
|
LIST OF THE USED BOOKS …………………………………………… |
165 |