“A mathematician is one who knows how to find analogies between statements. The best mathematician is the one who establishes analogies of proofs. The stronger one can notice the analogies of theories. But there are those who see analogies between analogies.
Stefan Banach
Math statistics for dummies
Most often, mathematical statistics is studied along with probability theory(course "Probability Theory and Mathematical Statistics", TViMS). Useful materials on the theory of probability (online textbook, calculators, examples of solutions, etc.) you.
Topics: 1. General population and sampling 2. Comparison of means 3. Correlation and regression.
Online resources
- Klokov S.A., Problems in Probability Theory and Mathematical Statistics. For students of mathematical specialties, problems with answers, some with solutions.
- Manita AD, Theory of Probability and Mathematical Statistics. The book is aimed at students of the natural faculties of Moscow State University. M.V. Lomonosov. In addition to information about the printed version of the textbook, you will find on this site the full text of the book, including brief statistical tables.
Main content sections: Events and their probabilities. Discrete random variables and their distributions. General random variables. Joint distribution of general random variables. Limit laws of probability theory. Survey of methods of mathematical statistics. Least square method. Confidence intervals. statistical hypotheses. Tables (standard normal law, chi-square distribution quantiles, Student's distribution quantiles).
- Chernova NI, Lectures on mathematical statistics Semester course of lectures. Very detailed and clear, recommended for economics students.
- Electronic textbook on mathematical statistics.
Tutorial includes: 1) Course of lectures on mathematical statistics: V.V. Shelomovsky. Mathematical Statistics (Murmansk: MGPU, 2005. - 128 p.), 2) A cycle of laboratory work performed using Maple, allowing you to better understand the calculation methods, 3) A cycle of tests to test knowledge.
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OGE (GIA) USE preparation for the olympiads school course Algebra Analytic geometry higher mathematics+8 Geometry Combinatorics Linear algebra Math statistics Mathematical analysis Applied math Probability theory Trigonometry
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A very effective tutor and talented teacher - he knows how to present the program of higher mathematics of the university in such a way that the course of mathematics from a nightmare has become annoying Expand necessity - despite the fact that from the school course the student confidently knew only the program of grades 5-6. All reviews (46)
Analytic geometry Calculus of variations Vector Analysis +33 higher mathematics Geometry Discrete Math Differential geometry Differential Equations Combinatorics Linear algebra Linear geometry Linear programming Math statistics Mathematical physics Mathematical models Mathematical analysis Optimal decision methods Optimization methods Optimal control Applied math Sopromat Tensor Analysis Theoretical mechanics Probability theory Graph theory Game theory Optimization theory Number theory Topology Trigonometry TFKT Partial differential equations Equations of mathematical physics financial mathematics functional analysis Econometrics
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Doctor of Physical and Mathematical Sciences. Leading Researcher at Moscow State University (Faculty of Mechanics and Mathematics), Professor at the Faculty of Additional Education Expand MGIMO, was a member of the examination committees in mathematics of Moscow State University, MGIMO, MGUDT.
Alexey Vasilievich is exactly the teacher we have been looking for for a long time. He knows how to find an approach to the student and competently present the educational material. All reviews (29)
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Prize-winner of the Olympiad Lomonosov 2007 in subjects - oral and written mathematics, composition. Participant of the interfaculty special course of Olympiad problems Expand Department of Mathematical Analysis of the Mekh-mat of Moscow State University. Experience in conducting circles of small fur-mat 2007-2012. Optional mathematics at lyceum 1553. Teacher of algebra, geometry, computer science, English at lyceum 1553 in 2011. Accompanying the education of children in language camps in England and Malta 2011-2012. Three years of experience in retail management in the central office of the largest bank in the CIS. I conduct classes using a Wacom graphics tablet and an online whiteboard (paid, which has the ability to use several people at the same time, simultaneous editing, video and sound are joint). After the lesson, the links to the room remain - the student always has access to what was written in the lesson and has access to the notes for the entire course, all materials written on the board are also sent to the client in PDF format. It is used for communication both Skype and the online room itself. The number of students prepared for exams is more than 100; Prepared for exams students of various universities of Moscow State University Mechanics and Mathematics, Faculty of Physics, Faculty of Economics, Moscow State Pedagogical University, Plekhanov, Financial Academy under the President, MGIMO, MEPhI, etc. I prepare children for the All-Russian, Lomonosov and Vuzovsky Olympiads under Bauman and Mifi, MIPT. Teaching is my main activity. I also prepare for admission to English and Swiss colleges. Passing a unified A-level exam in English in mathematics and physics. I prepare schoolchildren for passing the English OGE and the Unified State Examination.
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OGE (GIA) USE school course Algebra Analytic geometry higher mathematics Geometry +12 Discrete Math Differential Equations Linear algebra Linear geometry Math statistics Mathematical analysis In English Probability theory Graph Theory Game Theory Trigonometry Econometrics
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Graduate of the mech-mat of Moscow State University. There is experience in the banking sector as an analyst, experience as a system analyst in the field of IT development. Knowledge Expand programming, relational databases (sql). The first category in chess. There is a successful experience of working with all categories of students: Schoolchildren (OGE, Unified State Examination, improving academic performance) Students (almost all sections of higher mathematics and mechanics) Adults (classes "for oneself", help with work issues).
Course of Probability Theory and Mathematical Statistics. Sevastyanov B.A.
M.: Science. Ch. ed. Phys.-Math. lit., 1982.- 256 p.
The book is based on a one-year course of lectures given by the author for a number of years at the Department of Mathematics of the Faculty of Mechanics and Mathematics of Moscow State University. The basic concepts and facts of probability theory are introduced initially for a finite scheme. The mathematical expectation is generally defined in the same way as the Lebesgue integral, but the reader is not expected to have any prior knowledge of Lebesgue integration.
The book contains the following sections: independent tests and Markov chains, de Moivre-Laplace and Poisson limit theorems, random variables, characteristic and generating functions, law of large numbers, central limit theorem, basic concepts of mathematical statistics, testing of statistical hypotheses, statistical estimates, confidence intervals .
For undergraduate students of universities and technical colleges studying probability theory.
Format: djvu/zip
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TABLE OF CONTENTS
Preface 7
Chapter 1 Probability Space 9
§ 1. The subject of the theory of probability 9
§ 2. Events 12
§ 3. Probability space 16
§ 4. Finite probability space. Classic definition of probability 19
§ 5 Geometric probabilities 23
Tasks 24
Chapter 2. Conditional probabilities. Independence 26
§ 6. Conditional probabilities 26
§ 7. Total probability formula 28
§ 8. Bayes Formulas 29
§ 9. Independence of events 30
§ 10. Independence of partitions, algebras and a-algebras.... 33
§ 11. Independent tests 35
Tasks 39
Chapter 3. Random Variables (Final Scheme). 41
§ 12. Random variables. Indicators 41
§ 13. Mathematical expectation 45
§ 14. Multidimensional laws of distribution 50
§ 15. Independence of random variables 53
§ 10. Euclidean space of random variables. . . . 5th
§ 17. Conditional expectations 5E
§ 18. Chebyshev's inequality. Law of large numbers.... 61
Tasks 64
Chapter 4. Limit theorems in the Bernoulli scheme. 65
§ 19. Binomial distribution 65
§ 20. Poisson's theorem 66
§ 21. Local limit theorem of De Moivre - Laplace. . 70
§ 22. Integral limit theorem of De Moivre - Laplace 71
§ 23. Applications of limit theorems. 73
Tasks 76
Chapter 5. Markov Chains 77
§ 24. Markov dependence test 77
§ 25. Transition probabilities 78
§ 26. The theorem on limiting probabilities 80
Tasks 83
Chapter 6. Random variables (general case) 84
§ 27. Random variables and their distributions 84
§ 28. Multivariate distributions 92
§ 29. Independence of random variables 96
Tasks 98
Chapter 7. Expectation 100
§ 30. Definition of mathematical expectation 100
§ 31. Formulas for calculating the mathematical expectation 108
Tasks 115
Chapter 8 Generating Functions 117
§ 32. Integer random variables and their generating functions 117
§ 33. Factorial moments 118
§ 34. Multiplicative property 120
§ 35. Continuity theorem 123
§ 36. Branching processes 125
Tasks 127
Chapter 9 Characteristic Functions 129
§ 37. Definition and elementary properties of characteristic functions 129
§ 38. Inversion formulas for characteristic functions 136
§ 39. Continuous correspondence theorem between the set of characteristic functions and the set of distribution functions 140
Tasks 145
Chapter 10. Central Limit Theorem 146
§ 40. Central limit theorem for identically distributed independent terms 146
§ 41. Lyapunov's theorem 147
§ 42. Applications of the central limit theorem 150
Tasks 153
Chapter 11
§ 43. Definition and elementary properties 154
§ 44. Conversion formula 158
§ 45. Limit theorems for characteristic functions 159
§ 46. Multivariate normal distribution and related distributions 164
Tasks 173
Chapter 12
§ 47. Borel-Cantelli Lemma. Law "0 or 1" Kolmogorov 174
§ 48 Various types of convergence of random variables. . . 177
§ 49. Strong law of large numbers 181
Tasks 188
Chapter 13. Statistics 189
§ 50. Main tasks of mathematical statistics .... 189
§ 51. Sampling method 190
Tasks 194
Chapter 14. Statistical tests 195
§ 52. Statistical hypotheses 195
§ 53. Significance level and power of the test 197
§ 54. The optimal Neumann-Pearson criterion .... 199
§ 55. Optimal criteria for testing hypotheses about the parameters of normal and binomial distributions 201
§ 56. Criteria for testing complex hypotheses 2E4
§ 57. Non-parametric tests 206
Tasks 211
Chapter 15 Parameter Estimations 213
§ 58. Statistical estimates and their properties 213
§ 59. Conditional laws of distribution 216
§ 60. Sufficient statistics 220
§ 61. Efficiency of assessments 223
§ 62. Methods for finding estimates 228
Tasks 232
Chapter 16. Confidence intervals 234
§ 63. Determination of confidence intervals 234
§ 64. Confidence intervals for the parameters of the normal distribution 236
§ 65. Confidence intervals for the probability of success in the Bernoulli scheme 240
Tasks 244
Answers to problems 245
Normal distribution tables 251
Literature 253
Index 254
Ministry of the Russian Federation for Communications and Informatization
Siberian State University of Telecommunications and Informatics
N. I. Chernova
MATHEMATICAL
STATISTICS
Tutorial
Novosibirsk
Associate Professor, Cand. Phys.-Math. Sciences N. I. Chernova. Mathematical statistics: Textbook / SibGUTI. - Novosibirsk, 2009. - 90 p.
The textbook contains a semi-annual course of lectures on mathematical statistics for students of economic specialties. The textbook complies with the requirements of the State Educational Standard for professional educational programs in the specialty 080116 - "Mathematical Methods in Economics".
Chair MMBP Tab. 7, drawings - 9, list of lit. - 8 names
Reviewers: A.P. Kovalevsky, Ph.D. Phys.-Math. Sci., Associate Professor of the Department of Higher Mathematics, NSTU V. I. Lotov, Doctor of Physics and Mathematics. Sciences, Professor of the Department
theory of probability and mathematical statistics NSU
For the specialty 080116 - "Mathematical Methods in Economics"
Approved by the editorial and publishing council of SibGUTI as a textbook
c Siberian State University
telecommunications and informatics, 2009
Preface. . . . . . . . . . | ||
I. Basic concepts of mathematical statistics. . . . . . . . | ||
Problems of mathematical statistics . . . . . . . . . . . . . . . . . | ||
Sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | ||
Selected characteristics. . . . . . . . . . . . . . . . . . . . | ||
Properties of the empirical distribution function. . . . . . . . . | ||
§ 5. Properties of sample moments. . . . . . . . . . . . . . . . . . . 12
§ 6. Histogram as an estimate of density. . . . . . . . . . . . . . . . . 14
§ 7. Questions and exercises. . . . . . . . . . . . . . . . . . . . . . . . 15
CHAPTER II. Point Estimation. . . . . . . . . . . . . . . . . . . . . . 17
§ 1. Point estimates and their properties. . . . . . . . . . . . . . . . . . . 17
§ 2. Method of moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Properties of estimates of the method of moments. . . . . . . . . . . . . . . . . | |||
Maximum Likelihood Method. . . . . . . . . . . . . . . | |||
Asymptotic Normality of Estimates. . . . . . . . . . . . . . | |||
Questions and exercises. . . . . . . . . . . . . . . . . . . . . . . . | |||
Grade Comparison. . . . . . . . . . . . . . . . . . . . . . . | |||
Root Mean Square Approach to Comparing Estimates. . . . . . . . . | |||
Rao-Cramer inequality. . . . . . . . . . . . . . . . . . . . . | |||
Questions and exercises. . . . . . . . . . . . . . . . . . . . . . . . | |||
IV. interval estimation. . . . . . . . . . . . . . . . . . . | |||
Confidence intervals. . . . . . . . . . . . . . . . . . . . . . | |||
Principles for constructing confidence intervals. . . . . . . . | |||
Questions and exercises. . . . . . . . . . . . . . . . . . . . . . . . | |||
Distributions related to the normal . . . . . . . . . . | |||
Basic statistical distributions. . . . . . . . . . . . . . | |||
Transformations of normal samples. . . . . . . . . . . . . . . | |||
Confidence intervals for the normal distribution. . . | |||
§ 1. Hypotheses and criteria. . . . . . . . . . . . . . . . . . . . . . . . . 47
§ 2. Questions and exercises. . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter VII. Consent Criteria. . . . . . . . . . . . . . . . . . . . . . 51
§ 1. General view of the goodness of fit criteria. . . . . . . . . . . . . . . . . . . 51
§ 2. Testing simple hypotheses about parameters. . . . . . . . . . . . . . 53
§ 3. Criteria for testing the distribution hypothesis. . . . . . . . 56
§ 4. Criteria for testing parametric hypotheses. . . . . . . . 59
§ 5. Criteria for testing homogeneity. . . . . . . . . . . . . . . 61
§ 6. Criterion χ 2 for testing independence. . . . . . . . . . . . . 70
§ 7. Questions and exercises. . . . . . . . . . . . . . . . . . . . . . . . 71
§ 2. Maximum likelihood method.. . . . . . . . . . . . . . . 74
§ 3. Method of least squares.. . . . . . . . . . . . . . . . . . . 75
FOREWORD
The textbook contains a full course of lectures on mathematical statistics for students studying in the specialty "Mathematical Methods in Economics" of the Siberian State University of Telecommunications and Informatics. The content of the course fully complies with the educational standards for the preparation of bachelors in the specified specialty.
The course in mathematical statistics is based on the semester course in probability theory and is the basis for the yearly course in econometrics. As a result of studying the subject, students must master the mathematical methods of studying various models of mathematical statistics.
The course consists of eight chapters. The first chapter is the main one for understanding the subject. It introduces the reader to the basic concepts of mathematical statistics. The second chapter is devoted to methods for point estimation of unknown distribution parameters: moments and maximum likelihood.
The third chapter deals with the comparison of estimates in the root mean square sense. Here, the Rao-Cramer inequality is also studied as a means of checking the effectiveness of estimates.
The fourth chapter deals with the interval estimation of parameters, which ends in the next chapter with the construction of intervals for the parameters of the normal distribution. To do this, special statistical distributions are introduced, which are then used in the goodness-of-fit tests in the eighth chapter. Chapter six gives the necessary basic concepts of hypothesis testing theory, so the reader should study it very carefully.
Finally, chapters seven and eight provide a list of the most commonly used consent criteria in practice. In the ninth chapter, simple models and methods of regression analysis are considered and the main properties of the estimates obtained are proved.
Almost every chapter ends with a list of exercises in the text of the chapter. The application contains tables with a list of the main characteristics of discrete and absolutely continuous distributions, tables of basic statistical distributions.
FOREWORD |
There is a detailed index at the end of the book. The list of references lists textbooks that can be used in addition to the course, and collections of tasks for practical exercises.
The numbering of paragraphs in each chapter is separate. Formulas, examples, statements, etc. are numbered consecutively. When referring to an object from another chapter, for the convenience of the reader, the page number on which the object is contained is indicated. When referring to an object from the same chapter, only the number of the formula, example, statement is given. The end of proofs is marked with .
CHAPTER I
BASIC CONCEPTS OF MATHEMATICAL STATISTICS
Mathematical statistics is based on the methods of probability theory, but solves other problems. In probability theory, random variables with a given distribution or random experiments are considered, the properties of which are completely known. But where does knowledge about distributions come from in practical experiments? What is the probability, for example, that a coat of arms appears on a given coin? To determine this probability, we can flip the coin many times. But in any case, conclusions will have to be drawn from the results of a finite number of observations. So, observing 5,035 coats of arms after 10,000 tosses of a coin, it is impossible to draw an accurate conclusion about the probability of a coat of arms falling out: even if this probability differs from 0.5, the coat of arms can fall out 5035 times. Accurate conclusions about the distribution can only be made when an infinite number of tests have been carried out, which is not feasible. Mathematical statistics allows, based on the results of a finite number of experiments, to draw more or less accurate conclusions about the distributions of random variables observed in these experiments.
§ 1. Problems of mathematical statistics
Suppose we repeat the same random experiment under the same conditions. As a result of each repetition of the experiment, a certain set of data (numerical or otherwise) is observed.
This raises the following questions.
1. If one random variable is observed, how can one make the most accurate conclusion about its distribution based on a set of its values in several experiments?
2. If the manifestation of two or more signs is observed, what can be said about the type and strength of the dependence of the observed random variables?
It is often possible to make some assumptions about the observed distribution or about its properties. In this case, according to experimental data, it is required to confirm or refute these assumptions (“hypotheses”). At the same time, we must remember that the answer "yes" or "no" can only be given with a certain degree of certainty, and the longer we can continue the experiment, the more accurate the conclusions can be. Sometimes it is possible to assert in advance the presence
8 CHAPTER I. BASIC CONCEPTS OF MATHEMATICAL STATISTICS
some properties of the observed experiment - for example, about the functional dependence between the observed quantities, about the normality of the distribution, about its symmetry, about the presence of density in the distribution or about its discrete nature, etc.
So, mathematical statistics works where there is a random experiment, the properties of which are partially or completely unknown, and where we can reproduce this experiment under the same conditions some (or better, any) number of times.
Experimental results can be quantitative or qualitative. Quantitative results can, for example, be summed up. Thus, one of their meaningful characteristics is the arithmetic mean of observations. It is pointless to add up qualitative results, although they can be put into numerical form. Let's say the month of birth of the interviewee is a qualitative, not a quantitative observation: although it can be given as a number, the arithmetic mean of these numbers carries as much reasonable information as the message that, on average, a person was born between June and July.
In the first chapters, we will study working with quantitative observational results.
§ 2. Selection
Let ξ : Ω → R be a random variable observed in a random experiment. Carrying out this experiment n times under the same conditions, we will get the numbers X1 , X2 , . . . , Xn - values of the observed random variable in the first, second, etc. experiments. The random variable ξ has some distribution F, which is partially or completely unknown to us.
Let us consider in more detail the set X = (X1 , . . . , Xn ), called the sample.
In a series of experiments already performed, a sample is a set of numbers. But before the experiment is done, it makes sense to consider the sample as a set of random variables (independent and distributed in the same way as ξ ). Indeed, before conducting experiments, we cannot say what values the elements of the sample will take: these will be some of the values of the random variable ξ. Therefore, it makes sense to consider that before the experiment Xi is a random variable equally distributed with ξ , and after the experiment it is the number that we observe in the i-th experiment, i.e. one of the possible values of the random variable Xi .
Definition 1. A sample X = (X1 , . . . , Xn ) of size n from a distribution F is a set of n independent and identically distributed random variables that have a distribution F.
Sample elements are often transformed for the convenience of working with a large data set - they are ordered or grouped.
If the elements of the sample are X1 , . . . , Xn sort in ascending order, we get a set of new random variables, called a variational series:
X(1) 6 X(2) 6 . . . 6 X(n−1) 6 X(n) .
Here X(1) = min(X1 , . . . , Xn ), X(n) = max(X1 , . . . , Xn ). The element X(k) is called the k -th member of the variational series or the k -th order statistic.
When grouping data, several groups of sample element values are distinguished, the number of elements in each group is counted, and then only this new data set is dealt with. Both grouping and ordering data discard some of the information contained in the sample.
The task of mathematical statistics is to draw conclusions from a sample about the unknown distribution F, from which it is extracted. The distribution is characterized by a distribution function, density or table, a set of numerical characteristics: E ξ = E X1 , Dξ = D X1 , Eξ k = E X1 k . Based on the sample, one must be able to build approximations for all these characteristics. Such approximations are called estimates. The term "score" has nothing to do with inequalities. An estimate for some unknown characteristic of a distribution is a random variable constructed from a sample, which in some sense is an approximation of this unknown characteristic of the distribution.
Example 1. A six-sided die is rolled 100 times. The first face fell out 25 times, the second and fifth - 14 times each, the third - 21 times, the fourth - 15 times, the sixth - 11 times. We are dealing with a numerical sample, which, for convenience, is grouped by the number of points dropped.
According to the results of the experiment, it is impossible to determine the probabilities p1 , . . . , p6 face drops. We can only say that numerical estimates have been obtained for these probabilities: 0.25 for p1, 0.14 for p2 and for p5, etc.
Even without conducting such an experiment, we could say in advance that the estimate for the unknown probability p1 will be a random variable
and the estimate for the probability p2 is the random variable
In this series of experiments, these random variables took the values 0.25 and 0.14, respectively. In another series, their meanings will change.
CHAPTER I. BASIC CONCEPTS OF MATHEMATICAL STATISTICS |
§ 3. Selected characteristics
From the theory of probability, we know a universal tool for the approximate calculation of all kinds of mathematical expectations: the law of large numbers. This law guarantees that the arithmetic means of independent and identically distributed terms in some sense approach the expectation of a typical term (if, of course, this mathematical expectation exists).
Therefore, as an approximation (estimate) for the unknown mathematical expectation E X1, you can use the arithmetic mean of all sample elements: the sample mean
X1 + . . . +Xn | ||||||||||||||||||||||||||
As an estimate for E X1 k, the sample k -th moment | ||||||||||||||||||||||||||
X1 k + . . . + Xn k | ||||||||||||||||||||||||||
Xi k = | ||||||||||||||||||||||||||
and as an estimate for the variance D X1 = E (X1 − E X1 )2 = E X1 2 − (E X1 )2 |
||||||||||||||||||||||||||
sample variance is used | ||||||||||||||||||||||||||
S2 =n 1 | (Xi − X)2 = X2 − X | |||||||||||||||||||||||||
In general, the value | ||||||||||||||||||||||||||
g(X1 ) + . . . + g(Xn ) | ||||||||||||||||||||||||||
g(Xi ) = | ||||||||||||||||||||||||||
can be used to estimate the quantity E g(X1 ).
Similarly, Bernoulli's law of large numbers allows us to estimate different probabilities. For example, the probability of an event (X1< 3} можно заменить на долю успешных испытаний в схеме Бернулли: если для каждого элемента выборки успехом считать событие {Xi < 3}, то доля успехов
p = amount of Xi< 3n
will converge (in probability) to the probability of success P(X1< 3). Оценивать неизвестную функцию распределения F (y) = P(X1 < y) мож-
but with the help of the empirical distribution function
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120 minutes/4000 rub. - Items: Mathematics, Calculus, Probability Theory, Linear Algebra
- City: Moscow
- Nearest metro stations: Kitay-gorod, Lubyanka
- Home visit: available
- Status: Professor
- Education: Ural Pedagogical Institute, Faculty of Physics and Mathematics, graduated in 1982, diploma with honors. Candidate of Physical and Mathematical Sciences, Associate Professor of the State University.
- Lesson cost: 1500 r.-2000 r./60 min. depending on the class.
- Items: Mathematics, Calculus, Linear Algebra, Probability Theory
- City: Moscow
- Nearest metro station: Novogireevo
- Home visit: available
- Status: school teacher
- Education: Sverdlovsk Pedagogical Institute, specialty: mathematics, computer science and computer science, graduated in 1991.
Experienced teacher in mathematical statistics.
Professional and high-quality preparation for the 9th grade of the HSE Lyceum in 2019. Intensive work on the variants of the HSE Comprehensive Tests, as well as on assignments that strictly correspond to the exam variants! Careful development of methods for solving all tasks of the Comprehensive Test! The student will be well prepared!
Systematization of knowledge for grades 5 - 11. Effective and significant pull-ups in the program (algebra and geometry). Ensuring consistently high academic performance (for "4" and "5"). Thorough preparation for the OGE - 2019. Learning to solve problems of the 1st and 2nd parts of the OGE options ...
Private tutor in mathematical statistics.
Schoolchildren in grades 5-11, applicants (Preparation at Moscow State University or for tasks C5 and C6 at the Unified State Examination), students (classes in the general course of higher mathematics: mathematical analysis, analytical geometry, linear algebra, probability theory).
I give quite serious classes on author's materials, individually selected tasks for each student. In addition, I analyze complex Olympiad numbers and C6 with the Unified State Examination.
The minimum price for a lesson is 90 min. 3300 rub.
If preparation at Moscow State University or for tasks C5 and C6 on the Unified State Examination - within 3800-4000 rubles.
Professional math tutor. Guaranteed quality of work. Individual approach and selection of methods for each student...
- Lesson cost: 2200 rub. / 60 min
- Items: Mathematics, Calculus, Probability Theory, Linear Algebra
- City: Moscow
- Nearest metro station: Schukinskaya
- Home visit: No
- Status: Private teacher
- Education: Higher pedagogical education: Faculty of Mathematics, Moscow State Pedagogical University. Graduated in 1996.
Qualified tutor in mathematical statistics.
Subjects: Mathematics (school and higher, OGE and EGE), Physics (school, OGE and EGE), Probability theory, Mathematical statistics, Combinatorics.
Pupils, applicants, students. Preparation for any university, USE, Olympiad. Subjects: mathematics, physics, mathematical analysis, linear algebra, analytical geometry, probability theory, mathematical statistics, random processes.
Teacher of preparatory courses at the university.
- Lesson cost: My rate at home in Dolgoprudny is 3000 rubles/60 min. , at the student's home - 3700 rubles / 60 min. , remote classes (Skype) - 2700 rubles / 60 min.
- Items: Mathematics, Physics, Probability Theory, Calculus
- Cities: Moscow, Lobnya, Dolgoprudny, Dmitrov
- Nearest metro stations: Altufievo, River Station
- Home visit: available
- Status: Professor
- Education: Moscow Institute of Physics and Technology (MIPT), Faculty of Control and Applied Mathematics, Ph.D.
Experienced math tutor.
Mathematics and physics for middle and high school students, students, adults, preparation for the OGE and the USE. Classes with university applicants. Individual lessons are the most effective. Great teaching experience guarantees the successful study of the most complex issues.
- Lesson cost: Mathematics and Physics: 90 min. / 900 rubles for schoolchildren.
Students and adults 90 min. / 1200 rubles. - Items: Mathematics, Calculus, Physics
- Cities: Moscow, Zhukovsky, Zhukovsky, Zhukovsky, Zhukovsky
- Nearest metro stations: Kotelniki, Vykhino
- Home visit: available
- Status: Private teacher
- Education: Moscow State University M. V. Lomonosov, Faculty of Physics, Department of Mathematics for the Faculty of Physics, 1976. Russian Academy of Entrepreneurship, 1994
