Stochastics (eBook)

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Hans-Otto Georgii, Ludwig-Maximilians-Universität Munich, Germany.

Preface 5
Mathematics and Chance 11
I Probability Theory 15
1 Principles of Modelling Chance 17
1.1 Probability Spaces 17
1.2 Properties and Construction of Probability Measures 24
1.3 Random Variables 30
Problems 34
2 Stochastic Standard Models 37
2.1 The Uniform Distributions 37
2.2 Urn Models with Replacement 40
2.3 Urn Models without Replacement 45
2.4 The Poisson Distribution 49
2.5 Waiting Time Distributions 50
2.6 The Normal Distributions 56
Problems 58
3 Conditional Probabilities and Independence 61
3.1 Conditional Probabilities 61
3.2 Multi-Stage Models 67
3.3 Independence 74
3.4 Existence of Independent Random Variables, Product Measures 80
3.5 The Poisson Process 85
3.6 Simulation Methods 89
3.7 Tail Events 93
Problems 96
4 Expectation and Variance 102
4.1 The Expectation 102
4.2 Waiting Time Paradox and Fair Price of an Option 110
4.3 Variance and Covariance 117
4.4 Generating Functions 120
Problems 124
5 The Law of Large Numbers and the Central Limit Theorem 129
5.1 The Law of Large Numbers 129
5.2 Normal Approximation of Binomial Distributions 141
5.3 The Central Limit Theorem 148
5.4 Normal versus Poisson Approximation 153
Problems 156
6 Markov Chains 161
6.1 The Markov Property 161
6.2 Absorption Probabilities 165
6.3 Asymptotic Stationarity 169
6.4 Recurrence 181
Problems 191
II Statistics 199
7 Estimation 201
7.1 The Approach of Statistics 201
7.2 Facing the Choice 205
7.3 The Maximum Likelihood Principle 209
7.4 Bias and Mean Squared Error 215
7.5 Best Estimators 217
7.6 Consistent Estimators 224
7.7 Bayes Estimators 228
Problems 232
8 Confidence Regions 237
8.1 Definition and Construction 237
8.2 Confidence Intervals in the Binomial Model 243
8.3 Order Intervals 249
Problems 253
9 Around the Normal Distributions 256
9.1 The Multivariate Normal Distributions 256
9.2 The X2-, F- and t-Distributions 259
Problems 266
10 Hypothesis Testing 270
10.1 Decision Problems 270
10.2 Neyman-Pearson Tests 275
10.3 Most Powerful One-Sided Tests 281
10.4 Parameter Tests in the Gaussian Product Model 284
Problems 294
11 Asymptotic Tests and Rank Tests 299
11.1 Normal Approximation of Multinomial Distributions 299
11.2 The Chi-Square Test of Goodness of Fit 306
11.3 The Chi-Square Test of Independence 313
11.4 Order and Rank Tests 319
Problems 330
12 Regression Models and Analysis of Variance 335
12.1 Simple Linear Regression 335
12.2 The Linear Model 339
12.3 The Gaussian Linear Model 344
12.4 Analysis of Variance 352
Problems 361
Solutions 367
Tables 395
References 401
List of Notation 405
Index 409

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"The textbook is based on a series of lectures taught by the author for many years at the Mathematical Institute of the University of Munich. The material of the book covers two one-semester courses in probability and mathematical statistics, respectively. All chapters are equipped with exercises of varying degrees of difficulty that help to clarify the concepts.

The first part of the book is an introduction to probability theory. The material is presented using little of the measure-theoretical background but rather application-oriented examples that preserve its introductory character. Topics range from classical probability distributions to conditional distributions and limit theorems. A short introduction to Markov chains is also given. The second part of the book gives an introduction to mathematical statistics and describes main statistical procedures: parameter and interval estimation, hypothesis testing, linear regression and basics of the analysis of variance approach.

The book can be used by undergraduate mathematics majors but also by science and engineering students who wish not only to apply probability and statistics but also to understand how the methods work."

Vladimir P. Kurenok, MathSciNet