Ronald Meester - A Natural Introduction to Probability Theory (2

Ronald Meester

A Natural Introduction to Probability Theory, 2e, 2008,
Birkhäuser, 201 pages.

Contents
========

1 Experiments
1.1 Defnitions and Examples
1.2 Counting and Combinatorics
1.3 Properties of Probability Measures
1.4 Conditional Probabilities
1.5 Independence
1.6 A First Law of Large Numbers
1.7 Exercises

2 Random Variables and Random Vectors
2.1 Random Variables
2.2 Independence
2.3 Expectation and Variance
2.4 Random Vectors
2.5 Conditional Distributions and Expectations
2.6 Generating Functions
2.7 Exercises

3 Random Walk
3.1 Random Walk and Counting
3.2 The Arc-Sine Law
3.3 Exercises

4 Limit Theorems
4.1 The Law of Large Numbers
4.2 The Central Limit Theorem
4.3 Exercises

I Intermezzo
I.1 Uncountable Sample Spaces
I.2 An Event Without a Probability?!
I.3 Random Variables on Uncountable Sample Spaces

5 Continuous Random Variables and Vectors
5.1 Experiments
5.2 Properties of Probability Measures
5.3 Continuous Random Variables
5.4 Expectation
5.5 Random Vectors and Independence
5.6 Functions of Random Variables and Vectors
5.7 Sums of Random Variables
5.8 More About the Expectation; Variance
5.9 Random Variables Which are Neither Discrete Nor Continuous
5.10 Conditional Distributions and Expectations
5.11 The Law of Large Numbers
5.12 Exercises

6 Infnitely Many Repetitions
6.1 Infnitely Many Coin Flips and Random Points in (0, 1]
6.2 A More General Approach to Infnitely Many Repetitions
6.3 The Strong Law of Large Numbers
6.4 Random Walk Revisited
6.5 Branching Processes
6.6 Exercises

7 The Poisson Process
7.1 Building a Model
7.2 Basic Properties
7.3 The Waiting Time Paradox
7.4 The Strong Law of Large Numbers
7.5 Exercises

8 Limit Theorems
8.1 Weak Convergence
8.2 Characteristic Functions
8.3 Expansion of the Characteristic Function
8.4 The Law of Large Numbers
8.5 The Central Limit Theorem
8.6 Exercises

9 Extending the Probabilities
9.1 General Probability Measures


A Interpreting Probabilities
B Further Reading
C Answers to Selected Exercises

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