Probability: an intermediate level course
Authors
Description
This book is not intended to be introductory, although it can eventually be used as such. It is necessary for the reader to have some notion of classical discrete distributions based on counting permutations, combinations, etc., such as binomial, hypergeometric and multinomial distributions.
Prerequisites for following the text are a basic course in Differential and Integral Calculus and some familiarity with the basic concepts of sets and functions.
The lists of exercises are a very important part of the book and include purely computational exercises, to help the reader train in the basic calculation of probabilities, and others that understand and develop ideas covered in the text.
Name: Probability: an intermediate level course
Author(s): e Barry R. James
Pages: 296
Publication: IMPA, 2023
ISBN: 978-85-244-0455-9
Edition: 5
Foreword
Index of Notations
1 Basic definitions
1.1 Mathematical model for an experiment
1.2 Conditional probability
1.3 Independence
Exercises
2 Random variables
2.1 Random variables and distribution functions
2.2 Types of random variables
2.3 The distribution of a random variable
2.4 Random vectors
2.5 Independence
2.6 Distribution functions of random variables and vectors
2.7 The Jacobian method
2.8 Additional observations – random variables and vectors
Exercises
3 Mathematical hope
3.1 Preliminaries: the Stieljes integral
3.2 Hope
3.3 Properties of hope
3.4 Hope for functions of random variables
3.5 Moments
3.6 Hope for functions of random vectors
3.7 Convergence theorems
Exercises
4 Conditional distribution and conditional expectation
4.1 Conditional distribution of X given discrete Y
4.2 Conditional distribution of X given Y: general case
4.3 Formal definitions and existence theorems
4.4 Examples
4.5 Conditional expectation
Exercises
5 The Law of Large Numbers
5.1 Introduction to the Weak and Strong Laws of Large Numbers
5.2 Sequences of Events and the Borel-Cantelli Lemma
5.3 The Strong Law
Exercises
6 Characteristic functions and convergence in distributions
6.1 Characteristic functions
6.2 Convergence in distributions
6.3 Characteristic function of a random vector
6.4 Observations and complements
Exercises
7 The Central Limit Theorem
7.1 The Central Limit Theorem for sequences of random variables
7.2 The multivariate normal distribution
7.3 The Central Limit Theorem – multivariate case
Exercises
Bibliography
Alphabetical Index