初等概率论

PREFACE TO THE FOURTH EDITION

PROLOGUE TO INTRODUCTION TO MATHEMATICAL FINANCE

1 SET

1.1 Sample sets

1.2 Operations with sets

1.3 Various relations

1.4 Indicator

Exercises

2 PROBABILITY

2.1 Examples of probability

2.2 Definition and illustrations

2.3 Deductions from the axioms

2.4 Independent events

2.5 Arithmetical density

Exercises

3 COUNTING

3.1 Fundamental rule

3.2 Diverse ways of sampling

3.3 Allocation models; binomial coefficients

3.4 How to solve it

Exercises

4 RANDOM VARIABLES

4.1 What is a random variable?

4.2 How do random variables come about?

4.3 Distribution and expectation

4.4 Integer-valued random variables

4.5 Random variables with densities

4.6 General case

Exercises

APPENDIX 1： BOREL FIELDS AND GENERAL RANDOM VARIABLES

5 CONDITIONING AND INDEPENDENCE

5.1 Examples of conditioning

5.2 Basic formulas

5.3 Sequential sampling

5.4 P61yas urn scheme

5.5 Independence and relevance

5.6 Genetical models

Exercises

6 MEAN， VARIANCE， AND TRANSFORMS

6.1 Basic properties of expectation

6.2 The density case

6.3 Multiplication theorem; variance and covariance

6.4 Multinomial distribution

6.5 Generating function and the like

Exercises

7 POISSON AND NORMAL DISTRIBUTIONS

7.1 Models for Poisson distribution

7.2 Poisson process

7.3 From binomial to normal

7.4 Normal distribution

7.5 Central limit theorem

7.6 Law of large numbers

Exercises

APPENDIX 2： STIRLINGS FORMULA AND DE MOIVRE-LAPLACES THEOREM

8 FROM RANDOM WALKS TO MARKOV CHAINS

8.1 Problems of the wanderer or gambler

8.2 Limiting schemes

8.3 Transition probabilities

8.4 Basic structure of Markov chains

8.5 Further developments

8.6 Steady state

8.7 Winding up (or down?)

Exercises

APPENDIX 3： MARTINGALE

9 MEAN-VARIANCE PRICING MODEL

9.1 An investments primer

9.2 Asset return and risk

9.3 Portfolio allocation

9.4 Diversification

9.5 Mean-variance optimization

9.6 Asset return distributions

9.7 Stable probability distributions

Exercises

APPENDIX 4： PARETO AND STABLE LAWS

10 OPTION PRICING THEORY

10.1 Options basics

10.2 Arbitrage-free pricing： 1-period model

10.3 Arbitrage-free pricing： N-period model

10.4 Fundamental asset pricing theorems

Exercises

GENERAL REFERENCES

ANSWERS TO PROBLEMS

VALUES OF THE STANDARD NORMAL DISTRIBUTION FUNCTION

INDEX

本书是一部介绍概率论及其应用的入门教程。其原始版本面世已经有30余年，但仍然是本科一二年级的经典概率教程。在第4版中增加了两章讲述应用和数学金融。传承前面版本详细、严谨的风格，讲述了有价证券和期货理论的基本知识。书中用最初等的方法讲述了概率测度、随机变量、分布以及期望等基本概念。离散和连续的案例都有所涉及，在讲述后者的时候运用了微积分知识。配以大量的典型例子重点讲述概率推理，集中介绍了组合问题、Poison过程、随机漫步、遗传模型和Markov链。每章末都附有习题及其解答。目次：集合；概率；计数；随机变量；附录。
读者对象：数学专业的本科生以及广大概率论爱好者。