Elementary Probability Theory

by Chung, Kai Lai; Aitsahlia, Farid

• ISBN13: 9780387955780
• ISBN10: 038795578X
• Edition: 4th
• Format: Hardcover
• Copyright: 2003-03-01
• Publisher: Springer Verlag

Summary

This is an introductory textbook on probability theory and its applications. Basic concepts such as probability measure, random variable, distribution, and expectation are fully treated without technical complications. Both the discrete and continuous cases are covered, the elements of calculus being used in the latter case.The emphasis is on essential probabilistic reasoning, amply motivated, explained, and illustrated with a large number of carefully selected examples. Special topics include combinatorial problems, urn schemes, Poisson processes, random walks, genetic models, and Markov chains. Problems with solutions are provided at the end of each chapter. Its easy style and full discussion make this a useful text not only for mathematics and statistics majors, but also for students in engineering and physical, biological, and social sciences. This edition adds two new chapters covering applications to mathematical finance. Elements of modern portfolio and option theories are presented in a detailed and rigorous manner. The approach distinguishes this text from other more mathematically advanced treatises or more technical manuals.Kai Lai Chung is Professor Emeritus of Mathematics at Stanford University. Farid AitSahlia is a Senior Scientist with DemandTec, where he develops econometric and optimization methods for demand-based pricing models. He is also a visiting scholar in the department of statistics at Stanford University, where he obtained his Ph.D.in operations research.

Table of Contents

Preface to the Fourth Edition	xi
Prologue to Introduction to Mathematical Finance	xiii
Set	1	(19)
Sample sets	1	(2)
Operations with sets	3	(4)
Various relations	7	(6)
Indicator	13	(7)
Exercises	17	(3)
Probability	20	(26)
Examples of probability	20	(4)
Definition and illustrations	24	(7)
Deductions from the axioms	31	(4)
Independent events	35	(4)
Arithmetical density	39	(7)
Exercises	42	(4)
Counting	46	(28)
Fundamental rule	46	(3)
Diverse ways of sampling	49	(6)
Allocation models; binomial coefficients	55	(7)
How to solve it	62	(12)
Exercises	70	(4)
Random Variables	74	(43)
What is a random variable?	74	(4)
How do random variables come about?	78	(6)
Distribution and expectation	84	(6)
Integer-valued random variables	90	(5)
Random variables with densities	95	(10)
General case	105	(12)
Exercises	109	(6)
Appendix 1: Borel Fields and General Random Variables	115	(2)
Conditioning and Independence	117	(47)
Examples of conditioning	117	(5)
Basic formulas	122	(9)
Sequential sampling	131	(5)
Polya's urn scheme	136	(5)
Independence and relevance	141	(11)
Genetical models	152	(12)
Exercises	157	(7)
Mean, Variance, and Transforms	164	(39)
Basic properties of expectation	164	(5)
The density case	169	(4)
Multiplication theorem; variance and covariance	173	(7)
Multinomial distribution	180	(7)
Generating function and the like	187	(16)
Exercises	195	(8)
Poisson and Normal Distributions	203	(51)
Models for Poisson distribution	203	(8)
Poisson process	211	(11)
From binomial to normal	222	(7)
Normal distribution	229	(4)
Central limit theorem	233	(6)
Law of large numbers	239	(15)
Exercises	246	(5)
Appendix 2: Stirling's Formula and De Moivre--Laplace's Theorem	251	(3)
From Random Walks to Markov Chains	254	(75)
Problems of the wanderer or gambler	254	(7)
Limiting schemes	261	(5)
Transition probabilities	266	(9)
Basic structure of Markov chains	275	(9)
Further developments	284	(7)
Steady state	291	(12)
Winding up (or down?)	303	(26)
Exercises	314	(11)
Appendix 3: Martingale	325	(4)
Mean-Variance Pricing Model	329	(30)
An investments primer	329	(2)
Asset return and risk	331	(4)
Portfolio allocation	335	(1)
Diversification	336	(1)
Mean-variance optimization	337	(9)
Asset return distributions	346	(2)
Stable probability distributions	348	(11)
Exercises	351	(4)
Appendix 4: Pareto and Stable Laws	355	(4)
Option Pricing Theory	359	(20)
Options basics	359	(7)
Arbitrage-free pricing: 1-period model	366	(6)
Arbitrage-free pricing: N-period model	372	(4)
Fundamental asset pricing theorems	376	(3)
Exercises	377	(2)
General References	379	(2)
Answers to Problems	381	(12)
Values of the Standard Normal Distribution Function	393	(4)
Index	397