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Stochastics - Introduction to Probability and Statistics
of: Hans-Otto Georgii
Walter de Gruyter GmbH & Co.KG, 2013
ISBN: 9783110293609 , 416 Pages
2. Edition
Format: PDF
Copy protection: DRM
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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