Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions

Foreword......Page 6
Contents......Page 8
1 Relations Among Random Events 1......Page 12
2 A Direct Method for Evaluating Probabilities 4......Page 15
3 Geometric Probabilities 6......Page 17
4 Conditional Probability. The Multiplication Theorem for Probabilities 12......Page 23
5 The Addition Theorem for Probabilities 16......Page 27
6 The Total Probability Formula 22......Page 33
7 Computation of the Probabilities of Hypotheses After a Trial (Bayes\' Formula) 26......Page 37
8 Evaluation of Probabilities of Occurrence of an Event in Repeated Independent Trials 30......Page 41
9 The Multinomial Distribution. Recursion Formulas. Generating Functions 36......Page 47
10 The Probability Distribution Series, the Distribution Polygon and the Distribution Function of a Discrete Random Variable 43......Page 54
11 The Distribution Function and the Probability Density Function of a Continuous Random Variable 48......Page 59
12 Numerical Characteristics of Discrete Random Variables 54......Page 65
13 Numerical Characteristics of Continuous Random Variables 62......Page 73
14 Poisson\'s Law 67......Page 78
15 The Normal Distribution Law 70......Page 81
16 Characteristic Functions 74......Page 85
17 The Computation of the Total Probability and the Probability Density in Terms of Conditional Probability 80......Page 91
18 Distribution Laws and Numerical Characteristics of Systems of Random Variables 84......Page 95
19 The Normal Distribution Law in the Plane and in Space. The Multidimensional Normal Distribution 91......Page 102
20 Distribution Laws of Subsystems of Continuous Random Variables and Conditional Distribution Laws 99......Page 110
21 Numerical Characteristics of Functions of Random Variables 107......Page 118
22 The Distribution Laws of Functions of Random Variables 115......Page 126
23 The Characteristic Functions of Systems and Functions of Random Variables 124......Page 135
24 Convolution of Distribution Laws 128......Page 139
25 The Linearization of Functions of Random Variables 136......Page 147
26 The Convolution of Two-Dimensional and Three-Dimensional Normal Distribution Laws by Use of the Notion of Deviation Vectors 145......Page 156
27 The Entropy of Random Events and Variables 157......Page 168
28 The Quantity of Information 163......Page 174
29 The Law of Large Numbers 171......Page 182
30 The de Moivre-Laplace and Lyapunov Theorems 176......Page 187
31 General Properties of Correlation Functions and Distribution Laws of Random Functions 181......Page 192
32 Linear Operations with Random Functions 185......Page 196
33 Problems On Passages 192......Page 203
34 Spectral Decomposition of Stationary Random Functions 198......Page 209
35 Computation of Probability Characteristics of Random Functions at the Output of Dynamical Systems 205......Page 216
36 Optimal Dynamical Systems 216......Page 227
37 The Method of Envelopes 226......Page 237
38 Markov Chains 231......Page 242
39 The Markov Processes with a Discrete Number of States 246......Page 257
40 Continuous Markov Processes 256......Page 267
41 Determination of the Moments of Random Variables From Experimental Data 275......Page 286
42 Confidence Levels and Confidence Intervals 286......Page 297
43 Tests of Goodness-of-Fit 300......Page 311
44 Data Processing by the Method of Least Squares 325......Page 336
45 Statistical Methods of Quality Control 346......Page 357
46 Determination of Probability Characteristics of Random Functions From Experimental Data 368......Page 379
2 A Direct Method for Evaluating Probabilities 375......Page 386
3 Geometric Probabilities 376......Page 387
4 Conditional Probability. The Multiplication Theorem for Probabilities 379......Page 390
5 The Addition Theorem for Probabilities 381......Page 392
6 The Total Probability Formula 383......Page 394
7 Computation of the Probabilities of Hypotheses After a Trial (Bayes\' Formula) 384......Page 395
8 Evaluation of Probabilities of Occurrence of an Event in Repeated Independent Trials 385......Page 396
9 The Multinomial Distribution. Recursion Formulas. Generating Functions 387......Page 398
10 The Probability Distribution Series, the Distribution Polygon and the Distribution Function of a Discrete Random Variable 391......Page 402
11 The Distribution Function and the Probability Density Function of a Continuous Random Variable 393......Page 404
12 Numerical Characteristics of Discrete Random Variables 394......Page 405
13 Numerical Characteristics of Continuous Random Variables 395......Page 406
15 The Normal Distribution Law 397......Page 408
16 Characteristic Functions 398......Page 409
17 The Computation of the Total Probability and the Probability Density in Terms of Conditional Probability 399......Page 410
18 Distribution Laws and Numerical Characteristics of Systems of Random Variables 400......Page 411
19 The Normal Distribution Law in the Plane and in Space. The Multidimensional Normal Distribution 402......Page 413
20 Distribution Laws of Subsystems of Continuous Random Variables and Conditional Distribution Laws 404......Page 415
21 Numerical Characteristics of Functions of Random Variables 406......Page 417
22 The Distribution Laws of Functions of Random Variables 408......Page 419
23 The Characteristic Functions of Systems and Functions of Random Variables 411......Page 422
24 Convolution of Distribution Laws 412......Page 423
25 The Linearization of Functions of Random Variables 415......Page 426
26 The Convolution of Two-Dimensional and Three-Dimensional Normal Distribution Laws by Use of the Notion of Deviation Vectors 416......Page 427
27 The Entropy of Random Events and Variables 418......Page 429
28 The Quantity of Information 420......Page 431
29 The Law of Large Numbers 423......Page 434
31 General Properties of Correlation Functions and Distribution Laws of Random Functions 425......Page 436
32 Linear Operations with Random Functions 427......Page 438
33 Problems On Passages 429......Page 440
34 Spectral Decomposition of Stationary Random Functions 431......Page 442
35 Computation of Probability Characteristics of Random Functions at the Output of Dynamical Systems 434......Page 445
36 Optimal Dynamical Systems 439......Page 450
37 The Method of Envelopes 443......Page 454
38 Markov Chains 445......Page 456
39 The Markov Processes with a Discrete Number of States 452......Page 463
40 Continuous Markov Processes 456......Page 467
41 Determination of the Moments of Random Variables From Experimental Data 459......Page 470
42 Confidence Levels and Confidence Intervals 461......Page 472
43 Tests of Goodness-of-Fit 462......Page 473
44 Data Processing by the Method of Least Squares 465......Page 476
45 Statistical Methods of Quality Control 466......Page 477
46 Determination of Probability Characteristics of Random Functions From Experimental Data 468......Page 479
Sources of Tables Referred to in the Text 471......Page 482
Bibliography 475......Page 486
Index 479......Page 490