Probability

Preface

1 Recap of measure theory

2 Foundations of probability theory

2.1 Probability spaces

2.2 Conditional probability

2.3 Random variables

2.4 Expectation

2.5 The classical laws

2.6 Cumulative distribution function

2.7 The \(\sigma\)-algebra generated by a random variable

2.8 Moments and inequalities

3 Independence

3.1 Independent events

3.2 Intermezzo: monotone class lemma*

3.3 Independent \(\sigma\)-algebras and random variables

3.4 The Borel-Cantelli lemma

3.5 Sums of independent random variables

4 Convergence of random variables

4.1 Notions of convergence

4.2 Convergence in law

4.3 Characteristic function

4.4 The central limit theorem

5 Markov chains and random walks

5.1 Definition and basic properties

5.2 The Markov property

5.3 Recurrence and transience

5.4 Stationary and reversible measures

5.5 The Green function

5.6 Existence and uniqueness of stationary measures

5.7 Positive and null recurrence

5.8 Asymptotic behaviour

5.9 Markov chain Monte Carlo and the Metropolis–Hastings algorithm

6 Introduction to statistics

6.1 Estimators

6.2 Confidence intervals

6.3 Hypothesis testing

7 The strong law of large numbers

Probability

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