## Probability Theory |

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### Contents

Random variables | 24 |

Expectation | 30 |

Variance | 40 |

Higher moments and inequalities for deviations | 46 |

The Law of Large Numbers Limit Theorems | 50 |

The Poisson theorem and the simplest flow of events | 65 |

On random walks | 78 |

Foundations of the Theory | 88 |

Some particular distributions on the real line | 144 |

Convergence of distributions | 163 |

Conditional Distributions | 200 |

eral case Properties of conditional expectation The formula | 216 |

Some Kinds of Dependence | 226 |

Martingales | 233 |

Markov chains | 243 |

Limit Theorems | 260 |

### Common terms and phrases

a-algebra absolutely continuous arbitrary assertion assume Borel sets called Cauchy distribution ch.f chain characteristic function conditional expectation consider convergence Corollary corresponding countable cr-algebra deduce defined Definition degenerate distributions denote dependent distribution F distribution function elementary event equal Example exists F(dx finite formula Hence his/her holds identically distributed implies independent r.v.'s independent random variables inequality infinite integral large numbers law of large Lebesgue Lebesgue integral Lebesgue measure Lemma Let F Limit theorems martingale matrix means nonnegative normal distribution Observe parameter particles Poisson Poisson distribution preceding probability measure probability space probability theory Proof of Theorem prove r.vec random vector reader is invited real line relation representation right-hand side satisfying scheme Section 16 segment sequence of r.v.'s set-function stable distribution standard normal Subsection sufficiently summands symmetric tion trials uniformly uniquely valid values variance zero