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OGE (GIA) Unified State Exam preparation for the Olympics school course Algebra Analytic geometry Higher mathematics+8 Geometry Combinatorics Linear algebra Math statistics Mathematical analysis Applied Mathematics Probability theory Trigonometry
Children 6-7 years old Schoolchildren of grades 1-11 Students Adults
m. Ozernaya m. Yugo-Zapadnaya m. Kuntsevskaya (Filyovskaya)
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Very effective tutor and talented teacher- knows how to present a program like this higher mathematics University that the mathematics course from a nightmare has become annoying Expand necessity - despite the fact that from school course The student confidently knew only the 5th-6th grade curriculum. 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 solution methods Optimization methods Optimal control Applied Mathematics Sopromat Tensor analysis Theoretical mechanics Probability theory Graph theory Game theory Optimization theory Number theory Topology Trigonometry TFKP Partial differential equations Equations of mathematical physics Financial mathematics Functional analysis Econometrics
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m. Dmitry Donskoy Boulevard
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Doctor of Physics mathematical sciences. Leading Researcher Moscow State University (Faculty of Mechanics and Mathematics), Faculty Professor additional education Expand MGIMO, was a member of the examination commissions in mathematics of Moscow State University, MGIMO, MGUDT.
Alexey Vasilievich is exactly the teacher we have been looking for for a long time. Knows how to find an approach to a student and competently present educational material. All reviews (29)
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Prize-winner of the 2007 Lomonosov Olympiad in the subjects - oral and written mathematics, composition. Participant of the interfaculty special course olympiad problems Expand Department of Mathematical Analysis of Mechanics and Mathematics of Moscow State University. Experience in running small fur-mat clubs 2007-2012. Optional mathematics at Lyceum 1553. Teacher of algebra, geometry, computer science, in English at Lyceum 1553 in 2011. Supporting the education of children in language camps in England and Malta 2011-2012. Three years of retail management experience in central office largest bank in the CIS. I conduct classes using a Wacom graphics tablet and an online whiteboard (paid, which has the ability to be used by several people at the same time, simultaneous editing, joint video and sound). After the lesson, links to the room remain - the student always has access to what was written in the lesson and has access to notes for the entire duration of the course, all materials written on the board are also sent to the client in PDF format. Both Skype and the online room itself are used for communication. The number of students prepared for exams is more than 100, prepared for the OGE, Unified State Exam admission in lyceums at MEPhI, Moscow State University. Prepared students for exams from various universities of Moscow State University of 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 am also preparing for admission to English and Swiss colleges. Change unified exam A-level in English in mathematics and physics. Preparing schoolchildren for passing English OGE and Unified State Exam.
I studied with Alexey Alexandrovich, and in a month I managed to prepare with him for a retake in mathematical analysis. Explained the subject to me clearly and clearly, Expand I passed without any problems thanks to him. All reviews (52)
OGE (GIA) Unified State Exam 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 Faculty of Mechanics and Mathematics of Moscow State University. I have experience working in the banking industry as an analyst, and experience working as a systems analyst in the field of IT development. Knowledge Expand programming, relational databases (sql). First category in chess. I have successful experience working with all categories of students: Schoolchildren (OGE, Unified State Exam, improving academic performance) Students (almost all sections of higher mathematics and mechanics) Adults (classes for oneself, help with work questions).
Course on probability theory and mathematical statistics. Sevastyanov B.A.
M.: Science. Ch. ed. physics and mathematics lit., 1982.- 256 p.
The book is based on a year-long course of lectures given by the author over a number of years at the mathematics department of the Faculty of Mechanics and Mathematics of Moscow State University. Basic concepts and facts of probability theory are introduced initially for the final scheme. Mathematical expectation in general case is defined in the same way as the Lebesgue integral, but the reader is not expected to know any preliminary information on Lebesgue integration.
The book contains the following sections: independent tests and Markov chains, 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 statistical hypotheses, statistical estimates, confidence intervals.
For junior university and college students studying probability theory.
Format: djvu/zip
Size: 2.5 7 MB
/Download file
TABLE OF CONTENTS
Preface 7
Chapter 1. Probability space 9
§ 1. Subject of probability theory 9
§ 2. Events 12
§ 3. Probability space 16
§ 4. Finite probability space. Classic definition probabilities 19
§ 5 Geometric probabilities 23
Problems 24
Chapter 2. Conditional probabilities. Independence 26
§ 6. Conditional probabilities 26
§ 7. Formula full probability 28
§ 8. Bayes formulas 29
§ 9. Independence of events 30
§ 10. Independence of partitions, algebras and a-algebras.... 33
§ eleven. Independent tests 35
Problems 39
Chapter 3. Random variables (finite scheme). 41
§ 12. Random variables. Indicators 41
§ 13. Mathematical expectation 45
§ 14. Multidimensional distribution laws 50
§ 15. Independence of random variables 53
§ 10. Euclidean space of random greatnesses. . . . 5th
§ 17. Conditional mathematical expectations 5E
§ 18. Chebyshev's inequality. Law large numbers.... 61
Problems 64
Chapter 4. Limit theorems in Bernoulli's scheme. 65
§ 19. Binomial distribution 65
§ 20. Poisson's theorem 66
§ 21. Local limit theorem of Moivre - Laplace. . 70
§ 22. Integral limit theorem of Moivre - Laplace 71
§ 23. Applications of limit theorems. 73
Problems 76
Chapter 5. Markov Chains 77
§ 24. Markov dependence test 77
§ 25. Transitional probabilities 78
§ 26. Theorem on limiting probabilities 80
Problems 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
Problems 98
Chapter 7. Expectation 100
§ 30. Definition mathematical expectation 100
§ 31. Formulas for calculating mathematical expectation 108
Problems 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
Problems 127
Chapter 9. Characteristic functions 129
§ 37. Definition and simplest properties characteristic functions 129
§ 38. Inversion formulas for characteristic functions 136
§ 39. Theorem on continuous correspondence between the set of characteristic functions and the set of distribution functions 140
Problems 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
Problems 153
Chapter 11. Multidimensional characteristic functions.154
§ 43. Definition and simplest properties 154
§ 44. Circulation formula 158
§ 45. Limit theorems for characteristic functions 159
§ 46. Multivariate normal distribution and related distributions 164
Problems 173
Chapter 12. Strengthened Law of Large Numbers 174
§ 47. Borel-Cantelli lemma. Kolmogorov’s “0 or 1” law 174
§ 48 Different kinds convergence of random variables. . . 177
§ 49. Strengthened law of large numbers 181
Problems 188
Chapter 13. Statistics 189
§ 50. Main tasks mathematical statistics.... 189
§ 51. Sampling method 190
Problems 194
Chapter 14. Statistical criteria 195
§ 52. Statistical hypotheses 195
§ 53. Level of significance and power of criterion 197
§ 54. The optimal Neyman-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. Nonparametric criteria 206
Problems 211
Chapter 15. Parameter Estimates 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
Problems 232
Chapter 16. Confidence intervals 234
§ 63. Determination of confidence intervals 234
§ 64. Confidence intervals for parameters normal distribution 236
§ 65. Confidence intervals for the probability of success in the Bernoulli scheme 240
Problems 244
Answers to problems 245
Normal distribution tables 251
Literature 253
Subject index 254
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Ministry Russian Federation on communications and information
Siberian State University of Telecommunications and Informatics
N. I. Chernova
MATHEMATICAL
STATISTICS
Novosibirsk
Associate Professor, Candidate of Sciences physics and mathematics Sciences N.I. Chernova. Mathematical statistics: Textbook / SibGUTI. - Novosibirsk, 2009. - 90 p.
The textbook contains a six-month course of lectures on mathematical statistics for students of economic specialties. The textbook meets the requirements of the State educational standard for professional educational programs specialty 080116 - “Mathematical methods in economics.”
Department of IMBP Table. 7, drawings - 9, list of literature. - 8 names
Reviewers: A. P. Kovalevsky, Ph.D. physics and mathematics Sciences, Associate Professor of the Department of Higher Mathematics of NSTU V. I. Lotov, Doctor of Physics and Mathematics. Sciences, Professor of the Department
theory of probability and mathematical statistics NSU
For specialty 080116 - “Mathematical methods in economics”
Approved by the editorial and publishing council of SibGUTI as a teaching aid
c Siberian State University
telecommunications and information science, 2009
Preface. . . . . . . . . . | ||
I. Basic concepts of mathematical statistics. . . . . . . . | ||
Problems of mathematical statistics . . . . . . . . . . . . . . . . . | ||
Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | ||
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 method of moments estimators. . . . . . . . . . . . . . . . . | |||
Maximum likelihood method. . . . . . . . . . . . . . . | |||
Asymptotic normality of estimates. . . . . . . . . . . . . . | |||
Questions and Exercises. . . . . . . . . . . . . . . . . . . . . . . . | |||
Comparison of ratings. . . . . . . . . . . . . . . . . . . . . . . | |||
A 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 associated with normal. . . . . . . . . . | |||
Basic statistical distributions. . . . . . . . . . . . . . | |||
Transformations of normal samples. . . . . . . . . . . . . . . | |||
Confidence intervals for normal distribution. . . | |||
§ 1. Hypotheses and criteria. . . . . . . . . . . . . . . . . . . . . . . . . 47
§ 2. Questions and exercises. . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter VII. Consent criteria. . . . . . . . . . . . . . . . . . . . . . 51
§ 1. General form agreement 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 checking homogeneity. . . . . . . . . . . . . . . 61
§ 6. χ 2 criterion for checking independence. . . . . . . . . . . . . 70
§ 7. Questions and exercises. . . . . . . . . . . . . . . . . . . . . . . . 71
§ 2. Maximum likelihood method.. . . . . . . . . . . . . . . 74
§ 3. Least squares method.. . . . . . . . . . . . . . . . . . . 75
PREFACE
The tutorial contains full course lectures on mathematical statistics for students studying in the specialty “Mathematical methods in economics” at the Siberian State University of Telecommunications and Informatics. The course content is fully consistent educational standards training of bachelors in the specified specialty.
The course in mathematical statistics builds on a semester-long course in probability theory and is the basis for a year-long course in econometrics. As a result of studying the subject, students should master mathematical methods research various models 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 looks at comparing estimates in a root mean square sense. The Rao-Cramer inequality is also studied here as a means of checking the effectiveness of estimates.
The fourth chapter discusses interval parameter estimation, which ends in the next chapter with the construction of intervals for normal distribution parameters. To do this, special statistical distributions are introduced, which are then used in the goodness-of-fit tests in Chapter Eight. Chapter six gives the necessary basic concepts of the theory of hypothesis testing, 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. The ninth chapter discusses simple models and methods regression analysis and the main properties of the obtained estimates are proved.
Almost every chapter ends with a list of exercises based on the text of the chapter. The appendix contains tables with a list of the main characteristics of discrete and absolutely continuous distributions, tables of basic statistical distributions.
PREFACE |
A detailed subject index is provided at the end of the book. The bibliography lists textbooks that can be used to supplement the course and collections of problems for practical exercises.
The numbering of paragraphs in each chapter is separate. Formulas, examples, statements, etc. have continuous numbering. When referring to an object from another chapter, the page number on which the object is contained is indicated for the reader's convenience. When referring to an object from the same chapter, only the number of the formula, example, statement is given. The end of the evidence is marked with a symbol.
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 given distribution or random experiments whose properties are entirely known. But where does knowledge about distributions in practical experiments come from? With what probability, for example, does a coat of arms appear on a given coin? To determine this probability, we can toss a coin many times. But in any case, conclusions will have to be drawn based on the results. finite number observations. Thus, observing 5,035 coats of arms after 10,000 coin tosses, one cannot make an accurate conclusion about the probability of the coat of arms being dropped: even if this probability differs from 0.5, the coat of arms can appear 5,035 times. Accurate conclusions about the distribution can only be made when an infinite number of tests are 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 in 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 a more accurate conclusion about its distribution be made from 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 its properties. In this case, based on experimental data, it is necessary to confirm or refute these assumptions (“hypotheses”). It must be remembered 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 confirm the availability in advance
8 CHAPTER I. BASIC CONCEPTS OF MATHEMATICAL STATISTICS
some properties of the observed experiment - for example, about functional dependence between 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 are able to reproduce this experiment under the same conditions some (or better, any) number of times.
The results of experiments can be quantitative or qualitative character. Quantitative results can, for example, be added. Thus, one of their meaningful characteristics is the arithmetic mean of observations. It makes no sense to add up qualitative results, although they can be expressed in numerical form. Let's say, the month of birth of the respondent is qualitative, not quantitative observation: Although it can be specified as a number, the arithmetic mean of these numbers carries as much reasonable information as the message that the average person was born between June and July.
In the first chapters we will study working with quantitative results observations.
§ 2. Sampling
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 obtain 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 take a closer look at the set X = (X1, . . . , Xn), called a sample.
In a series of experiments that have already been carried out, a sample is a set of numbers. But before the experiment is carried out, 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 sample elements 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, identically distributed with ξ, and after the experiment, it is the number that we observe in the i-th experiment, i.e. one of possible values random variable Xi.
Definition 1. A sample X = (X1, . . . , Xn) of volume n from distribution F is a set of n independent and identically distributed random variables having distribution F.
Select items are often transformed to make it easier to work with a large set of data - ordered or grouped.
If the sample elements are X1, . . . , Xn are ordered in ascending order, and a set of new random variables is obtained, called a variation 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 kth term variation series or the kth order statistic.
When grouping data, you select several groups of sample element values, count the number of elements in each group, and then deal only with this new set of data. 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 drawn. 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. Using a sample, you need to be able to build approximations for all these characteristics. Such approximations are called estimates. The term "assessment" has nothing to do with inequalities. An estimate for some unknown distribution characteristic is a random variable constructed from a sample, which in some sense is an approximation of this unknown distribution characteristic.
Example 1. A six-sided die is tossed 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 drawn.
Based on these experimental results, it is impossible to determine the probabilities p1, . . . , p6 loss of edges. We can only say that numerical estimates for these probabilities have been obtained: 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 probability p2 will be the random variable
In this series of experiments, these random variables took values of 0.25 and 0.14, respectively. In another series their meanings will change.
CHAPTER I. BASIC CONCEPTS OF MATHEMATICAL STATISTICS |
§ 3. Selected characteristics
From probability theory we know universal remedy for approximate calculation of all possible 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 mathematical 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: sample mean
X1 + . . . +Xn | ||||||||||||||||||||||||||
The sample kth moment is suitable as an estimate for E X1 k | ||||||||||||||||||||||||||
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 |
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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 value of 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 successful tests in the Bernoulli scheme: if for each element of the sample the event (Xi< 3}, то доля успехов
p = quantity 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 using the empirical distribution function