Stochastics: Introduction to Probability and Statistics

✍ Scribed by Hans-Otto Georgii

• Publisher: De Gruyter
• Year: 2012
• Tongue: English
• Leaves: 420
• Edition: 2nd rev. and ext. ed.
• Category: Library

✦ Synopsis

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.

• Lots of examplesof real life problems
• Problem sets with solution hints

• Suitable for atwo semester course or self-study

✦ Table of Contents

Preface
Mathematics and Chance
I Probability Theory
1 Principles of Modelling Chance
1.1 Probability Spaces
1.2 Properties and Construction of Probability Measures
1.3 Random Variables
Problems
2 Stochastic Standard Models
2.1 The Uniform Distributions
2.2 Urn Models with Replacement
2.3 Urn Models without Replacement
2.4 The Poisson Distribution
2.5 Waiting Time Distributions
2.6 The Normal Distributions
Problems
3 Conditional Probabilities and Independence
3.1 Conditional Probabilities
3.2 Multi-Stage Models
3.3 Independence
3.4 Existence of Independent Random Variables, Product Measures
3.5 The Poisson Process
3.6 Simulation Methods
3.7 Tail Events
Problems
4 Expectation and Variance
4.1 The Expectation
4.2 Waiting Time Paradox and Fair Price of an Option
4.3 Variance and Covariance
4.4 Generating Functions
Problems
5 The Law of Large Numbers and the Central Limit Theorem
5.1 The Law of Large Numbers
5.2 Normal Approximation of Binomial Distributions
5.3 The Central Limit Theorem
5.4 Normal versus Poisson Approximation
Problems
6 Markov Chains
6.1 The Markov Property
6.2 Absorption Probabilities
6.3 Asymptotic Stationarity
6.4 Recurrence
Problems
II Statistics
7 Estimation
7.1 The Approach of Statistics
7.2 Facing the Choice
7.3 The Maximum Likelihood Principle
7.4 Bias and Mean Squared Error
7.5 Best Estimators
7.6 Consistent Estimators
7.7 Bayes Estimators
Problems
8 Confidence Regions
8.1 Definition and Construction
8.2 Confidence Intervals in the Binomial Model
8.3 Order Intervals
Problems
9 Around the Normal Distributions
9.1 The Multivariate Normal Distributions
9.2 The X2-, F- and t-Distributions
Problems
10 Hypothesis Testing
10.1 Decision Problems
10.2 Neyman-Pearson Tests
10.3 Most Powerful One-Sided Tests
10.4 Parameter Tests in the Gaussian Product Model
Problems
11 Asymptotic Tests and Rank Tests
11.1 Normal Approximation of Multinomial Distributions
11.2 The Chi-Square Test of Goodness of Fit
11.3 The Chi-Square Test of Independence
11.4 Order and Rank Tests
Problems
12 Regression Models and Analysis of Variance
12.1 Simple Linear Regression
12.2 The Linear Model
12.3 The Gaussian Linear Model
12.4 Analysis of Variance
Problems
Solutions
Tables
References
List of Notation
Index