Gmurman v. Probability Theory and Mathematics. Gmurman V.E

Name: Probability theory and math statistics. 2003.

The book (8th ed. - 2002) contains basically all the material of the program on probability theory and mathematical statistics. great attention devoted to statistical methods of processing experimental data. At the end of each chapter there are problems with answers.
It is intended for university students and persons using probabilistic and statistical methods when solving practical problems.

The subject of probability theory. The events (phenomena) observed by us can be divided into the following three types: reliable, impossible and random.
A certain event is called a certain event that will definitely occur if a certain set of conditions S is fulfilled. For example, if a vessel contains water at normal atmospheric pressure and a temperature of 20 °, then the event “the water in the vessel is in liquid state” is reliable. In this example, given Atmosphere pressure and water temperature constitute a set of conditions S.
An event is called impossible if the set of conditions S is met.

TABLE OF CONTENTS
Introduction 14
PART ONE. RANDOM EVENTS
Chapter first. Basic concepts of probabilistic theorists 17
§ 1. Trials and events 17
§ 2. Types random events 17
§ 3. Classical definition of probability 18
§ 4. Basic forms of combinatorics 22
§ 5. Examples of direct calculation of probabilities 23
§ 6. Relative frequency. Relative frequency stability 24
§ 7. Boundedness of the classical definition of probability.
Statistical Probability 26
§ 8. geometric probabilities 27
Tasks 30
Chapter two. Probability addition theorem 31
§ 1. The theorem of addition of probabilities Not joint events 31
§ 2. Complete group of events 33
§ 3. Opposite events 34
§ 4. The principle of the practical impossibility of unlikely events 35
Tasks 36
Chapter three. Probability multiplication theorem 37
§ 1. Product of events 37
§ 2 Conditional Probability 37
§ 3 Probability multiplication theorem 38
§ 4 Independent events Multiplication theorem for independent events 40
§ 5 Probability of occurrence of at least one event 44
Tasks 47
Chapter Four Consequences of the addition and multiplication theorems 4S
§ 1 The theorem of addition of probabilities of joint events 48
§ 2 Formula full probability 50
§ 3 Probability of hypotheses Bayes Formula 52
Tasks 53
Chapter Five Repeating Tests 55
§ 1 Bernoulli formula 55
§ 2 Local theorem Laplace 57
§ 3 Laplace integral theorem 59
§ 4 Probability of deviation of the relative frequency from the constant probability in independent tests 61
Tasks 63
PART TWO. RANDOM VALUES
Chapter Six Types of random variables. Specifying a discrete random variable 64
§ 1 Random variable 64
§ 2 Discrete and continuous random variables 65
§ 3 Probability distribution law of a discrete random variable 65
§ 4 Binomial distribution 66
§ 5 Poisson distribution 68
§ 6 The simplest flow of events 69
§ 7 Geometric distribution 72
§ 8 Hypergeometric distribution 73
Tasks 74
Chapter Seven Mathematical expectation of a discrete random variable 75
§ 1 Numerical characteristics of discrete random variables 75
§ 2 Mathematical expectation of a discrete random variable 76
§ 3 Probabilistic meaning mathematical expectation 77
§ 4 Properties of mathematical expectation 78
§ 5 Mathematical expectation of the number of occurrences of an event in independent trials S3
Tasks 84
Chapter Eight Dispersion of a Discrete Random Variable 85
§ 1 Expediency of introducing a numerical characteristic of the scattering of a random variable 85
§ 2 Deviation of a random variable from its mathematical expectation 86
§ 3 Dispersion of a discrete random variable 87
§ 4 Formula for calculating the variance 89
§ 5 Properties of dispersion 90
§ 6 Variance in the number of occurrences of an event in independent trials 92
§ 7 Standard deviation 94
§ 8 Standard deviation of the sum of mutually independent random variables 95
§ 9 Equally distributed mutually independent random variables 95
§ 10 Initial and central theoretical moments 98
Tasks 100
Chapter Nine Law big numbers 101
§ 1 Preliminary remarks 101
§ 2 Chebyshev's inequality 101
§3 Chebyshev's theorem 103
§ 4 Essence of Chebyshev's theorem 106
§ 5 Significance of Chebyshev's theorem for practice 107
§ 6 Bernoulli theorem 108
Tasks 110
Chapter Ten The Probability Distribution Function of a Random Variable 111
§ 1 Definition of the distribution function 111
§ 2 Properties of the distribution function 112
§ 3 Graph of the distribution function 114
Tasks 115
Chapter Eleven The Probability Density of a Continuous Random Variable 116
§ 1 Determination of the distribution density 116
§ 2 Probability of hitting a continuous random variable in specified interval 116
§ 3. Finding the distribution function from a known distribution density 118
5 4. Properties of the distribution density 119
§ 5. The probabilistic meaning of the distribution density 121
§ 6. The law of uniform distribution of probabilities 122
Tasks 124
Chapter twelve. Normal distribution 124
§ I. Numerical characteristics of continuous random variables 124
§ 2. Normal distribution 127
§ 3. Normal curve 130
§ 4. Influence of the parameters of the normal distribution on the shape of the normal curve 131
§ 5. Probability of falling into a given interval of a normal random variable 132
§ 6. Calculation of the probability of a given deviation 133
§ 7. Three Sigma Rule 134
§ 8. The concept of Lyapunov's theorem. The wording of the central limit theorem 135
§ 9. Estimation of the deviation of the theoretical distribution from the normal one. Asymmetry and kurtosis 137
§ 10. Function of one random argument and its distribution 139
§ 11. Mathematical expectation of a function of one random argument 141
§ 12. Function of two random arguments. Distribution of the sum of independent terms. Stability of the Normal Distribution 143
§ 13. Distribution "chi square * 145
§ 14. Distribution of Student 146
§ 15. Distribution / "Fischer-Snedekor 147
Tasks 147
Chapter thirteen. Demonstration raspoedeyam 149
§ 1. Definition of the exponential distribution 149
§ 2. The probability of an exponentially distributed random variable falling into a given interval 150
§ 3. Numerical characteristics of the exponential distribution 151
§ 4. Reliability function 152
§ 5. The exponential law of reliability 153
§6. characteristic property exponential law of reliability 154
Tasks 155
Chapter fourteen. System of two random trends 155
§ 1. The concept of a system of several random variables 155
§ 2. Probability distribution law for a discrete two-dimensional random variable 156
§ 3. The distribution function of a two-dimensional random variable 158
§ 4. Properties of the distribution function of a two-dimensional random variable 159
§ 5. The probability of a random point falling into a half-strip 161
§ 6. The probability of a random point falling into a rectangle 162
§ 7. Density of the joint probability distribution of a continuous two-dimensional random variable (two-dimensional probability density) 163
§ 8. Finding the distribution function of a system from a known distribution density 163
§ 9. The probabilistic meaning of the two-dimensional probability density 164
§ 10. The probability of a random point falling into an arbitrary region 165
§ 11. Properties of the two-dimensional probability density 167
§ 12. Finding the probability densities of the components of a two-dimensional random variable 168
§ 13. Conditional distribution laws for the components of a system of discrete random variables 169
§ 14. Conditional distribution laws for the components of a system of continuous random variables 171
§ 15. Conditional expectation 173
§ 16. Dependent and independent random variables 174
§ 17. Numerical characteristics of systems of two random variables. correlation moment. Correlation coefficient 176
§ 18. Correlated™ and dependence of random variables 179
§ 19. normal law distributions on the plane 181
§ 20. Linear Regression. Straight Lines RMS Regression 182
§ 21. Linear correlation. Normal correlation 184
Tasks 185
PART THREE. ELEMENTS OF MATHEMATICAL STATISTICS
Chapter fifteen. Sampling method 187
§ 1. Problems of mathematical statistics 187
§ 2. Brief historical reference 188
§ 3. General and sampling frame 188
§ 4. Repeated and non-repetitive sampling. Representative Sample 189
§ 5 Methods of selection 190
§ 6 Statistical distribution samples 192
§ 7 Empirical distribution function 192
§ 8 Polygon and histogram 194
Tasks 196
Chapter Sixteen Statistical Estimation of Distribution Parameters 197
§ 1 Statistical estimates distribution parameters 197
§ 2 Unbiased, efficient and consistent estimators 198
§ 3 General average 194
§ 4 Sample mean 200
§ 5 Estimation of the general mean from the sample mean Stability of sample means 201
§ 6 Group and overall averages 203
§ 7 Deviation from the general average and its property 204
§ 8 General variance 205
§ 9 Sample variance 206
§ 10 Formula for calculating the variance 207
§ 11 Group, intragroup. intergroup and total variance 207
§ 12 Addition of variances 210
§ 13 Estimation of the general variance from the corrected sample 211
§ 14 Accuracy of measurement, confidence level(reliability) Confidence interval 213
§ 15 Confidence intervals for estimating the expectation of a normal distribution with known about 2)4
§ 16 Confidence intervals for estimating the mathematical expectation of a normal distribution with an unknown o 216
§17 Evaluation true value measured value 219
§ 18 Confidence intervals for estimating the mean standard deviation about normal distribution 220
§ 19 Evaluation of measurement accuracy 223
§ 20 Estimation of probability (binomial distribution) by relative frequency 224
§ 21 Method of moments for point estimation of distribution parameters 226
§ 22 Maximum likelihood method 229
§ 23 Other characteristics variation series 234
Tasks 235
Chapter 17
§ 1 Conditional options 237
§2 Ordinary, initial and central empirical moments 238
§ 3 Conditional empirical moments Finding central moments conditional 239
§ 4 Method of products for calculating the sample mean and variance 241
§ 5 Reduction initial options to equally spaced 243
§ 6 Empirical and leveling (theoretical) frequencies 245
§ 7 Construction of a normal curve from experimental data 249
§ 8 Estimation of the deviation of the empirical distribution from the normal Skewness and kurtosis 250
Tasks 252
Chapter Eighteen Elements of the Theory of Correlation 253
§ 1 Functional, statistical and correlation dependencies 253
§ 2 Conditional averages 254
§ 3 Sample regression equations 254
§ 4 Finding the parameters of the sample equation of a straight line of mean square regression from ungrouped data 255
§ 5 correlation table 257
§ 6 Finding the parameters of the sample equation of a straight regression line from grouped data 259
§ 7 Sample correlation coefficient 261
§ 8 Method of calculation sampling rate correlations 262
§ 9 Example for finding a sample equation of a straight regression line 267
§ 10 Preliminary considerations for the introduction of a measure of any correlation 268
§ 11 Sample correlation 270
§12 Properties of the sample correlation relation 272
§ 13 Correlation ratio as a measure of correlation Advantages and disadvantages of this measure 274
§ 14 The simplest cases of curvilinear correlation 275
§ 15 The concept of multiple correlation 276
Tasks 278
Chapter Nineteen Statistical verification statistical hypotheses 281
§ 1 Statistical hypothesis Null and competing, simple and complex hypotheses 281
§ 2 Errors of the first and second kind 282
§ 3 Statistical test criterion null hypothesis Observed value of criterion 283
§ 4 Critical region Hypothesis acceptance area Critical points 284
§ 5 Finding the right-handed critical region 285
§ 6 Finding left-sided and two-sided critical regions 286
§ 7 Additional information about the choice of the critical region Power of the criterion 287
§ 8 Comparison of two variances of normal populations 288
§ 9 Comparison of the amended sample variance with a hypothetical general variance normal set 293
§ 10 Comparison of two means of normal populations whose variances are known (independent samples) 297
§ 11 Comparison of two means of randomly distributed populations (large independent samples) 303
§ 12 Comparison of two averages of normal populations whose variances are unknown and the same (small independent samples) 305
§ 13 Comparison of the sample mean with the hypothetical general mean of a normal population 308
§ 14 Relationship between a two-sided critical region and confidence interval 312
§ 15 Determining the minimum sample size when comparing sample and hypothetical general averages 313
§ 16 Example for finding the power of criterion 313
§ 17 Comparison of two means of normal populations with unknown variances (dependent samples) 314
§ 18 Comparison of the observed relative frequency with the hypothetical probability of occurrence of the event 317
§19 Comparison of two probabilities binomial distributions 319
§ 20 Comparison of several variances of normal populations for samples of different sizes Bartlett's test 322
§ 21 Comparison of several variances of normal populations over samples of the same size Cochran's test 325
§ 22 Testing the hypothesis of the significance of the sample correlation coefficient 327
§ 23 Testing the Hypothesis of Normal Distribution population Pearson goodness-of-fit test 329
§ 24 Method for calculating the theoretical frequencies of the normal distribution 333
§ 25 Spearman's sample coefficient of rank correlation and testing of the hypothesis about its significance 335
§ 26 Kendall's sample rank correlation coefficient and testing of the hypothesis about its significance 341
§ 27 The Wilcoxon test and testing the hypothesis of homogeneity of two samples 343
Tasks 346
Chapter Twenty One-way analysis of variance 349
§ I Comparison of several means The concept of analysis of variance 349
§ 2 Total, factor and residual sums of squared deviations 350
§ 3 Relationship between general, factorial and residual amounts 354
§ 4 General, factor and residual dispersion 355
§ 5 Comparison of several averages by the method analysis of variance 355
§ 6 Unequal number of trials at different levels 358
Tasks 361
PART FOUR. MONTE CARLO METHOD. MARKOV CHAINS
Chapter 21 Monte Carlo Modeling (Playing) of Random Greatness 363
§ 1 Subject of the Monte Carlo method 363
§ 2 Error estimate of the Monte Carlo method 364
§ 3 Random numbers 366
§ 4 Playing a Discrete Random Variable 366
§ 5 Enactment opposite events 368
§ 6 Enactment of a complete group of events 369
§ 7 Playing a continuous random variable Method inverse functions 371
§ 8 Method of superposition 375
§ 9 Approximate playing of a normal random variable 377
Tasks 379
Chapter twenty-two Initial information about Markov chains. 380
§ 1 Markov chain 380
§ 2 Homogeneous chain Markov Transition Probabilities Transition Matrix 381
§ Markov equality 383
Tasks 385
PART FIVE. RANDOM FUNCTIONS
Chapter Twenty-Three Random Functions 386
§ 1 Main tasks 386
§ 2 Definition random function 386
§ 3 Correlation theory of random functions 388
§ 4 Mathematical expectation of a random function 390
§ 5 Properties of the expectation of a random function 390
§ 6 Variance of a random function 391
§ 7 Properties of the dispersion of a random function 392
§ 8 Expediency of introduction of correlation function 393
§ 9 Correlation function of a random function 394
§ 10 Properties of the correlation function 395
§ 11 Standardized correlation function 398
§ 12 Cross-correlation function 399
§ 13 Properties of the cross-correlation function 400
§ 14 Normalized cross-correlation function 401
§ 15 Characteristics of the sum of random functions 402
§ 16 Derivative of a random function and its characteristics 405
§ 17 The integral of a random function and its characteristics 409
§ 18 Complex random variables and their numerical characteristics 413
§ 19 Complex random functions and their characteristics 415
Tasks 417
Chapter Twenty Four Stationary Random Functions 419
§1 Definition of a stationary random function 419
§ 2 Properties of the correlation function of a stationary random function 421
§ 3 Normalized correlation function of a stationary random function 421
§ 4 Stationary related random functions 423
§ 5 Correlation function of the derivative of a stationary random function 424
§ 6 Cross-correlation function of a stationary random function and its derivative 425
§ 7 Correlation function of the integral of a stationary random function 426
§ 8 Determining the characteristics of ergodic stationary random functions from experiment 428
Tasks 430
Chapter Twenty-Five Elements of the Spectral Theory of Stationary Random Functions 431
§ 1 Representation of a stationary random function in the form harmonic vibrations with random amplitudes and random phases 431
§ 2 Discrete spectrum of a stationary random function 435
§ 3 Continuous spectrum of a stationary random function Spectral density 437
§ 4 Normalized Spectral Density 441
§ 5 Mutual spectral density of stationary and stationary related random functions 442
§ 6 Delta function 443
§ 7 Stationary white noise 444
§ 8 Transformation of a stationary random function of a stationary linear dynamic system 446
Tasks 449
Addition 451
Applications 461
Index 474

Many generations of students both in our country and abroad are well aware of this manual, which has become a classic. educational publication. Its value lies in the fact that difficult questions theory of probability and mathematical statistics are presented in a logical sequence and in an accessible form. A large number of examples allows you to better understand the material, and the tasks given at the end of each chapter to consolidate the knowledge gained.

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Library > Books on Mathematics > Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

Agekyan T.A. Fundamentals of Error Theory for Astronomers and Physicists (2nd ed.). M.: Nauka, 1972 (djvu)
Agekyan T.A. The theory of probability for astronomers and physicists. M.: Nauka, 1974 (djvu)
Anderson T. Statistical analysis of time series. M.: Mir, 1976 (djvu)
Bakelman I.Ya. Werner A.L. Kantor B.E. Introduction to differential geometry "in the large". Moscow: Nauka, 1973 (djvu)
Bernstein S.N. Probability Theory. M.-L.: GI, 1927 (djvu)
Billingsley P. Convergence of probability measures. Moscow: Nauka, 1977 (djvu)
Box J. Jenkins G. Analysis of time series: forecast and management. Issue 1. M.: Mir, 1974 (djvu)
Box J. Jenkins G. Analysis of time series: forecast and management. Issue 2. M.: Mir, 1974 (djvu)
Borel E. Probability and reliability. Moscow: Nauka, 1969 (djvu)
Van der Waerden B.L. Math statistics. M.: IL, 1960 (djvu)
Vapnik V.N. Recovery of dependences on empirical data. M.: Nauka, 1979 (djvu)
Wentzel E.S. Introduction to operations research. M.: Soviet radio, 1964 (djvu)
Wentzel E.S. Elements of Game Theory (2nd ed.). Series: Popular lectures on mathematics. Issue 32. M.: Nauka, 1961 (djvu)
Venztel E.S. Probability Theory (4th ed.). Moscow: Nauka, 1969 (djvu)
Venztel E.S., Ovcharov L.A. Probability Theory. Tasks and exercises. Moscow: Nauka, 1969 (djvu)
Vilenkin N.Ya., Potapov V.G. Taskbook-workshop on the theory of probability with elements of combinatorics and mathematical statistics. M.: Enlightenment, 1979 (djvu)
Gmurman V.E. A Guide to Problem Solving in Probability and Mathematical Statistics (3rd ed.). M.: Higher. school, 1979 (djvu)
Gmurman V.E. Probability Theory and Mathematical Statistics (4th ed.). M.: graduate School 1972 (djvu)
Gnedenko B.V., Kolmogorov A.N. Limit distributions for sums of independent random variables. M.-L.: GITTL, 1949 (djvu)
Gnedenko B.V., Khinchin A.Ya. An Elementary Introduction to Probability Theory (7th ed.). Moscow: Nauka, 1970 (djvu)
Oak J.L. Probabilistic processes. M.: IL, 1956 (djvu)
David G. Order statistics. M.: Nauka, 1979 (djvu)
Ibragimov I.A., Linnik Yu.V. Independent and stationary quantities. Moscow: Nauka, 1965 (djvu)
Idie W., Dryyard D., James F., Roos M., Sadoulé B. Statistical methods in experimental physics. Moscow: Atomizdat, 1976 (djvu)
Kassandrova O.N., Lebedev V.V. Processing of observation results. Moscow: Nauka, 1970 (djvu)
Katz M. Probability and related issues in physics. M.: Mir, 1965 (djvu)
Katz M. Several probabilistic problems of physics and mathematics. Moscow: Nauka, 1967 (djvu)
Katz M. Statistical independence in probability theory, analysis and number theory. M.: IL, 1963 (djvu)
Kamalov M.K. Distribution quadratic forms in samples from a normal population. Tashkent: Academy of Sciences of the Uzbek SSR, 1958 (djvu)
Kendall M., Moran P. Geometric probabilities. M.: Nauka, 1972 (djvu)
Kendall M., Stewart A. Vol. 1. Theory of distributions. Moscow: Nauka, 1965 (djvu)
Kendall M., Stuart A. Volume 2. Statistical Inference and Relationships. Moscow: Nauka, 1973 (djvu)
Kendall M., Stuart A. Volume 3. Multidimensional statistical analysis and time series. M.: Nauka, 1976 (djvu)
Kolmogorov A.N. Basic concepts of probability theory (2nd ed.) M.: Nauka, 1974 (djvu)
Kolchin V.F., Sevastyanov B.A., Chistyakov V.P. Random placements. M.: Nauka, 1976 (djvu)
Cramer G. Mathematical Methods statistics (2nd ed.). M.: Mir, 1976 (djvu)
Leman E. Verification of statistical hypotheses. M.: Science. 1979
Linnik Yu.V., Ostrovsky I.V. Decompositions of random variables and vectors. M.: Nauka, 1972 (djvu)
Likholetov I.I., Matskevich I.P. A guide to problem solving higher mathematics, Probability Theory and Mathematical Statistics (2nd ed.). Mn.: Vysh. school, 1969 (djvu)
Loev M. Probability Theory. M.: IL, 1962 (djvu)
Malakhov A.H. Cumulant analysis of random non-Gaussian processes and their transformations. M.: Sov. radio, 1978 (djvu)
Meshalkin L.D. Collection of problems in the theory of probability. Moscow: Moscow State University, 1963 (djvu)
Mitropolsky A.K. Theory of moments. M.-L.: GIKSL, 1933 (djvu)
Mitropolsky A.K. Technique statistical calculations(2nd ed.). Moscow: Nauka, 1971 (djvu)
Mosteller F., Rourke R., Thomas J. Probability. M.: Mir, 1969 (djvu)
Nalimov V.V. Application of mathematical statistics in the analysis of matter. M.: GIFML, 1960 (djvu)
Neveu J. Mathematical Foundations probability theory. M.: Mir, 1969 (djvu)
Preston K. Mathematics. New in Foreign Science No.7. Gibbs states on countable sets. M.: Mir, 1977

Many generations of students both in our country and abroad are well aware of this manual, which has become a classic educational publication. Its value lies in the fact that the complex issues of probability theory and mathematical statistics are presented in a logical sequence and in an accessible form. A large number of examples allows you to better understand the material, and the tasks given at the end of each chapter consolidate your knowledge.

Step 1. Choose books in the catalog and click the "Buy" button;

Step 2. Go to the "Basket" section;

Step 3. Specify the required quantity, fill in the data in the Recipient and Delivery blocks;

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9th ed., Ster.-M.: Higher School, 2004.- 404 p.

The manual (8th edition - 2003) contains the necessary theoretical information and formulas, solutions are given typical tasks, tasks are placed for independent decision accompanied by answers and instructions. Much attention is paid to methods statistical processing experimental data.

For university students. It may be useful to persons applying probabilistic and statistical methods in solving practical problems.

Format: pdf/zip

Size: 17.8 MB

Download: Links removed at the request of Urayt, see below.urait.ru/catalog

See also: Theory of Probability and Mathematical Statistics. Gmurman V.E. (2003, 479s.)

TABLE OF CONTENTS
PART ONE. RANDOM EVENTS
Chapter first. Definition of Probability 8
§ 1. Classical and statistical definitions probabilities... 8
§ 2. Geometric probabilities 12
Chapter two. Key Theorem 18
§ 1. Theorem of addition and multiplication of probabilities 18
§ 2. Probability of occurrence of at least one event 29
§ 3. Total probability formula 31
§ 4. Bayes formula 32
Chapter three. Repeat Test 37
§ 1. Bernoulli formula 37
§ 2. Local and integral theorem Laplace 39
§ 3. Deviation of relative frequency from constant probability in independent tests 43
§ 4. The most probable number of occurrences of an event in independent trials 46
§ 5. Generating function 50
PART TWO. RANDOM VALUES
Chapter Four. Discrete random variables 52
§ 1. Law of distribution of probabilities of a discrete random variable. Binomial and Poisson laws 52
§ 2. The simplest flow of events 60
§ 3. Numerical characteristics of discrete random variables. 63
§ 4. Theoretical points 79
Chapter five. Law of Large Numbers 82
§ 1. Chebyshev's inequality 82
§ 2. Chebyshev's theorem 85
Chapter six. Probability density functions of random variables
§ 1. The probability distribution function of a random variable 87
§ 2. Density of the probability distribution of a continuous random variable 91
§ 3. Numerical characteristics of continuous random variables 94
§ 4. Uniform distribution 106
§ 5. Normal distribution 109
§ 6. The exponential distribution and its numerical characteristics 114
§ 7. Reliability function 119
Chapter seven. Distribution of a function of one and two random arguments 121
§ 1. Function of one random argument 121
§ 2. Function of two random arguments 132
Chapter eight. System of two random variables 137
§ 1. Law of distribution of a two-dimensional random variable 137
§ 2. Conditional laws of probability distribution of components of a discrete two-dimensional random variable 142
§ 3. Finding densities and conditional distribution laws for the components of a continuous two-dimensional random variable .... 144
§ 4. Numerical characteristics of a continuous system of two random variables 146
PART THREE. ELEMENTS OF MATHEMATICAL STATISTICS
Chapter nine. Sampling method 151
§ 1. Statistical distribution of the sample 151
§ 2. Empirical distribution function 152
§ 3. Polygon and histogram 152
Chapter ten. Statistical estimates of distribution parameters 157
§ 1. Point estimates 157
§ 2. Method of moments 163
§ 3. Maximum likelihood method 169
§ 4. Interval Estimates 174
Chapter Eleven. Methods for calculating the summary characteristics of the sample 181
§ 1. The method of products for calculating the sample mean and variance 181
§ 2. The method of sums for calculating the sample mean and variance 184
§ 3. Skewness and kurtosis of the empirical distribution 186
Chapter twelve. Elements of the theory of correlation 190
§1. Linear Correlation 190
§ 2. Curvilinear correlation 196
§ 3. Rank correlation 201
Chapter thirteen. Statistical testing of statistical hypotheses 206
§ 1. Basic information 206
§ 2. Comparison of two variances of normal populations 207
§ 3. Comparison of the corrected sample variance with the hypothetical general variance of a normal population 210
§ 4. Comparison of two mean general populations whose variances are known (large independent samples). 213
§ 5. Comparison of two averages of normal populations, the variances of which are unknown and the same (small independent samples) 215
§ 6. Comparison of the sample mean with the hypothetical general mean of a normal population 218
§ 7. Comparison of two means of normal populations with unknown variances (dependent samples) 226
§ 8. Comparison of the observed relative frequency with the hypothetical probability of the occurrence of the event 229
§ 9. Comparison of several variances of normal populations for samples of different sizes. Bartlett's test 231
§ 10. Comparison of several variances of normal general populations for samples of the same size. Cochran's test 234
§eleven. Comparison of two probabilities of binomial distributions 237
§ 12. Testing the hypothesis about the significance of the sample correlation coefficient 239
§ 13. Testing the hypothesis about the significance of the Spearman sample rank correlation coefficient 244
§ 14. Testing the hypothesis about the significance of Kendall's sample rank correlation coefficient 246
§ 15. Testing the hypothesis of homogeneity of two samples by the Wilcoxon test 247
§ 16. Testing the hypothesis about the normal distribution of the general population according to the Pearson criterion 251
§ 17. Graphical verification of the hypothesis about the normal distribution of the general population. Straight Diagram Method 25 9
§ 18. Testing the hypothesis about the exponential distribution of the general population 268
§ 19. Testing the hypothesis about the distribution of the general population according to the binomial law 272
§ 20. Testing the hypothesis of a uniform distribution of the general population 275
§ 21. Testing the hypothesis about the distribution of the general population according to the Poisson law 279
Chapter fourteen. One-way analysis of variance .......... 283
§ 1. same number tests at all levels 283
§ 2. Unequal number of tests at different levels 289
PART FOUR. SIMULATION OF RANDOM VARIABLES
Chapter fifteen. Simulation (playing) of random variables by the Monte Carlo method.................................................................. ................ 294
§ 1. Playing a Discrete Random Variable 294
§ 2. Playing a complete group of events 295
§ 3. Playing a continuous random variable 297
§ 4. Approximate simulation of a normal random variable 302
§ 5. Playing a two-dimensional random variable 303
§ 6. Evaluation of the reliability of the simplest systems by the Monte Carlo method 307
§ 7. Calculation of systems queuing with Monte Carlo failures 311
§ 8. Calculation definite integrals Monte Carlo method 317
PART FIVE. RANDOM FUNCTIONS
Chapter sixteen. Correlation theory of random functions.... 330
§ 1. Basic concepts. Characteristics of random functions... 330
§ 2. Characteristics of the sum of random functions 337
§ 3. Characteristics of the derivative of a random function 339
§ 4. Characteristics of the integral of a random function 342
Chapter seventeen. Stationary random functions 347
§ 1. Characteristics of a stationary random function 347
§ 2. Stationarily coupled random functions 351
§ 3. Correlation function of the derivative of a stationary random function 352
§ 4. Correlation function of the integral of a stationary random function 355
§ 5. Mutual correlation function of a differentiable stationary random function and its derivatives 357
§ 6. Spectral density of a stationary random function 360
§ 7. Transformation of a stationary random function by a stationary linear dynamical system 369
Answers 373
Applications 387