book

Probability Theory, 2nd Edition

Name: Probability Theory, 2nd Edition
Author: Werner Linde
ISBN: 9783111325170

by Werner Linde

June 2024

Intermediate to advanced

500 pages

14h 7m

English

De Gruyter

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Preface
1 Probabilities
1.1 Probability spaces1.1.1 Sample spaces1.1.2 σ-fields of events∗1.1.3 Probability measures1.2 Basic properties of probability measures1.3 Discrete probability measures1.4 Special discrete probability measures1.4.1 Dirac measure1.4.2 Uniform distribution on a finite set1.4.3 Binomial distribution1.4.4 Multinomial distribution1.4.5 Poisson distribution1.4.6 Hypergeometric distribution1.4.7 Geometric distribution1.4.8 Negative binomial distribution1.5 Continuous probability measures1.6 Special continuous distributions1.6.1 Uniform distribution on an interval1.6.2 Normal distribution1.6.3 Gamma distribution1.6.4 Exponential distribution1.6.5 Erlang distribution1.6.6 Chi-squared distribution1.6.7 Beta distribution1.6.8 Cauchy distribution1.7 Distribution function1.8 Multivariate continuous distributions1.8.1 Multivariate density functions1.8.2 Multivariate uniform distribution1.9 Products of probability spaces∗1.9.1 Product σ-fields and measures1.9.2 Product measures: discrete case1.9.3 Product measures: continuous case1.10 Problems
2 Conditional probabilities and independence
2.1 Conditional probabilities2.2 Independence of events2.3 Problems
3 Random variables and their distribution
3.1 Transformation of random values3.2 Probability distribution of a random variable3.3 Special random variables3.4 Random vectors3.5 Joint and marginal distributions3.5.1 Marginal distributions: discrete case3.5.2 Marginal distributions: continuous case3.6 Independence of random variables3.6.1 Independence of discrete random variables3.6.2 Independence of continuous random variables3.7 Order statistics∗3.8 Problems
4 Operations on random variables
4.1 Mappings of random variables4.2 Linear transformations4.3 Coin tossing versus uniform distribution4.3.1 Binary fractions4.3.2 Binary fractions of random numbers4.3.3 Random numbers generated by coin tossing4.4 Simulation of random variables4.5 Addition of random variables4.5.1 Sums of discrete random variables4.5.2 Sums of continuous random variables4.6 Sums of certain random variables4.7 Products and quotients of random variables4.7.1 Student’s t-distribution4.7.2 F-distribution4.8 Problems
5 Expected value, variance, and covariance
5.1 Expected value5.1.1 Expected value of discrete random variables5.1.2 Expected value of certain discrete random variables5.1.3 Expected value of continuous random variables5.1.4 Expected value of certain continuous random variables5.1.5 Properties of the expected value5.2 Variance5.2.1 Higher moments of random variables5.2.2 Variance of random variables5.2.3 Variance of certain random variables5.3 Covariance and correlation5.3.1 Covariance5.3.2 Correlation coefficient5.4 Some paradoxes and examples5.4.1 Boy or girl paradox5.4.2 Randomly chosen entries5.4.3 Secretary problem5.4.4 Two-envelope paradox5.5 Gambler’s ruin5.6 Problems
6 Normally distributed random vectors
6.1 Representation and density6.2 Expected value and covariance matrix6.3 Problems
7 Limit theorems
7.1 Laws of large numbers7.1.1 Chebyshev’s inequality7.1.2 Infinite sequences of independent random variables∗7.1.3 Borel–Cantelli lemma∗7.1.4 Weak law of large numbers7.1.5 Strong law of large numbers7.2 Central limit theoremSpecial cases of the central limit theorem7.3 Problems
8 Mathematical statistics
8.1 Statistical models8.1.1 Nonparametric statistical models8.1.2 Parametric statistical models8.2 Statistical hypothesis testing8.2.1 Hypotheses and tests8.2.2 Power function and significance tests8.3 Tests for binomial distributed populations8.4 Tests for normally distributed populations8.4.1 Fisher’s theorem8.4.2 Quantiles8.4.3 Z-tests or Gauss tests8.4.4 t-tests8.4.5 χ2-tests for the variance8.4.6 Two-sample Z-tests8.4.7 Two-sample t-tests8.4.8 F-tests8.5 Point estimators8.5.1 Maximum likelihood estimation8.5.2 Unbiased estimators8.5.3 Risk function8.6 Confidence regions and intervals8.6.1 Construction of confidence regions8.6.2 Normally distributed samples8.6.3 Binomial distributed populations8.6.4 Hypergeometric distributed populations8.7 ProblemsA AppendixA.1 NotationsA.2 Elements of set theoryA.2.1 Set operationsA.2.2 Preimages of setsA.2.3 ProblemsA.3 CombinatoricsA.3.1 Binomial coefficientsA.3.2 Drawing balls out of an urnA.3.3 Multinomial coefficientsA.3.4 ProblemsA.4 Vectors and matricesA.5 Some analytic toolsBibliography
Subject Index

Overview

This book is intended as an introduction to Probability Theory and Mathematical Statistics for students in mathematics, the physical sciences, engineering, and related fields. It is based on the author’s 25 years of experience teaching probability and is squarely aimed at helping students overcome common difficulties in learning the subject. The focus of the book is an explanation of the theory, mainly by the use of many examples. Whenever possible, proofs of stated results are provided. All sections conclude with a short list of problems. The book also includes several optional sections on more advanced topics. This textbook would be ideal for use in a first course in Probability Theory.

Contents:
Probabilities
Conditional Probabilities and Independence
Random Variables and Their Distribution
Operations on Random Variables
Expected Value, Variance, and Covariance
Normally Distributed Random Vectors
Limit Theorems

Introduction to Stochastic Processes
Mathematical Statistics
Appendix
Bibliography
Index

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