A First Course in Probability and Markov Chains

Contents

Preface

1     Combinatorics

1.1   Binomial coefficients

1.1.1    Pascal triangle

1.1.2    Some properties of binomial coefficients

1.1.3    Generalized binomial coefficients and binomial series

1.1.4    Inversion formulas

1.1.5    Exercises

1.2   Sets, permutations and functions

1.2.1    Sets

1.2.2    Permutations

1.2.3    Multisets

1.2.4    Lists and functions

1.2.5    Injective functions

1.2.6    Monotone increasing functions

1.2.7    Monotone nondecreasing functions

1.2.8    Surjective functions

1.2.9    Exercises

1.3   Drawings

1.3.1    Ordered drawings

1.3.2    Simple drawings

1.3.3    Multiplicative property of drawings

1.3.4    Exercises

1.4   Grouping

1.4.1    Collocations of pairwise different objects

1.4.2    Collocations of identical objects

1.4.3    Multiplicative property

1.4.4    Collocations in statistical physics

1.4.5    Exercises

2     Probability measures

2.1   Elementary probability

2.1.1    Exercises

2.2   Basic facts

2.2.1    Events

2.2.2    Probability measures

2.2.3    Continuity of measures

2.2.4    Integral with respect to a measure

2.2.5    Probabilities on finite and denumerable sets

2.2.6    Probabilities on denumerable sets

2.2.7    Probabilities on uncountable sets

2.2.8    Exercises

2.3   Conditional probability

2.3.1    Definition

2.3.2    Bayes formula

2.3.3    Exercises

2.4   Inclusion–exclusion principle

2.4.1    Exercises

3     Random variables

3.1   Random variables

3.1.1    Definitions

3.1.2    Expected value

3.1.3    Functions of ...