## Stochastics: Introduction to Probability and Statistics This book is a translation of the third edition of the well accepted German textbook 'Stochastik', which presents the fundamental ideas and results of both probability theory and statistics, and comprises the material of a one-year course. The stochastic concepts, models and methods are motivated by examples and problems and then developed and analysed systematically. |

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### Contents

Mathematics and Chance | 1 |

Principles of Modelling Chance | 7 |

Stochastic Standard Models | 26 |

Conditional Probabilities and Independence | 50 |

Expectation and Variance | 90 |

The Law of Large Numbers and the Central Limit Theorem | 117 |

Markov Chains | 149 |

Estimation | 187 |

Around the Normal Distributions | 241 |

Hypothesis Testing | 255 |

Asymptotic Tests and Rank Tests | 283 |

Regression Models and Analysis of Variance | 318 |

Tables | 349 |

355 | |

363 | |

Confidence Regions | 222 |

### Common terms and phrases

a-algebra alternative arbitrary assumption asymptotic balls Bernoulli sequence beta distribution binomial distribution Borel called central limit theorem confidence interval confidence region consider convergence in distribution Corollary corresponding countable defined Definition density q depend determine discrete density distributed with parameter distribution density distribution function distribution Q equation error level Example exists expectation Figure finite gamma distribution Gaussian product model Hence i.i.d. random variables implies independent random variables inequality infinite large numbers law of large Lebesgue density lemma likelihood function likelihood ratio Markov chain maximum likelihood estimator means n-fold null hypothesis observations obtain outcomes Poisson distributions Poisson process probability measure probability space Proof properties quantiles random vector real random variable regression rejection region Remark sample space space Q standard normal distribution statistical model stochastic Suppose test problem transition matrix unbiased estimator uniform distribution uniformly unknown variance