Georgii Stochastics

Introduction to Probability and Statistics
2. revidierte and ext. ed
ISBN: 978-3-11-029360-9
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 416 Seiten

Reihe: De Gruyter Textbook

This textbook, now in its second revised and extended edition, presents the fundamental ideas and results of both probability theory and statistics. It comprises the material of a one-year course, which is addressed to students of mathematics and to scientists with an interest in the mathematical side of stochastics.
The stochastic concepts, models and methods are motivated by examples and then developed and analysed systematically. Some measure theory is included, but this is done at an elementary level that is in accordance with the introductory character of the book. A large number of problems, now in part with solutions, offer applications and supplements to the text.

Zielgruppe

Students, Lecturers, and Reseachers in Mathematics, Economics, Engeneering and Nature Sciences; Academic Libraries

Autoren/Hrsg.

Georgii, Hans-Otto

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik

Inhaltsverzeichnis

1;Preface;5
2;Mathematics and Chance;11
3;I Probability Theory;15
3.1;1 Principles of Modelling Chance;17
3.1.1;1.1 Probability Spaces;17
3.1.2;1.2 Properties and Construction of Probability Measures;24
3.1.3;1.3 Random Variables;30
3.1.4;Problems;34
3.2;2 Stochastic Standard Models;37
3.2.1;2.1 The Uniform Distributions;37
3.2.2;2.2 Urn Models with Replacement;40
3.2.3;2.3 Urn Models without Replacement;45
3.2.4;2.4 The Poisson Distribution;49
3.2.5;2.5 Waiting Time Distributions;50
3.2.6;2.6 The Normal Distributions;56
3.2.7;Problems;58
3.3;3 Conditional Probabilities and Independence;61
3.3.1;3.1 Conditional Probabilities;61
3.3.2;3.2 Multi-Stage Models;67
3.3.3;3.3 Independence;74
3.3.4;3.4 Existence of Independent Random Variables, Product Measures;80
3.3.5;3.5 The Poisson Process;85
3.3.6;3.6 Simulation Methods;89
3.3.7;3.7 Tail Events;93
3.3.8;Problems;96
3.4;4 Expectation and Variance;102
3.4.1;4.1 The Expectation;102
3.4.2;4.2 Waiting Time Paradox and Fair Price of an Option;110
3.4.3;4.3 Variance and Covariance;117
3.4.4;4.4 Generating Functions;120
3.4.5;Problems;124
3.5;5 The Law of Large Numbers and the Central Limit Theorem;129
3.5.1;5.1 The Law of Large Numbers;129
3.5.2;5.2 Normal Approximation of Binomial Distributions;141
3.5.3;5.3 The Central Limit Theorem;148
3.5.4;5.4 Normal versus Poisson Approximation;153
3.5.5;Problems;156
3.6;6 Markov Chains;161
3.6.1;6.1 The Markov Property;161
3.6.2;6.2 Absorption Probabilities;165
3.6.3;6.3 Asymptotic Stationarity;169
3.6.4;6.4 Recurrence;181
3.6.5;Problems;191
4;II Statistics;199
4.1;7 Estimation;201
4.1.1;7.1 The Approach of Statistics;201
4.1.2;7.2 Facing the Choice;205
4.1.3;7.3 The Maximum Likelihood Principle;209
4.1.4;7.4 Bias and Mean Squared Error;215
4.1.5;7.5 Best Estimators;217
4.1.6;7.6 Consistent Estimators;224
4.1.7;7.7 Bayes Estimators;228
4.1.8;Problems;232
4.2;8 Confidence Regions;237
4.2.1;8.1 Definition and Construction;237
4.2.2;8.2 Confidence Intervals in the Binomial Model;243
4.2.3;8.3 Order Intervals;249
4.2.4;Problems;253
4.3;9 Around the Normal Distributions;256
4.3.1;9.1 The Multivariate Normal Distributions;256
4.3.2;9.2 The X2-, F- and t-Distributions;259
4.3.3;Problems;266
4.4;10 Hypothesis Testing;270
4.4.1;10.1 Decision Problems;270
4.4.2;10.2 Neyman-Pearson Tests;275
4.4.3;10.3 Most Powerful One-Sided Tests;281
4.4.4;10.4 Parameter Tests in the Gaussian Product Model;284
4.4.5;Problems;294
4.5;11 Asymptotic Tests and Rank Tests;299
4.5.1;11.1 Normal Approximation of Multinomial Distributions;299
4.5.2;11.2 The Chi-Square Test of Goodness of Fit;306
4.5.3;11.3 The Chi-Square Test of Independence;313
4.5.4;11.4 Order and Rank Tests;319
4.5.5;Problems;330
4.6;12 Regression Models and Analysis of Variance;335
4.6.1;12.1 Simple Linear Regression;335
4.6.2;12.2 The Linear Model;339
4.6.3;12.3 The Gaussian Linear Model;344
4.6.4;12.4 Analysis of Variance;352
4.6.5;Problems;361
5;Solutions;367
6;Tables;395
7;References;401
8;List of Notation;405
9;Index;409

Über Autor(innen)

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